課題#
Plimpton 322の数表を生成する#
%%ai vioras
Plimton 322について
Plimpton 322は、古代バビロニアの粘土板で、一般的に古代の数学的テキストとして認識されています。これは紀元前1800年から1600年頃に書かれたと考えられています。
この粘土板は、現在はニューヨークのコロンビア大学の蔵書となっていて、ジョージ・アーサー・プリンプトンが1936年に購入したコレクションの一部です。それで”Plimpton 322”という名前がつけられました。
Plimpton 322は、ピタゴラスの定理のバビロニア版を含む15のピタゴラスの三つ組をリスト化しています。これらの三つ組は、3つの正の整数$a$, $b$, $c$で表され、$a^2 + b^2 = c^2$という関係を満たします。しかし、バビロニア人はこの関係を直接的に表現するのではなく、代わりに$a/c$と$b/c$という2つの比に焦点を当てています。
この粘土板がどのような目的で作成されたのかについては学者の間で意見が分かれています。一部の研究者は、それが教育目的で使用されたと考えています。他の研究者は、より複雑な幾何学的問題を解くためのツールであった可能性を示唆しています。しかし、その真の目的は未だに完全には解明されていません。
%%ai vioras
ピタゴラスの三数を探索するプログラムを書いてください
Pythonでピタゴラスの三つ組を探索する簡単なプログラムを書くことができます。以下にその一例を示します。
def find_pythagorean_triplets(limit):
triplets = []
for a in range(1, limit):
for b in range(a, limit):
c = (a**2 + b**2)**0.5
if c.is_integer() and c <= limit:
triplets.append((a, b, int(c)))
return triplets
limit = 100
triplets = find_pythagorean_triplets(limit)
for triplet in triplets:
print(triplet)
このプログラムは、指定された上限までの全てのピタゴラスの三つ組を探索します。aとbは1から上限までの範囲でループします。cはピタゴラスの定理を使用して計算され、その値が整数であるかどうか（つまり、実際にピタゴラスの三つ組であるかどうか）を確認します。三つ組が見つかった場合、それらはリストに追加されます。
最後に、見つかった全ての三つ組が出力されます。上記の例では、上限を100と設定していますが、この値は必要に応じて変更できます。
def find_pythagorean_triplets(limit):
triplets = []
for a in range(1, limit):
for b in range(a, limit):
c = (a**2 + b**2)**0.5
if c.is_integer() and c <= limit:
triplets.append((a, b, int(c)))
return triplets
limit = 100
triplets = find_pythagorean_triplets(limit)
for triplet in triplets:
print(triplet)
(3, 4, 5)
(5, 12, 13)
(6, 8, 10)
(7, 24, 25)
(8, 15, 17)
(9, 12, 15)
(9, 40, 41)
(10, 24, 26)
(11, 60, 61)
(12, 16, 20)
(12, 35, 37)
(13, 84, 85)
(14, 48, 50)
(15, 20, 25)
(15, 36, 39)
(16, 30, 34)
(16, 63, 65)
(18, 24, 30)
(18, 80, 82)
(20, 21, 29)
(20, 48, 52)
(21, 28, 35)
(21, 72, 75)
(24, 32, 40)
(24, 45, 51)
(24, 70, 74)
(25, 60, 65)
(27, 36, 45)
(28, 45, 53)
(28, 96, 100)
(30, 40, 50)
(30, 72, 78)
(32, 60, 68)
(33, 44, 55)
(33, 56, 65)
(35, 84, 91)
(36, 48, 60)
(36, 77, 85)
(39, 52, 65)
(39, 80, 89)
(40, 42, 58)
(40, 75, 85)
(42, 56, 70)
(45, 60, 75)
(48, 55, 73)
(48, 64, 80)
(51, 68, 85)
(54, 72, 90)
(57, 76, 95)
(60, 63, 87)
(60, 80, 100)
(65, 72, 97)
%%ai vioras
Plimton 322の数表にある範囲で
Plimpton 322の数表は、最大で約169が記録されていると言われています。したがって、この上限を考慮に入れてピタゴラスの三つ組を探索するプログラムを修正しましょう。
def find_pythagorean_triplets(limit):
triplets = []
for a in range(1, limit):
for b in range(a, limit):
c = (a**2 + b**2)**0.5
if c.is_integer() and c <= limit:
triplets.append((a, b, int(c)))
return triplets
limit = 170
triplets = find_pythagorean_triplets(limit)
for triplet in triplets:
print(triplet)
上記のプログラムは、1から169までのピタゴラスの三つ組をすべて探索し、それらを出力します。ただし、Plimpton 322に記録された三つ組は、特定の比率を持つことが指摘されています（すなわち、それらは「原始的」なピタゴラスの三つ組ではない可能性があります）。そのため、このプログラムが生成するすべての三つ組がPlimpton 322に記録されているわけではありません。
def find_pythagorean_triplets(limit):
triplets = []
for a in range(1, limit):
for b in range(a, limit):
c = (a**2 + b**2)**0.5
if c.is_integer() and c <= limit:
triplets.append((a, b, int(c)))
return triplets
limit = 170
triplets = find_pythagorean_triplets(limit)
for triplet in triplets:
print(triplet)
(3, 4, 5)
(5, 12, 13)
(6, 8, 10)
(7, 24, 25)
(8, 15, 17)
(9, 12, 15)
(9, 40, 41)
(10, 24, 26)
(11, 60, 61)
(12, 16, 20)
(12, 35, 37)
(13, 84, 85)
(14, 48, 50)
(15, 20, 25)
(15, 36, 39)
(15, 112, 113)
(16, 30, 34)
(16, 63, 65)
(17, 144, 145)
(18, 24, 30)
(18, 80, 82)
(20, 21, 29)
(20, 48, 52)
(20, 99, 101)
(21, 28, 35)
(21, 72, 75)
(22, 120, 122)
(24, 32, 40)
(24, 45, 51)
(24, 70, 74)
(24, 143, 145)
(25, 60, 65)
(26, 168, 170)
(27, 36, 45)
(27, 120, 123)
(28, 45, 53)
(28, 96, 100)
(30, 40, 50)
(30, 72, 78)
(32, 60, 68)
(32, 126, 130)
(33, 44, 55)
(33, 56, 65)
(35, 84, 91)
(35, 120, 125)
(36, 48, 60)
(36, 77, 85)
(36, 105, 111)
(36, 160, 164)
(39, 52, 65)
(39, 80, 89)
(40, 42, 58)
(40, 75, 85)
(40, 96, 104)
(42, 56, 70)
(42, 144, 150)
(44, 117, 125)
(45, 60, 75)
(45, 108, 117)
(48, 55, 73)
(48, 64, 80)
(48, 90, 102)
(48, 140, 148)
(50, 120, 130)
(51, 68, 85)
(51, 140, 149)
(54, 72, 90)
(55, 132, 143)
(56, 90, 106)
(56, 105, 119)
(57, 76, 95)
(60, 63, 87)
(60, 80, 100)
(60, 91, 109)
(60, 144, 156)
(63, 84, 105)
(64, 120, 136)
(65, 72, 97)
(65, 156, 169)
(66, 88, 110)
(66, 112, 130)
(69, 92, 115)
(72, 96, 120)
(72, 135, 153)
(72, 154, 170)
(75, 100, 125)
(78, 104, 130)
(80, 84, 116)
(80, 150, 170)
(81, 108, 135)
(84, 112, 140)
(84, 135, 159)
(85, 132, 157)
(87, 116, 145)
(88, 105, 137)
(90, 120, 150)
(93, 124, 155)
(96, 110, 146)
(96, 128, 160)
(99, 132, 165)
(100, 105, 145)
(102, 136, 170)
(119, 120, 169)
%%ai vioras
Plimpton 322は15行というわりに多すぎませんか？
その通りです。Plimpton 322には15行のエントリがありますが、上記のプログラムは指定された範囲内で可能な全てのピタゴラスの三つ組を生成します。その結果、出力される三つ組の数が多すぎるかもしれません。
Plimpton 322に記されているのは、「原始的」でないピタゴラスの三つ組です。つまり、三つ組の各数値が互いに素（共通の約数を持たない）である必要はないということです。さらに、これらの三つ組はある特定の比率を持つことが指摘されています。したがって、全てのピタゴラスの三つ組がPlimpton 322に記録されているわけではありません。
具体的には、Plimpton 322に記録された三つ組は、短辺と斜辺の比が特定のパターンに従っています。したがって、この特性をプログラムに組み込むことで、Plimpton 322に似たような特性を持つピタゴラスの三つ組を見つけることができるかもしれません。ただし、これにはさらに詳細な数学的分析が必要であり、全ての三つ組が厳密にPlimpton 322と一致するわけではないかもしれません。
%%ai vioras
$p$, $q$を互いに素な正の整数とすると、
Plimpton 322の最も左のカラムは$(1/2 (p/q + q/p))^2$に対応すると考える事ができます。
このとき、Plimpton 322を生成する$p$と$q$の組み合わせを求めるプログラムを書いてください
指定された条件に基づいて、Plimpton 322の最も左のカラムを生成する$p$と$q$の組み合わせを探索するプログラムをPythonで書くことができます。以下にその一例を示します。
from math import gcd
def find_plimpton_pairs(limit):
pairs = []
for p in range(1, limit+1):
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1: # ensure p and q are coprime
value = ((p/q + q/p) / 2)**2
pairs.append(((p, q), value))
return pairs
limit = 50 # adjust this value as needed
pairs = find_plimpton_pairs(limit)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
このプログラムは、指定された上限までのすべての$p$と$q$の組み合わせを探索します。$p$と$q$は、$p$が$q$より大きい、互いに素（gcdが1）である組み合わせとします。それぞれの組み合わせについて、与えられた式を計算し、結果とともにリストに追加します。
最後に、得られたすべての組み合わせとその値が出力されます。この例では上限を50と設定していますが、この値は必要に応じて変更できます。
from math import gcd
def find_plimpton_pairs(limit):
pairs = []
for p in range(1, limit+1):
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1: # ensure p and q are coprime
value = ((p/q + q/p) / 2)**2
pairs.append(((p, q), value))
return pairs
limit = 50 # adjust this value as needed
pairs = find_plimpton_pairs(limit)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
Pair: (2, 1), Value: 1.5625
Pair: (3, 1), Value: 2.777777777777778
Pair: (3, 2), Value: 1.173611111111111
Pair: (4, 1), Value: 4.515625
Pair: (4, 3), Value: 1.0850694444444442
Pair: (5, 1), Value: 6.760000000000001
Pair: (5, 2), Value: 2.1025
Pair: (5, 3), Value: 1.2844444444444443
Pair: (5, 4), Value: 1.050625
Pair: (6, 1), Value: 9.506944444444445
Pair: (6, 5), Value: 1.033611111111111
Pair: (7, 1), Value: 12.755102040816327
Pair: (7, 2), Value: 3.582908163265306
Pair: (7, 3), Value: 1.90702947845805
Pair: (7, 4), Value: 1.3472576530612241
Pair: (7, 5), Value: 1.1175510204081633
Pair: (7, 6), Value: 1.0239512471655328
Pair: (8, 1), Value: 16.50390625
Pair: (8, 3), Value: 2.3129340277777777
Pair: (8, 5), Value: 1.2376562500000001
Pair: (8, 7), Value: 1.0179368622448979
Pair: (9, 1), Value: 20.753086419753085
Pair: (9, 2), Value: 5.574845679012346
Pair: (9, 4), Value: 1.815007716049383
Pair: (9, 5), Value: 1.3871604938271607
Pair: (9, 7), Value: 1.0644998740236837
Pair: (9, 8), Value: 1.0139371141975309
Pair: (10, 1), Value: 25.502499999999998
Pair: (10, 3), Value: 3.3002777777777776
Pair: (10, 7), Value: 1.132704081632653
Pair: (10, 9), Value: 1.011141975308642
Pair: (11, 1), Value: 30.752066115702483
Pair: (11, 2), Value: 8.070764462809917
Pair: (11, 3), Value: 3.879706152433425
Pair: (11, 4), Value: 2.4236828512396698
Pair: (11, 5), Value: 1.7616528925619837
Pair: (11, 6), Value: 1.414657943067034
Pair: (11, 7), Value: 1.2185866081969978
Pair: (11, 8), Value: 1.104887654958678
Pair: (11, 9), Value: 1.0408121620242834
Pair: (11, 10), Value: 1.009111570247934
Pair: (12, 1), Value: 36.501736111111114
Pair: (12, 5), Value: 1.9834027777777774
Pair: (12, 7), Value: 1.3197633219954648
Pair: (12, 11), Value: 1.007590105601469
Pair: (13, 1), Value: 42.751479289940825
Pair: (13, 2), Value: 11.068417159763314
Pair: (13, 3), Value: 5.2077580539119
Pair: (13, 4), Value: 3.164293639053254
Pair: (13, 5), Value: 2.2269822485207103
Pair: (13, 6), Value: 1.7268655489809335
Pair: (13, 7), Value: 1.4347301050597754
Pair: (13, 8), Value: 1.2548308062130178
Pair: (13, 9), Value: 1.1414274234787054
Pair: (13, 10), Value: 1.0704289940828404
Pair: (13, 11), Value: 1.028167636559245
Pair: (13, 12), Value: 1.0064205292570676
Pair: (14, 1), Value: 49.50127551020408
Pair: (14, 3), Value: 5.95592403628118
Pair: (14, 5), Value: 2.4918877551020406
Pair: (14, 9), Value: 1.2082545981355508
Pair: (14, 11), Value: 1.059295412379828
Pair: (14, 13), Value: 1.0055020528921628
Pair: (15, 1), Value: 56.75111111111111
Pair: (15, 2), Value: 14.566944444444445
Pair: (15, 4), Value: 4.033402777777778
Pair: (15, 7), Value: 1.702403628117914
Pair: (15, 8), Value: 1.450017361111111
Pair: (15, 11), Value: 1.0993204775022953
Pair: (15, 13), Value: 1.0206180144641683
Pair: (15, 14), Value: 1.004767573696145
Pair: (16, 1), Value: 64.5009765625
Pair: (16, 3), Value: 7.619900173611111
Pair: (16, 5), Value: 3.0844140625000005
Pair: (16, 7), Value: 1.8539740114795917
Pair: (16, 9), Value: 1.3692250192901234
Pair: (16, 11), Value: 1.1470896823347108
Pair: (16, 13), Value: 1.0437372873520712
Pair: (16, 15), Value: 1.0041710069444443
Pair: (17, 1), Value: 72.75086505190312
Pair: (17, 2), Value: 18.565960207612456
Pair: (17, 3), Value: 8.535563244905807
Pair: (17, 4), Value: 5.029465830449827
Pair: (17, 5), Value: 3.4116262975778544
Pair: (17, 6), Value: 2.5380863129565556
Pair: (17, 7), Value: 2.016877339170962
Pair: (17, 8), Value: 1.6842695717993081
Pair: (17, 9), Value: 1.4620445127942243
Pair: (17, 10), Value: 1.3090051903114186
Pair: (17, 11), Value: 1.2017787182933455
Pair: (17, 12), Value: 1.126303585159554
Pair: (17, 13), Value: 1.0737085645257056
Pair: (17, 14), Value: 1.0381726219899723
Pair: (17, 15), Value: 1.0157477893118032
Pair: (17, 16), Value: 1.0036798496972321
Pair: (18, 1), Value: 81.50077160493828
Pair: (18, 5), Value: 3.75929012345679
Pair: (18, 7), Value: 2.1908698664651047
Pair: (18, 11), Value: 1.2627856851341699
Pair: (18, 13), Value: 1.1096911753963037
Pair: (18, 17), Value: 1.0032706437694903
Pair: (19, 1), Value: 90.75069252077564
Pair: (19, 2), Value: 23.06527008310249
Pair: (19, 3), Value: 10.534010464758387
Pair: (19, 4), Value: 6.151705332409971
Pair: (19, 5), Value: 4.1273130193905825
Pair: (19, 6), Value: 3.031875192366882
Pair: (19, 7), Value: 2.3757702526994176
Pair: (19, 8), Value: 1.9544775796398892
Pair: (19, 9), Value: 1.6702917136896824
Pair: (19, 10), Value: 1.4717520775623265
Pair: (19, 11), Value: 1.3296627824454565
Pair: (19, 12), Value: 1.2264591028008618
Pair: (19, 13), Value: 1.1510596797193857
Pair: (19, 14), Value: 1.0961932556956302
Pair: (19, 15), Value: 1.0569282856263467
Pair: (19, 16), Value: 1.0298243810595566
Pair: (19, 17), Value: 1.0124222411793462
Pair: (19, 18), Value: 1.0029261140179886
Pair: (20, 1), Value: 100.50062500000001
Pair: (20, 3), Value: 11.616736111111113
Pair: (20, 7), Value: 2.5714413265306124
Pair: (20, 9), Value: 1.7851929012345682
Pair: (20, 11), Value: 1.4020712809917355
Pair: (20, 13), Value: 1.1973409763313612
Pair: (20, 17), Value: 1.0266457612456747
Pair: (20, 19), Value: 1.0026333102493075
Pair: (21, 1), Value: 110.75056689342404
Pair: (21, 2), Value: 28.064767573696145
Pair: (21, 4), Value: 7.399695294784581
Pair: (21, 5), Value: 4.9241723356009075
Pair: (21, 8), Value: 2.258937429138322
Pair: (21, 10), Value: 1.6591893424036284
Pair: (21, 11), Value: 1.4797511291017784
Pair: (21, 13), Value: 1.248171852567457
Pair: (21, 16), Value: 1.0757887790532878
Pair: (21, 17), Value: 1.0453200888198415
Pair: (21, 19), Value: 1.0100501881269592
Pair: (21, 20), Value: 1.0023823696145124
Pair: (22, 1), Value: 121.50051652892563
Pair: (22, 3), Value: 13.949093204775023
Pair: (22, 5), Value: 5.352913223140497
Pair: (22, 7), Value: 2.994697672457413
Pair: (22, 9), Value: 2.0356660034690344
Pair: (22, 13), Value: 1.303269719790699
Pair: (22, 15), Value: 1.1539967860422407
Pair: (22, 17), Value: 1.0679619806113987
Pair: (22, 19), Value: 1.0216469975504225
Pair: (22, 21), Value: 1.0021656734319075
Pair: (23, 1), Value: 132.75047258979208
Pair: (23, 2), Value: 33.56439035916824
Pair: (23, 3), Value: 15.198697752572988
Pair: (23, 4), Value: 8.773186436672967
Pair: (23, 5), Value: 5.801814744801511
Pair: (23, 6), Value: 4.190624343625289
Pair: (23, 7), Value: 3.2221364916476984
Pair: (23, 8), Value: 2.5966519966918713
Pair: (23, 9), Value: 2.1709958225396155
Pair: (23, 10), Value: 1.8697589792060492
Pair: (23, 11), Value: 1.6501585714508895
Pair: (23, 12), Value: 1.4864557078344884
Pair: (23, 13), Value: 1.3624120535564477
Pair: (23, 14), Value: 1.26737249720304
Pair: (23, 15), Value: 1.1941104809913885
Pair: (23, 16), Value: 1.1375845492674859
Pair: (23, 17), Value: 1.0941909066528872
Pair: (23, 18), Value: 1.0612981049732781
Pair: (23, 19), Value: 1.0369484052385465
Pair: (23, 20), Value: 1.0196609168241964
Pair: (23, 21), Value: 1.0082987196138697
Pair: (23, 22), Value: 1.0019772610101707
Pair: (24, 1), Value: 144.5004340277778
Pair: (24, 5), Value: 6.2708506944444435
Pair: (24, 7), Value: 3.460042871315192
Pair: (24, 11), Value: 1.7426000057392104
Pair: (24, 13), Value: 1.425421700361604
Pair: (24, 17), Value: 1.1237039239715494
Pair: (24, 19), Value: 1.0555759945367804
Pair: (24, 23), Value: 1.0018124146712875
Pair: (25, 1), Value: 156.75039999999998
Pair: (25, 2), Value: 39.5641
Pair: (25, 3), Value: 17.86471111111111
Pair: (25, 4), Value: 10.272025000000001
Pair: (25, 6), Value: 4.854677777777779
Pair: (25, 7), Value: 3.7083755102040814
Pair: (25, 8), Value: 2.96700625
Pair: (25, 9), Value: 2.461412345679012
Pair: (25, 11), Value: 1.839722314049587
Pair: (25, 12), Value: 1.6426694444444445
Pair: (25, 13), Value: 1.4921562130177513
Pair: (25, 14), Value: 1.3755938775510208
Pair: (25, 16), Value: 1.2127515625
Pair: (25, 17), Value: 1.1562574394463672
Pair: (25, 18), Value: 1.1118530864197533
Pair: (25, 19), Value: 1.0772254847645428
Pair: (25, 21), Value: 1.030708390022676
Pair: (25, 22), Value: 1.0164305785123968
Pair: (25, 23), Value: 1.006968620037807
Pair: (25, 24), Value: 1.0016673611111113
Pair: (26, 1), Value: 169.50036982248523
Pair: (26, 3), Value: 19.281106180144636
Pair: (26, 5), Value: 7.269245562130179
Pair: (26, 7), Value: 3.9671008936118826
Pair: (26, 9), Value: 2.616375374388195
Pair: (26, 11), Value: 1.9414427355860921
Pair: (26, 15), Value: 1.3343211702827085
Pair: (26, 17), Value: 1.1916537847300421
Pair: (26, 19), Value: 1.1016499614810928
Pair: (26, 21), Value: 1.0463116706248572
Pair: (26, 23), Value: 1.0151067941074485
Pair: (26, 25), Value: 1.0015390532544381
Pair: (27, 1), Value: 182.75034293552812
Pair: (27, 2), Value: 46.06387174211248
Pair: (27, 4), Value: 11.896111968449931
Pair: (27, 5), Value: 7.798573388203018
Pair: (27, 7), Value: 4.236191595979956
Pair: (27, 8), Value: 3.3696041237997254
Pair: (27, 10), Value: 2.3567935528120714
Pair: (27, 11), Value: 2.0476935460100445
Pair: (27, 13), Value: 1.636358471116306
Pair: (27, 14), Value: 1.4970623022871699
Pair: (27, 16), Value: 1.2997055576989025
Pair: (27, 17), Value: 1.229731204997128
Pair: (27, 19), Value: 1.1286473710809404
Pair: (27, 20), Value: 1.0927992112482856
Pair: (27, 22), Value: 1.0425303823872847
Pair: (27, 23), Value: 1.0259308527879558
Pair: (27, 25), Value: 1.005934705075446
Pair: (27, 26), Value: 1.0014250087255787
Pair: (28, 1), Value: 196.500318877551
Pair: (28, 3), Value: 22.280647675736965
Pair: (28, 5), Value: 8.34797193877551
Pair: (28, 9), Value: 2.9455821680524066
Pair: (28, 11), Value: 2.158418894417271
Pair: (28, 13), Value: 1.7136536197319165
Pair: (28, 15), Value: 1.442858560090703
Pair: (28, 17), Value: 1.2703563042864203
Pair: (28, 19), Value: 1.15805108400701
Pair: (28, 23), Value: 1.039196621465221
Pair: (28, 25), Value: 1.0128984693877552
Pair: (28, 27), Value: 1.0013231887405167
Pair: (29, 1), Value: 210.75029726516053
Pair: (29, 2), Value: 53.06368906064209
Pair: (29, 3), Value: 23.863786497555818
Pair: (29, 4), Value: 13.645381242568371
Pair: (29, 5), Value: 8.917431629013079
Pair: (29, 6), Value: 6.350979323556611
Pair: (29, 7), Value: 4.805382319396249
Pair: (29, 8), Value: 3.804181220273484
Pair: (29, 9), Value: 3.119757490348057
Pair: (29, 10), Value: 2.6322265160523184
Pair: (29, 11), Value: 2.2735723902084297
Pair: (29, 12), Value: 2.002875627559783
Pair: (29, 13), Value: 1.7943206523651047
Pair: (29, 14), Value: 1.630968053095198
Pair: (29, 15), Value: 1.5013291055621616
Pair: (29, 16), Value: 1.397388943593936
Pair: (29, 17), Value: 1.3134182819102318
Pair: (29, 18), Value: 1.2452336650959321
Pair: (29, 19), Value: 1.1897226952480395
Pair: (29, 20), Value: 1.1445310642092745
Pair: (29, 21), Value: 1.1078513054052377
Pair: (29, 22), Value: 1.0782771641395033
Pair: (29, 23), Value: 1.0547012850396393
Pair: (29, 24), Value: 1.0362420935724668
Pair: (29, 25), Value: 1.0221907253269915
Pair: (29, 26), Value: 1.011971958572846
Pair: (29, 27), Value: 1.005115081170923
Pair: (29, 28), Value: 1.0012319062583417
Pair: (30, 1), Value: 225.5002777777778
Pair: (30, 7), Value: 5.105447845804989
Pair: (30, 11), Value: 2.393115243342516
Pair: (30, 13), Value: 1.8783053911900067
Pair: (30, 17), Value: 1.3588244905805456
Pair: (30, 19), Value: 1.2235464758387196
Pair: (30, 23), Value: 1.0722752572988867
Pair: (30, 29), Value: 1.001149755581979
Pair: (31, 1), Value: 240.75026014568158
Pair: (31, 2), Value: 60.563540582726326
Pair: (31, 3), Value: 27.196785755578688
Pair: (31, 4), Value: 15.519787330905306
Pair: (31, 5), Value: 10.116503642039543
Pair: (31, 6), Value: 7.182976355648052
Pair: (31, 7), Value: 5.415808362887299
Pair: (31, 8), Value: 4.270555573621228
Pair: (31, 9), Value: 3.4871211829241666
Pair: (31, 10), Value: 2.928514568158169
Pair: (31, 11), Value: 2.517014817554029
Pair: (31, 12), Value: 2.2058637559255407
Pair: (31, 13), Value: 1.9655622533233996
Pair: (31, 14), Value: 1.7767538597124597
Pair: (31, 15), Value: 1.6263105561336573
Pair: (31, 16), Value: 1.5050738569849116
Pair: (31, 17), Value: 1.4064969808698409
Pair: (31, 18), Value: 1.3257995465114785
Pair: (31, 19), Value: 1.2594250564249498
Pair: (31, 20), Value: 1.2046832726326744
Pair: (31, 21), Value: 1.1595088260763895
Pair: (31, 22), Value: 1.1222948074061974
Pair: (31, 23), Value: 1.091775855726844
Pair: (31, 24), Value: 1.0669446070354955
Pair: (31, 25), Value: 1.0469910509885536
Pair: (31, 26), Value: 1.0312578890332431
Pair: (31, 27), Value: 1.0192072443970543
Pair: (31, 28), Value: 1.010395540890654
Pair: (31, 29), Value: 1.00445433747298
Pair: (31, 30), Value: 1.0010755578679615
Pair: (32, 1), Value: 256.500244140625
Pair: (32, 3), Value: 28.946641710069443
Pair: (32, 5), Value: 10.746103515625002
Pair: (32, 7), Value: 5.736452686543367
Pair: (32, 9), Value: 3.6802692177854937
Pair: (32, 11), Value: 2.645243494963843
Pair: (32, 13), Value: 2.056052665033284
Pair: (32, 15), Value: 1.6927094184027778
Pair: (32, 17), Value: 1.4563697894139274
Pair: (32, 19), Value: 1.297276039863227
Pair: (32, 21), Value: 1.1881648818381518
Pair: (32, 23), Value: 1.1130823376949432
Pair: (32, 25), Value: 1.0621878906250002
Pair: (32, 27), Value: 1.0291444964206105
Pair: (32, 29), Value: 1.0097217900007431
Pair: (32, 31), Value: 1.0010083185646461
Pair: (33, 1), Value: 272.7502295684114
Pair: (33, 2), Value: 68.56341827364557
Pair: (33, 4), Value: 17.519298094582187
Pair: (33, 5), Value: 11.395739210284663
Pair: (33, 7), Value: 6.067371301137535
Pair: (33, 8), Value: 4.768598628328742
Pair: (33, 10), Value: 3.2454568411386595
Pair: (33, 13), Value: 2.1497438070864643
Pair: (33, 14), Value: 1.9340260208766704
Pair: (33, 16), Value: 1.622246075814968
Pair: (33, 17), Value: 1.5083867933820747
Pair: (33, 19), Value: 1.3370293211642998
Pair: (33, 20), Value: 1.272452364554637
Pair: (33, 23), Value: 1.136091973177383
Pair: (33, 25), Value: 1.0790802571166207
Pair: (33, 26), Value: 1.0579249324878697
Pair: (33, 28), Value: 1.0272392875883134
Pair: (33, 29), Value: 1.016788793785875
Pair: (33, 31), Value: 1.0039138905849718
Pair: (33, 32), Value: 1.0009471938848715
Pair: (34, 1), Value: 289.5002162629758
Pair: (34, 3), Value: 32.61305747789312
Pair: (34, 5), Value: 12.065406574394464
Pair: (34, 7), Value: 6.408556069486617
Pair: (34, 9), Value: 4.085418535605963
Pair: (34, 11), Value: 2.91459757213532
Pair: (34, 13), Value: 2.2466076145042075
Pair: (34, 15), Value: 1.8331036139946177
Pair: (34, 19), Value: 1.378624950876554
Pair: (34, 21), Value: 1.2507007705042803
Pair: (34, 23), Value: 1.1607169138087792
Pair: (34, 25), Value: 1.097564359861592
Pair: (34, 27), Value: 1.0540891798501053
Pair: (34, 29), Value: 1.0255156881945613
Pair: (34, 31), Value: 1.0085571276316119
Pair: (34, 33), Value: 1.0008914641857392
Pair: (35, 1), Value: 306.7502040816327
Pair: (35, 2), Value: 77.06331632653061
Pair: (35, 3), Value: 34.52961451247165
Pair: (35, 4), Value: 19.643890306122447
Pair: (35, 6), Value: 9.014291383219955
Pair: (35, 8), Value: 5.298217474489796
Pair: (35, 9), Value: 4.297394809775761
Pair: (35, 11), Value: 3.05568561308821
Pair: (35, 12), Value: 2.656123866213152
Pair: (35, 13), Value: 2.3466199734331608
Pair: (35, 16), Value: 1.7485339604591839
Pair: (35, 17), Value: 1.6186681731516135
Pair: (35, 18), Value: 1.5113384983623082
Pair: (35, 19), Value: 1.422011419526259
Pair: (35, 22), Value: 1.231523444088379
Pair: (35, 23), Value: 1.1868816789475714
Pair: (35, 24), Value: 1.149235048185941
Pair: (35, 26), Value: 1.0909917280521677
Pair: (35, 27), Value: 1.0688715321519553
Pair: (35, 29), Value: 1.0357824747021283
Pair: (35, 31), Value: 1.0148009089171572
Pair: (35, 32), Value: 1.0080518574617345
Pair: (35, 33), Value: 1.0034662019077605
Pair: (35, 34), Value: 1.0008405126756585
Pair: (36, 1), Value: 324.5001929012346
Pair: (36, 5), Value: 13.4648225308642
Pair: (36, 7), Value: 7.121697058453012
Pair: (36, 11), Value: 3.20102699979594
Pair: (36, 13), Value: 2.4497600719555845
Pair: (36, 17), Value: 1.6768557232261099
Pair: (36, 19), Value: 1.4671442708867686
Pair: (36, 23), Value: 1.214521123596817
Pair: (36, 25), Value: 1.138963271604938
Pair: (36, 29), Value: 1.047485586309655
Pair: (36, 31), Value: 1.0225268897496183
Pair: (36, 35), Value: 1.0007938082640464
Pair: (37, 1), Value: 342.75018261504744
Pair: (37, 2), Value: 86.0632304601899
Pair: (37, 3), Value: 38.5294213132051
Pair: (37, 4), Value: 21.893546840759683
Pair: (37, 5), Value: 14.194565376186999
Pair: (37, 6), Value: 10.013518586153722
Pair: (37, 7), Value: 7.493642014877537
Pair: (37, 8), Value: 5.859343613038713
Pair: (37, 9), Value: 4.74010046082118
Pair: (37, 10), Value: 3.9407615047479916
Pair: (37, 11), Value: 3.3506088174392845
Pair: (37, 12), Value: 2.9030326779482185
Pair: (37, 13), Value: 2.5560098720181883
Pair: (37, 14), Value: 2.281966018693818
Pair: (37, 15), Value: 2.0621994967940913
Pair: (37, 16), Value: 1.8836635146548575
Pair: (37, 17), Value: 1.7370318040850166
Pair: (37, 18), Value: 1.6154944358773184
Pair: (37, 19), Value: 1.5139849739685032
Pair: (37, 20), Value: 1.4286710189919654
Pair: (37, 21), Value: 1.3566103334443103
Pair: (37, 22), Value: 1.2955137821538312
Pair: (37, 23), Value: 1.2435787854476863
Pair: (37, 24), Value: 1.199370295126207
Pair: (37, 25), Value: 1.1617344046749454
Pair: (37, 26), Value: 1.1297347543449416
Pair: (37, 27), Value: 1.1026051076101127
Pair: (37, 28), Value: 1.0797135645711902
Pair: (37, 29), Value: 1.060535259686849
Pair: (37, 30), Value: 1.044631320509699
Pair: (37, 31), Value: 1.0316324987135237
Pair: (37, 32), Value: 1.0212263242444304
Pair: (37, 33), Value: 1.0131469418938708
Pair: (37, 34), Value: 1.0071670087276092
Pair: (37, 35), Value: 1.0030911882649334
Pair: (37, 36), Value: 1.0007508916574228
Pair: (38, 1), Value: 361.5001731301939
Pair: (38, 3), Value: 40.61266928285626
Pair: (38, 5), Value: 14.944328254847644
Pair: (38, 7), Value: 7.875830318276896
Pair: (38, 9), Value: 4.970813669163161
Pair: (38, 11), Value: 3.504419827842769
Pair: (38, 13), Value: 2.6653536773262956
Pair: (38, 15), Value: 2.143398738073253
Pair: (38, 17), Value: 1.7991695741356668
Pair: (38, 21), Value: 1.3949445198208554
Pair: (38, 23), Value: 1.2740055323115271
Pair: (38, 25), Value: 1.1858063711911357
Pair: (38, 27), Value: 1.1214108139636505
Pair: (38, 29), Value: 1.0748533848702737
Pair: (38, 31), Value: 1.0420284805474445
Pair: (38, 33), Value: 1.0200355672056756
Pair: (38, 35), Value: 1.0067783650856466
Pair: (38, 37), Value: 1.000711364018057
Pair: (39, 1), Value: 380.75016436554904
Pair: (39, 2), Value: 95.56315746219592
Pair: (39, 4), Value: 24.268254848783695
Pair: (39, 5), Value: 15.714109138724524
Pair: (39, 7), Value: 8.26825799353272
Pair: (39, 8), Value: 6.45192564513478
Pair: (39, 10), Value: 4.318936554898092
Pair: (39, 11), Value: 3.662450214897767
Pair: (39, 14), Value: 2.472266668008426
Pair: (39, 16), Value: 2.027429143039119
Pair: (39, 17), Value: 1.8632455882921681
Pair: (39, 19), Value: 1.6126600629051089
Pair: (39, 20), Value: 1.5163712195923735
Pair: (39, 22), Value: 1.3651934215745405
Pair: (39, 23), Value: 1.3057584491349215
Pair: (39, 25), Value: 1.2111284681130834
Pair: (39, 28), Value: 1.1138753455030928
Pair: (39, 29), Value: 1.0903717358487324
Pair: (39, 31), Value: 1.0536368742564213
Pair: (39, 32), Value: 1.0396482127814757
Pair: (39, 34), Value: 1.0189425607811287
Pair: (39, 35), Value: 1.0117559607669497
Pair: (39, 37), Value: 1.0027739237718445
Pair: (39, 38), Value: 1.000674877659216
Pair: (40, 1), Value: 400.50015625
Pair: (40, 3), Value: 44.945850694444445
Pair: (40, 7), Value: 8.67092155612245
Pair: (40, 9), Value: 5.450927854938271
Pair: (40, 11), Value: 3.824691373966942
Pair: (40, 13), Value: 2.8932701553254443
Pair: (40, 17), Value: 1.9292392949826989
Pair: (40, 19), Value: 1.66443949099723
Pair: (40, 21), Value: 1.4759357284580497
Pair: (40, 23), Value: 1.3387999172967864
Pair: (40, 27), Value: 1.1626030949931414
Pair: (40, 29), Value: 1.107030506837099
Pair: (40, 31), Value: 1.0663893405306972
Pair: (40, 33), Value: 1.0374657082185492
Pair: (40, 37), Value: 1.0060903259678602
Pair: (40, 39), Value: 1.0006411283694938
Pair: (41, 1), Value: 420.7501487209994
Pair: (41, 2), Value: 105.5630948839976
Pair: (41, 3), Value: 47.195782933439084
Pair: (41, 4), Value: 26.76800453599048
Pair: (41, 5), Value: 17.31371802498512
Pair: (41, 6), Value: 12.178965067089695
Pair: (41, 7), Value: 9.083817941215749
Pair: (41, 8), Value: 7.075924393961928
Pair: (41, 9), Value: 5.700318005890086
Pair: (41, 10), Value: 4.717372099940511
Pair: (41, 11), Value: 3.991135736795787
Pair: (41, 12), Value: 3.4398186016921137
Pair: (41, 13), Value: 3.0118202394320086
Pair: (41, 14), Value: 2.6732819689446266
Pair: (41, 15), Value: 2.4012400026439287
Pair: (41, 16), Value: 2.1796741383477096
Pair: (41, 17), Value: 1.9971326179630264
Pair: (41, 18), Value: 1.8452535050418253
Pair: (41, 19), Value: 1.7178157046079616
Pair: (41, 20), Value: 1.6101133997620463
Pair: (41, 21), Value: 1.5185338065426446
Pair: (41, 22), Value: 1.4402660876790185
Pair: (41, 23), Value: 1.3730968491389925
Pair: (41, 24), Value: 1.315263990101791
Pair: (41, 25), Value: 1.2653506246281974
Pair: (41, 26), Value: 1.2222069932309942
Pair: (41, 27), Value: 1.184892231337249
Pair: (41, 28), Value: 1.152630426798917
Pair: (41, 29), Value: 1.1247770953391791
Pair: (41, 30), Value: 1.1007933439090491
Pair: (41, 31), Value: 1.0802257711671301
Pair: (41, 32), Value: 1.0626906940158385
Pair: (41, 33), Value: 1.0478616678930346
Pair: (41, 34), Value: 1.0354595375960511
Pair: (41, 35), Value: 1.0252444487610632
Pair: (41, 36), Value: 1.0170093905376723
Pair: (41, 37), Value: 1.0105749429993367
Pair: (41, 38), Value: 1.0057849790966662
Pair: (41, 39), Value: 1.0025031279321306
Pair: (41, 40), Value: 1.0006098490481854
Pair: (42, 1), Value: 441.500141723356
Pair: (42, 5), Value: 18.143543083900227
Pair: (42, 11), Value: 4.161776625250652
Pair: (42, 13), Value: 3.1334187027868343
Pair: (42, 17), Value: 2.066909606980047
Pair: (42, 19), Value: 1.7727687797187208
Pair: (42, 23), Value: 1.4086200485235052
Pair: (42, 25), Value: 1.2941770975056688
Pair: (42, 29), Value: 1.1435650855665294
Pair: (42, 31), Value: 1.09509312743481
Pair: (42, 37), Value: 1.0161522181309826
Pair: (42, 41), Value: 1.0005808044018718
Pair: (43, 1), Value: 462.7501352082207
Pair: (43, 2), Value: 116.06304083288263
Pair: (43, 3), Value: 51.86232798509705
Pair: (43, 4), Value: 29.392788331530557
Pair: (43, 5), Value: 18.993380205516495
Pair: (43, 6), Value: 13.345145273721533
Pair: (43, 7), Value: 9.940298672200088
Pair: (43, 8), Value: 7.7313095761222295
Pair: (43, 9), Value: 6.217741989330236
Pair: (43, 10), Value: 5.136020822065982
Pair: (43, 11), Value: 4.3366081285841345
Pair: (43, 12), Value: 3.7295394282194585
Pair: (43, 13), Value: 3.2580572898832245
Pair: (43, 14), Value: 2.884919178596263
Pair: (43, 15), Value: 2.5848662940929032
Pair: (43, 16), Value: 2.3402773669889125
Pair: (43, 17), Value: 2.1385561446288177
Pair: (43, 18), Value: 1.9705049943579778
Pair: (43, 19), Value: 1.829281081785617
Pair: (43, 20), Value: 1.7097082882639263
Pair: (43, 21), Value: 1.6078127663540627
Pair: (43, 22), Value: 1.5205027622704255
Pair: (43, 23), Value: 1.4453436742488914
Pair: (43, 24), Value: 1.3803972962111652
Pair: (43, 25), Value: 1.324105137912385
Pair: (43, 26), Value: 1.2752025323139642
Pair: (43, 27), Value: 1.2326545843562051
Pair: (43, 28), Value: 1.1956078368340306
Pair: (43, 29), Value: 1.1633533953822774
Pair: (43, 30), Value: 1.1352985097049457
Pair: (43, 31), Value: 1.1109444652986202
Pair: (43, 32), Value: 1.0898692335806517
Pair: (43, 33), Value: 1.0717137449523508
Pair: (43, 34), Value: 1.0561709452972803
Pair: (43, 35), Value: 1.042977009083785
Pair: (43, 36), Value: 1.031904236691171
Pair: (43, 37), Value: 1.022755276873646
Pair: (43, 38), Value: 1.0153583991646304
Pair: (43, 39), Value: 1.0095636036893265
Pair: (43, 40), Value: 1.005239403055706
Pair: (43, 41), Value: 1.0022701468292103
Pair: (43, 42), Value: 1.0005537865046865
Pair: (44, 1), Value: 484.5001291322314
Pair: (44, 3), Value: 54.27893996786042
Pair: (44, 5), Value: 19.863228305785125
Pair: (44, 7), Value: 10.383878499747006
Pair: (44, 9), Value: 6.48576835271911
Pair: (44, 13), Value: 3.385728672551225
Pair: (44, 15), Value: 2.6801658631772263
Pair: (44, 17), Value: 2.2120596993050987
Pair: (44, 19), Value: 1.887336957143838
Pair: (44, 21), Value: 1.654452982983827
Pair: (44, 23), Value: 1.483244787842335
Pair: (44, 25), Value: 1.3551076446280992
Pair: (44, 27), Value: 1.2580605791359156
Pair: (44, 29), Value: 1.1841055573844597
Pair: (44, 31), Value: 1.127738113922309
Pair: (44, 35), Value: 1.053289024287401
Pair: (44, 37), Value: 1.0303247567144989
Pair: (44, 39), Value: 1.0146218267940295
Pair: (44, 41), Value: 1.00499513584004
Pair: (44, 43), Value: 1.000528611065173
Pair: (45, 1), Value: 506.7501234567902
Pair: (45, 2), Value: 127.0629938271605
Pair: (45, 4), Value: 32.142600308641974
Pair: (45, 7), Value: 10.837682035777275
Pair: (45, 8), Value: 8.418057484567901
Pair: (45, 11), Value: 4.698822569125599
Pair: (45, 13), Value: 3.516426327708379
Pair: (45, 14), Value: 3.1071056941295034
Pair: (45, 16), Value: 2.509144000771605
Pair: (45, 17), Value: 2.2874091161519075
Pair: (45, 19), Value: 1.9469224718716864
Pair: (45, 22), Value: 1.6057241607999184
Pair: (45, 23), Value: 1.5223029708978035
Pair: (45, 26), Value: 1.3323473226678353
Pair: (45, 28), Value: 1.2425171642731168
Pair: (45, 29), Value: 1.2057891105532803
Pair: (45, 31), Value: 1.1454369805115554
Pair: (45, 32), Value: 1.12080451871142
Pair: (45, 34), Value: 1.080648575334273
Pair: (45, 37), Value: 1.038807816825835
Pair: (45, 38), Value: 1.0288602475975515
Pair: (45, 41), Value: 1.0086908879928906
Pair: (45, 43), Value: 1.0020682517743995
Pair: (45, 44), Value: 1.0005051142740535
Pair: (46, 1), Value: 529.500118147448
Pair: (46, 3), Value: 59.27884110480992
Pair: (46, 5), Value: 21.66295368620037
Pair: (46, 7), Value: 11.30170759229968
Pair: (46, 9), Value: 7.040434140820089
Pair: (46, 11), Value: 4.88619666765611
Pair: (46, 13), Value: 3.650144433507455
Pair: (46, 15), Value: 2.8776942869145143
Pair: (46, 17), Value: 2.36459443946599
Pair: (46, 19), Value: 2.008025189952296
Pair: (46, 21), Value: 1.75164950983544
Pair: (46, 25), Value: 1.420242155009452
Pair: (46, 27), Value: 1.3117810671064538
Pair: (46, 29), Value: 1.2283750834477816
Pair: (46, 31), Value: 1.16400795976938
Pair: (46, 33), Value: 1.1144293293825003
Pair: (46, 35), Value: 1.0765673585124031
Pair: (46, 37), Value: 1.048157296800198
Pair: (46, 39), Value: 1.0274997700746573
Pair: (46, 41), Value: 1.013299494854647
Pair: (46, 43), Value: 1.004555226296133
Pair: (46, 45), Value: 1.0004831501318583
Pair: (47, 1), Value: 552.7501131733816
Pair: (47, 2), Value: 138.56295269352648
Pair: (47, 3), Value: 61.8621296715457
Pair: (47, 4), Value: 35.017435774105934
Pair: (47, 5), Value: 22.592829334540518
Pair: (47, 6), Value: 15.84435201951612
Pair: (47, 7), Value: 11.775953658964717
Pair: (47, 8), Value: 9.136149346423721
Pair: (47, 9), Value: 7.327068278479174
Pair: (47, 10), Value: 6.033817338162064
Pair: (47, 11), Value: 5.077743565952957
Pair: (47, 12), Value: 4.351366411397818
Pair: (47, 13), Value: 3.7868777807838296
Pair: (47, 14), Value: 3.3397840236139724
Pair: (47, 15), Value: 2.9799084553090887
Pair: (47, 16), Value: 2.6861989481948845
Pair: (47, 17), Value: 2.443606761267605
Pair: (47, 18), Value: 2.2411434842870634
Pair: (47, 19), Value: 2.070633984116853
Pair: (47, 20), Value: 1.9258943526482575
Pair: (47, 21), Value: 1.8021770349908486
Pair: (47, 22), Value: 1.6957883133986058
Pair: (47, 23), Value: 1.603819569538946
Pair: (47, 24), Value: 1.5239552289246014
Pair: (47, 25), Value: 1.4543333635129017
Pair: (47, 26), Value: 1.3934430757980396
Pair: (47, 27), Value: 1.3400479768201043
Pair: (47, 28), Value: 1.2931284413946653
Pair: (47, 29), Value: 1.25183755353868
Pair: (47, 30), Value: 1.2154671545696896
Pair: (47, 31), Value: 1.1834214303513815
Pair: (47, 32), Value: 1.1551961834045383
Pair: (47, 33), Value: 1.1303624333378648
Pair: (47, 34), Value: 1.1085533426482728
Pair: (47, 35), Value: 1.0894537190158995
Pair: (47, 36), Value: 1.072791529740847
Pair: (47, 37), Value: 1.0583309993217862
Pair: (47, 38), Value: 1.0458669613981586
Pair: (47, 39), Value: 1.03522021114388
Pair: (47, 40), Value: 1.0262336605930287
Pair: (47, 41), Value: 1.018769142190202
Pair: (47, 42), Value: 1.0127047386028505
Pair: (47, 43), Value: 1.0079325420541023
Pair: (47, 44), Value: 1.0043567659911183
Pair: (47, 45), Value: 1.0018921471645177
Pair: (47, 46), Value: 1.0004625881746867
Pair: (48, 1), Value: 576.5001085069445
Pair: (48, 5), Value: 23.542712673611106
Pair: (48, 7), Value: 12.260418881094102
Pair: (48, 11), Value: 5.2734599187901745
Pair: (48, 13), Value: 3.9266216972797507
Pair: (48, 17), Value: 2.524438091719531
Pair: (48, 19), Value: 2.1347388739804556
Pair: (48, 23), Value: 1.6462470545184833
Pair: (48, 25), Value: 1.4894168402777779
Pair: (48, 29), Value: 1.2761532701232
Pair: (48, 31), Value: 1.2036508239753152
Pair: (48, 35), Value: 1.1031250885770973
Pair: (48, 37), Value: 1.0692910763381627
Pair: (48, 41), Value: 1.0250533562404984
Pair: (48, 43), Value: 1.012149080677994
Pair: (48, 47), Value: 1.0004433115317388
Pair: (49, 1), Value: 600.750104123282
Pair: (49, 2), Value: 150.5629164931279
Pair: (49, 3), Value: 67.19538155398213
Pair: (49, 4), Value: 38.01729097251145
Pair: (49, 5), Value: 24.51260308204915
Pair: (49, 6), Value: 17.177359549261883
Pair: (49, 8), Value: 9.885570140045814
Pair: (49, 9), Value: 7.918927812999728
Pair: (49, 10), Value: 6.5129123281965855
Pair: (49, 11), Value: 5.473342718770758
Pair: (49, 12), Value: 4.683396530380859
Pair: (49, 13), Value: 4.069371982581222
Pair: (49, 15), Value: 3.1912055162200934
Pair: (49, 16), Value: 2.871382122683257
Pair: (49, 17), Value: 2.6070812478652927
Pair: (49, 18), Value: 2.3863594001470583
Pair: (49, 19), Value: 2.2003308870611393
Pair: (49, 20), Value: 2.0422743127863394
Pair: (49, 22), Value: 1.7905816188846932
Pair: (49, 23), Value: 1.6897693068971735
Pair: (49, 24), Value: 1.6020757048567726
Pair: (49, 25), Value: 1.5254770512286546
Pair: (49, 26), Value: 1.4583311255911615
Pair: (49, 27), Value: 1.3992940755709355
Pair: (49, 29), Value: 1.301301330549449
Pair: (49, 30), Value: 1.260655398213707
Pair: (49, 31), Value: 1.224672255446807
Pair: (49, 32), Value: 1.1928038813580277
Pair: (49, 33), Value: 1.1645840098000184
Pair: (49, 34), Value: 1.1396139187968106
Pair: (49, 36), Value: 1.0980996376252692
Pair: (49, 37), Value: 1.0810035020105149
Pair: (49, 38), Value: 1.0660396147265507
Pair: (49, 39), Value: 1.053013194973276
Pair: (49, 40), Value: 1.0417535011453563
Pair: (49, 41), Value: 1.0321103565562733
Pair: (49, 43), Value: 1.0171588861590708
Pair: (49, 44), Value: 1.0116291614891864
Pair: (49, 45), Value: 1.0072693990672612
Pair: (49, 46), Value: 1.0039968873240435
Pair: (49, 47), Value: 1.0017376191337204
Pair: (49, 48), Value: 1.0004252152604238
Pair: (50, 1), Value: 625.5001000000001
Pair: (50, 3), Value: 69.94534444444444
Pair: (50, 7), Value: 13.260002040816326
Pair: (50, 9), Value: 8.224149382716048
Pair: (50, 11), Value: 5.677389256198348
Pair: (50, 13), Value: 4.215124852071007
Pair: (50, 17), Value: 2.6915297577854673
Pair: (50, 19), Value: 2.2674019390581717
Pair: (50, 21), Value: 1.961333560090703
Pair: (50, 23), Value: 1.7343744801512286
Pair: (50, 27), Value: 1.4302388203017835
Pair: (50, 29), Value: 1.3272629013079666
Pair: (50, 31), Value: 1.2464642039542144
Pair: (50, 33), Value: 1.182821028466483
Pair: (50, 37), Value: 1.0934376186997805
Pair: (50, 39), Value: 1.0630138724523341
Pair: (50, 41), Value: 1.03990249851279
Pair: (50, 43), Value: 1.0229205516495405
Pair: (50, 47), Value: 1.003833454051607
Pair: (50, 49), Value: 1.0004082049146192
条件を絞る: \(p > q\)#
%%ai vioras
$p$と$q$は対称なので$p>q$を仮定してください
その通りです、$p$と$q$は対称的なので、$p > q$であるという仮定を追加してプログラムを修正します。以下のプログラムでは、$q$の範囲が$p$より小さい値に制限されています。
from math import gcd
def find_plimpton_pairs(limit):
pairs = []
for p in range(1, limit+1):
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1: # ensure p and q are coprime
value = ((p/q + q/p) / 2)**2
pairs.append(((p, q), value))
return pairs
limit = 50 # adjust this value as needed
pairs = find_plimpton_pairs(limit)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
このプログラムは、指定された上限までのすべての$p$と$q$の組み合わせを探索します。$p$と$q$は、$p$が$q$より大きい、互いに素（gcdが1）である組み合わせとします。それぞれの組み合わせについて、与えられた式を計算し、結果とともにリストに追加します。
最後に、得られたすべての組み合わせとその値が出力されます。この例では上限を50と設定していますが、この値は必要に応じて変更できます。
from math import gcd
def find_plimpton_pairs(limit):
pairs = []
for p in range(1, limit+1):
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1: # ensure p and q are coprime
value = ((p/q + q/p) / 2)**2
pairs.append(((p, q), value))
return pairs
limit = 50 # adjust this value as needed
pairs = find_plimpton_pairs(limit)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
Pair: (2, 1), Value: 1.5625
Pair: (3, 1), Value: 2.777777777777778
Pair: (3, 2), Value: 1.173611111111111
Pair: (4, 1), Value: 4.515625
Pair: (4, 3), Value: 1.0850694444444442
Pair: (5, 1), Value: 6.760000000000001
Pair: (5, 2), Value: 2.1025
Pair: (5, 3), Value: 1.2844444444444443
Pair: (5, 4), Value: 1.050625
Pair: (6, 1), Value: 9.506944444444445
Pair: (6, 5), Value: 1.033611111111111
Pair: (7, 1), Value: 12.755102040816327
Pair: (7, 2), Value: 3.582908163265306
Pair: (7, 3), Value: 1.90702947845805
Pair: (7, 4), Value: 1.3472576530612241
Pair: (7, 5), Value: 1.1175510204081633
Pair: (7, 6), Value: 1.0239512471655328
Pair: (8, 1), Value: 16.50390625
Pair: (8, 3), Value: 2.3129340277777777
Pair: (8, 5), Value: 1.2376562500000001
Pair: (8, 7), Value: 1.0179368622448979
Pair: (9, 1), Value: 20.753086419753085
Pair: (9, 2), Value: 5.574845679012346
Pair: (9, 4), Value: 1.815007716049383
Pair: (9, 5), Value: 1.3871604938271607
Pair: (9, 7), Value: 1.0644998740236837
Pair: (9, 8), Value: 1.0139371141975309
Pair: (10, 1), Value: 25.502499999999998
Pair: (10, 3), Value: 3.3002777777777776
Pair: (10, 7), Value: 1.132704081632653
Pair: (10, 9), Value: 1.011141975308642
Pair: (11, 1), Value: 30.752066115702483
Pair: (11, 2), Value: 8.070764462809917
Pair: (11, 3), Value: 3.879706152433425
Pair: (11, 4), Value: 2.4236828512396698
Pair: (11, 5), Value: 1.7616528925619837
Pair: (11, 6), Value: 1.414657943067034
Pair: (11, 7), Value: 1.2185866081969978
Pair: (11, 8), Value: 1.104887654958678
Pair: (11, 9), Value: 1.0408121620242834
Pair: (11, 10), Value: 1.009111570247934
Pair: (12, 1), Value: 36.501736111111114
Pair: (12, 5), Value: 1.9834027777777774
Pair: (12, 7), Value: 1.3197633219954648
Pair: (12, 11), Value: 1.007590105601469
Pair: (13, 1), Value: 42.751479289940825
Pair: (13, 2), Value: 11.068417159763314
Pair: (13, 3), Value: 5.2077580539119
Pair: (13, 4), Value: 3.164293639053254
Pair: (13, 5), Value: 2.2269822485207103
Pair: (13, 6), Value: 1.7268655489809335
Pair: (13, 7), Value: 1.4347301050597754
Pair: (13, 8), Value: 1.2548308062130178
Pair: (13, 9), Value: 1.1414274234787054
Pair: (13, 10), Value: 1.0704289940828404
Pair: (13, 11), Value: 1.028167636559245
Pair: (13, 12), Value: 1.0064205292570676
Pair: (14, 1), Value: 49.50127551020408
Pair: (14, 3), Value: 5.95592403628118
Pair: (14, 5), Value: 2.4918877551020406
Pair: (14, 9), Value: 1.2082545981355508
Pair: (14, 11), Value: 1.059295412379828
Pair: (14, 13), Value: 1.0055020528921628
Pair: (15, 1), Value: 56.75111111111111
Pair: (15, 2), Value: 14.566944444444445
Pair: (15, 4), Value: 4.033402777777778
Pair: (15, 7), Value: 1.702403628117914
Pair: (15, 8), Value: 1.450017361111111
Pair: (15, 11), Value: 1.0993204775022953
Pair: (15, 13), Value: 1.0206180144641683
Pair: (15, 14), Value: 1.004767573696145
Pair: (16, 1), Value: 64.5009765625
Pair: (16, 3), Value: 7.619900173611111
Pair: (16, 5), Value: 3.0844140625000005
Pair: (16, 7), Value: 1.8539740114795917
Pair: (16, 9), Value: 1.3692250192901234
Pair: (16, 11), Value: 1.1470896823347108
Pair: (16, 13), Value: 1.0437372873520712
Pair: (16, 15), Value: 1.0041710069444443
Pair: (17, 1), Value: 72.75086505190312
Pair: (17, 2), Value: 18.565960207612456
Pair: (17, 3), Value: 8.535563244905807
Pair: (17, 4), Value: 5.029465830449827
Pair: (17, 5), Value: 3.4116262975778544
Pair: (17, 6), Value: 2.5380863129565556
Pair: (17, 7), Value: 2.016877339170962
Pair: (17, 8), Value: 1.6842695717993081
Pair: (17, 9), Value: 1.4620445127942243
Pair: (17, 10), Value: 1.3090051903114186
Pair: (17, 11), Value: 1.2017787182933455
Pair: (17, 12), Value: 1.126303585159554
Pair: (17, 13), Value: 1.0737085645257056
Pair: (17, 14), Value: 1.0381726219899723
Pair: (17, 15), Value: 1.0157477893118032
Pair: (17, 16), Value: 1.0036798496972321
Pair: (18, 1), Value: 81.50077160493828
Pair: (18, 5), Value: 3.75929012345679
Pair: (18, 7), Value: 2.1908698664651047
Pair: (18, 11), Value: 1.2627856851341699
Pair: (18, 13), Value: 1.1096911753963037
Pair: (18, 17), Value: 1.0032706437694903
Pair: (19, 1), Value: 90.75069252077564
Pair: (19, 2), Value: 23.06527008310249
Pair: (19, 3), Value: 10.534010464758387
Pair: (19, 4), Value: 6.151705332409971
Pair: (19, 5), Value: 4.1273130193905825
Pair: (19, 6), Value: 3.031875192366882
Pair: (19, 7), Value: 2.3757702526994176
Pair: (19, 8), Value: 1.9544775796398892
Pair: (19, 9), Value: 1.6702917136896824
Pair: (19, 10), Value: 1.4717520775623265
Pair: (19, 11), Value: 1.3296627824454565
Pair: (19, 12), Value: 1.2264591028008618
Pair: (19, 13), Value: 1.1510596797193857
Pair: (19, 14), Value: 1.0961932556956302
Pair: (19, 15), Value: 1.0569282856263467
Pair: (19, 16), Value: 1.0298243810595566
Pair: (19, 17), Value: 1.0124222411793462
Pair: (19, 18), Value: 1.0029261140179886
Pair: (20, 1), Value: 100.50062500000001
Pair: (20, 3), Value: 11.616736111111113
Pair: (20, 7), Value: 2.5714413265306124
Pair: (20, 9), Value: 1.7851929012345682
Pair: (20, 11), Value: 1.4020712809917355
Pair: (20, 13), Value: 1.1973409763313612
Pair: (20, 17), Value: 1.0266457612456747
Pair: (20, 19), Value: 1.0026333102493075
Pair: (21, 1), Value: 110.75056689342404
Pair: (21, 2), Value: 28.064767573696145
Pair: (21, 4), Value: 7.399695294784581
Pair: (21, 5), Value: 4.9241723356009075
Pair: (21, 8), Value: 2.258937429138322
Pair: (21, 10), Value: 1.6591893424036284
Pair: (21, 11), Value: 1.4797511291017784
Pair: (21, 13), Value: 1.248171852567457
Pair: (21, 16), Value: 1.0757887790532878
Pair: (21, 17), Value: 1.0453200888198415
Pair: (21, 19), Value: 1.0100501881269592
Pair: (21, 20), Value: 1.0023823696145124
Pair: (22, 1), Value: 121.50051652892563
Pair: (22, 3), Value: 13.949093204775023
Pair: (22, 5), Value: 5.352913223140497
Pair: (22, 7), Value: 2.994697672457413
Pair: (22, 9), Value: 2.0356660034690344
Pair: (22, 13), Value: 1.303269719790699
Pair: (22, 15), Value: 1.1539967860422407
Pair: (22, 17), Value: 1.0679619806113987
Pair: (22, 19), Value: 1.0216469975504225
Pair: (22, 21), Value: 1.0021656734319075
Pair: (23, 1), Value: 132.75047258979208
Pair: (23, 2), Value: 33.56439035916824
Pair: (23, 3), Value: 15.198697752572988
Pair: (23, 4), Value: 8.773186436672967
Pair: (23, 5), Value: 5.801814744801511
Pair: (23, 6), Value: 4.190624343625289
Pair: (23, 7), Value: 3.2221364916476984
Pair: (23, 8), Value: 2.5966519966918713
Pair: (23, 9), Value: 2.1709958225396155
Pair: (23, 10), Value: 1.8697589792060492
Pair: (23, 11), Value: 1.6501585714508895
Pair: (23, 12), Value: 1.4864557078344884
Pair: (23, 13), Value: 1.3624120535564477
Pair: (23, 14), Value: 1.26737249720304
Pair: (23, 15), Value: 1.1941104809913885
Pair: (23, 16), Value: 1.1375845492674859
Pair: (23, 17), Value: 1.0941909066528872
Pair: (23, 18), Value: 1.0612981049732781
Pair: (23, 19), Value: 1.0369484052385465
Pair: (23, 20), Value: 1.0196609168241964
Pair: (23, 21), Value: 1.0082987196138697
Pair: (23, 22), Value: 1.0019772610101707
Pair: (24, 1), Value: 144.5004340277778
Pair: (24, 5), Value: 6.2708506944444435
Pair: (24, 7), Value: 3.460042871315192
Pair: (24, 11), Value: 1.7426000057392104
Pair: (24, 13), Value: 1.425421700361604
Pair: (24, 17), Value: 1.1237039239715494
Pair: (24, 19), Value: 1.0555759945367804
Pair: (24, 23), Value: 1.0018124146712875
Pair: (25, 1), Value: 156.75039999999998
Pair: (25, 2), Value: 39.5641
Pair: (25, 3), Value: 17.86471111111111
Pair: (25, 4), Value: 10.272025000000001
Pair: (25, 6), Value: 4.854677777777779
Pair: (25, 7), Value: 3.7083755102040814
Pair: (25, 8), Value: 2.96700625
Pair: (25, 9), Value: 2.461412345679012
Pair: (25, 11), Value: 1.839722314049587
Pair: (25, 12), Value: 1.6426694444444445
Pair: (25, 13), Value: 1.4921562130177513
Pair: (25, 14), Value: 1.3755938775510208
Pair: (25, 16), Value: 1.2127515625
Pair: (25, 17), Value: 1.1562574394463672
Pair: (25, 18), Value: 1.1118530864197533
Pair: (25, 19), Value: 1.0772254847645428
Pair: (25, 21), Value: 1.030708390022676
Pair: (25, 22), Value: 1.0164305785123968
Pair: (25, 23), Value: 1.006968620037807
Pair: (25, 24), Value: 1.0016673611111113
Pair: (26, 1), Value: 169.50036982248523
Pair: (26, 3), Value: 19.281106180144636
Pair: (26, 5), Value: 7.269245562130179
Pair: (26, 7), Value: 3.9671008936118826
Pair: (26, 9), Value: 2.616375374388195
Pair: (26, 11), Value: 1.9414427355860921
Pair: (26, 15), Value: 1.3343211702827085
Pair: (26, 17), Value: 1.1916537847300421
Pair: (26, 19), Value: 1.1016499614810928
Pair: (26, 21), Value: 1.0463116706248572
Pair: (26, 23), Value: 1.0151067941074485
Pair: (26, 25), Value: 1.0015390532544381
Pair: (27, 1), Value: 182.75034293552812
Pair: (27, 2), Value: 46.06387174211248
Pair: (27, 4), Value: 11.896111968449931
Pair: (27, 5), Value: 7.798573388203018
Pair: (27, 7), Value: 4.236191595979956
Pair: (27, 8), Value: 3.3696041237997254
Pair: (27, 10), Value: 2.3567935528120714
Pair: (27, 11), Value: 2.0476935460100445
Pair: (27, 13), Value: 1.636358471116306
Pair: (27, 14), Value: 1.4970623022871699
Pair: (27, 16), Value: 1.2997055576989025
Pair: (27, 17), Value: 1.229731204997128
Pair: (27, 19), Value: 1.1286473710809404
Pair: (27, 20), Value: 1.0927992112482856
Pair: (27, 22), Value: 1.0425303823872847
Pair: (27, 23), Value: 1.0259308527879558
Pair: (27, 25), Value: 1.005934705075446
Pair: (27, 26), Value: 1.0014250087255787
Pair: (28, 1), Value: 196.500318877551
Pair: (28, 3), Value: 22.280647675736965
Pair: (28, 5), Value: 8.34797193877551
Pair: (28, 9), Value: 2.9455821680524066
Pair: (28, 11), Value: 2.158418894417271
Pair: (28, 13), Value: 1.7136536197319165
Pair: (28, 15), Value: 1.442858560090703
Pair: (28, 17), Value: 1.2703563042864203
Pair: (28, 19), Value: 1.15805108400701
Pair: (28, 23), Value: 1.039196621465221
Pair: (28, 25), Value: 1.0128984693877552
Pair: (28, 27), Value: 1.0013231887405167
Pair: (29, 1), Value: 210.75029726516053
Pair: (29, 2), Value: 53.06368906064209
Pair: (29, 3), Value: 23.863786497555818
Pair: (29, 4), Value: 13.645381242568371
Pair: (29, 5), Value: 8.917431629013079
Pair: (29, 6), Value: 6.350979323556611
Pair: (29, 7), Value: 4.805382319396249
Pair: (29, 8), Value: 3.804181220273484
Pair: (29, 9), Value: 3.119757490348057
Pair: (29, 10), Value: 2.6322265160523184
Pair: (29, 11), Value: 2.2735723902084297
Pair: (29, 12), Value: 2.002875627559783
Pair: (29, 13), Value: 1.7943206523651047
Pair: (29, 14), Value: 1.630968053095198
Pair: (29, 15), Value: 1.5013291055621616
Pair: (29, 16), Value: 1.397388943593936
Pair: (29, 17), Value: 1.3134182819102318
Pair: (29, 18), Value: 1.2452336650959321
Pair: (29, 19), Value: 1.1897226952480395
Pair: (29, 20), Value: 1.1445310642092745
Pair: (29, 21), Value: 1.1078513054052377
Pair: (29, 22), Value: 1.0782771641395033
Pair: (29, 23), Value: 1.0547012850396393
Pair: (29, 24), Value: 1.0362420935724668
Pair: (29, 25), Value: 1.0221907253269915
Pair: (29, 26), Value: 1.011971958572846
Pair: (29, 27), Value: 1.005115081170923
Pair: (29, 28), Value: 1.0012319062583417
Pair: (30, 1), Value: 225.5002777777778
Pair: (30, 7), Value: 5.105447845804989
Pair: (30, 11), Value: 2.393115243342516
Pair: (30, 13), Value: 1.8783053911900067
Pair: (30, 17), Value: 1.3588244905805456
Pair: (30, 19), Value: 1.2235464758387196
Pair: (30, 23), Value: 1.0722752572988867
Pair: (30, 29), Value: 1.001149755581979
Pair: (31, 1), Value: 240.75026014568158
Pair: (31, 2), Value: 60.563540582726326
Pair: (31, 3), Value: 27.196785755578688
Pair: (31, 4), Value: 15.519787330905306
Pair: (31, 5), Value: 10.116503642039543
Pair: (31, 6), Value: 7.182976355648052
Pair: (31, 7), Value: 5.415808362887299
Pair: (31, 8), Value: 4.270555573621228
Pair: (31, 9), Value: 3.4871211829241666
Pair: (31, 10), Value: 2.928514568158169
Pair: (31, 11), Value: 2.517014817554029
Pair: (31, 12), Value: 2.2058637559255407
Pair: (31, 13), Value: 1.9655622533233996
Pair: (31, 14), Value: 1.7767538597124597
Pair: (31, 15), Value: 1.6263105561336573
Pair: (31, 16), Value: 1.5050738569849116
Pair: (31, 17), Value: 1.4064969808698409
Pair: (31, 18), Value: 1.3257995465114785
Pair: (31, 19), Value: 1.2594250564249498
Pair: (31, 20), Value: 1.2046832726326744
Pair: (31, 21), Value: 1.1595088260763895
Pair: (31, 22), Value: 1.1222948074061974
Pair: (31, 23), Value: 1.091775855726844
Pair: (31, 24), Value: 1.0669446070354955
Pair: (31, 25), Value: 1.0469910509885536
Pair: (31, 26), Value: 1.0312578890332431
Pair: (31, 27), Value: 1.0192072443970543
Pair: (31, 28), Value: 1.010395540890654
Pair: (31, 29), Value: 1.00445433747298
Pair: (31, 30), Value: 1.0010755578679615
Pair: (32, 1), Value: 256.500244140625
Pair: (32, 3), Value: 28.946641710069443
Pair: (32, 5), Value: 10.746103515625002
Pair: (32, 7), Value: 5.736452686543367
Pair: (32, 9), Value: 3.6802692177854937
Pair: (32, 11), Value: 2.645243494963843
Pair: (32, 13), Value: 2.056052665033284
Pair: (32, 15), Value: 1.6927094184027778
Pair: (32, 17), Value: 1.4563697894139274
Pair: (32, 19), Value: 1.297276039863227
Pair: (32, 21), Value: 1.1881648818381518
Pair: (32, 23), Value: 1.1130823376949432
Pair: (32, 25), Value: 1.0621878906250002
Pair: (32, 27), Value: 1.0291444964206105
Pair: (32, 29), Value: 1.0097217900007431
Pair: (32, 31), Value: 1.0010083185646461
Pair: (33, 1), Value: 272.7502295684114
Pair: (33, 2), Value: 68.56341827364557
Pair: (33, 4), Value: 17.519298094582187
Pair: (33, 5), Value: 11.395739210284663
Pair: (33, 7), Value: 6.067371301137535
Pair: (33, 8), Value: 4.768598628328742
Pair: (33, 10), Value: 3.2454568411386595
Pair: (33, 13), Value: 2.1497438070864643
Pair: (33, 14), Value: 1.9340260208766704
Pair: (33, 16), Value: 1.622246075814968
Pair: (33, 17), Value: 1.5083867933820747
Pair: (33, 19), Value: 1.3370293211642998
Pair: (33, 20), Value: 1.272452364554637
Pair: (33, 23), Value: 1.136091973177383
Pair: (33, 25), Value: 1.0790802571166207
Pair: (33, 26), Value: 1.0579249324878697
Pair: (33, 28), Value: 1.0272392875883134
Pair: (33, 29), Value: 1.016788793785875
Pair: (33, 31), Value: 1.0039138905849718
Pair: (33, 32), Value: 1.0009471938848715
Pair: (34, 1), Value: 289.5002162629758
Pair: (34, 3), Value: 32.61305747789312
Pair: (34, 5), Value: 12.065406574394464
Pair: (34, 7), Value: 6.408556069486617
Pair: (34, 9), Value: 4.085418535605963
Pair: (34, 11), Value: 2.91459757213532
Pair: (34, 13), Value: 2.2466076145042075
Pair: (34, 15), Value: 1.8331036139946177
Pair: (34, 19), Value: 1.378624950876554
Pair: (34, 21), Value: 1.2507007705042803
Pair: (34, 23), Value: 1.1607169138087792
Pair: (34, 25), Value: 1.097564359861592
Pair: (34, 27), Value: 1.0540891798501053
Pair: (34, 29), Value: 1.0255156881945613
Pair: (34, 31), Value: 1.0085571276316119
Pair: (34, 33), Value: 1.0008914641857392
Pair: (35, 1), Value: 306.7502040816327
Pair: (35, 2), Value: 77.06331632653061
Pair: (35, 3), Value: 34.52961451247165
Pair: (35, 4), Value: 19.643890306122447
Pair: (35, 6), Value: 9.014291383219955
Pair: (35, 8), Value: 5.298217474489796
Pair: (35, 9), Value: 4.297394809775761
Pair: (35, 11), Value: 3.05568561308821
Pair: (35, 12), Value: 2.656123866213152
Pair: (35, 13), Value: 2.3466199734331608
Pair: (35, 16), Value: 1.7485339604591839
Pair: (35, 17), Value: 1.6186681731516135
Pair: (35, 18), Value: 1.5113384983623082
Pair: (35, 19), Value: 1.422011419526259
Pair: (35, 22), Value: 1.231523444088379
Pair: (35, 23), Value: 1.1868816789475714
Pair: (35, 24), Value: 1.149235048185941
Pair: (35, 26), Value: 1.0909917280521677
Pair: (35, 27), Value: 1.0688715321519553
Pair: (35, 29), Value: 1.0357824747021283
Pair: (35, 31), Value: 1.0148009089171572
Pair: (35, 32), Value: 1.0080518574617345
Pair: (35, 33), Value: 1.0034662019077605
Pair: (35, 34), Value: 1.0008405126756585
Pair: (36, 1), Value: 324.5001929012346
Pair: (36, 5), Value: 13.4648225308642
Pair: (36, 7), Value: 7.121697058453012
Pair: (36, 11), Value: 3.20102699979594
Pair: (36, 13), Value: 2.4497600719555845
Pair: (36, 17), Value: 1.6768557232261099
Pair: (36, 19), Value: 1.4671442708867686
Pair: (36, 23), Value: 1.214521123596817
Pair: (36, 25), Value: 1.138963271604938
Pair: (36, 29), Value: 1.047485586309655
Pair: (36, 31), Value: 1.0225268897496183
Pair: (36, 35), Value: 1.0007938082640464
Pair: (37, 1), Value: 342.75018261504744
Pair: (37, 2), Value: 86.0632304601899
Pair: (37, 3), Value: 38.5294213132051
Pair: (37, 4), Value: 21.893546840759683
Pair: (37, 5), Value: 14.194565376186999
Pair: (37, 6), Value: 10.013518586153722
Pair: (37, 7), Value: 7.493642014877537
Pair: (37, 8), Value: 5.859343613038713
Pair: (37, 9), Value: 4.74010046082118
Pair: (37, 10), Value: 3.9407615047479916
Pair: (37, 11), Value: 3.3506088174392845
Pair: (37, 12), Value: 2.9030326779482185
Pair: (37, 13), Value: 2.5560098720181883
Pair: (37, 14), Value: 2.281966018693818
Pair: (37, 15), Value: 2.0621994967940913
Pair: (37, 16), Value: 1.8836635146548575
Pair: (37, 17), Value: 1.7370318040850166
Pair: (37, 18), Value: 1.6154944358773184
Pair: (37, 19), Value: 1.5139849739685032
Pair: (37, 20), Value: 1.4286710189919654
Pair: (37, 21), Value: 1.3566103334443103
Pair: (37, 22), Value: 1.2955137821538312
Pair: (37, 23), Value: 1.2435787854476863
Pair: (37, 24), Value: 1.199370295126207
Pair: (37, 25), Value: 1.1617344046749454
Pair: (37, 26), Value: 1.1297347543449416
Pair: (37, 27), Value: 1.1026051076101127
Pair: (37, 28), Value: 1.0797135645711902
Pair: (37, 29), Value: 1.060535259686849
Pair: (37, 30), Value: 1.044631320509699
Pair: (37, 31), Value: 1.0316324987135237
Pair: (37, 32), Value: 1.0212263242444304
Pair: (37, 33), Value: 1.0131469418938708
Pair: (37, 34), Value: 1.0071670087276092
Pair: (37, 35), Value: 1.0030911882649334
Pair: (37, 36), Value: 1.0007508916574228
Pair: (38, 1), Value: 361.5001731301939
Pair: (38, 3), Value: 40.61266928285626
Pair: (38, 5), Value: 14.944328254847644
Pair: (38, 7), Value: 7.875830318276896
Pair: (38, 9), Value: 4.970813669163161
Pair: (38, 11), Value: 3.504419827842769
Pair: (38, 13), Value: 2.6653536773262956
Pair: (38, 15), Value: 2.143398738073253
Pair: (38, 17), Value: 1.7991695741356668
Pair: (38, 21), Value: 1.3949445198208554
Pair: (38, 23), Value: 1.2740055323115271
Pair: (38, 25), Value: 1.1858063711911357
Pair: (38, 27), Value: 1.1214108139636505
Pair: (38, 29), Value: 1.0748533848702737
Pair: (38, 31), Value: 1.0420284805474445
Pair: (38, 33), Value: 1.0200355672056756
Pair: (38, 35), Value: 1.0067783650856466
Pair: (38, 37), Value: 1.000711364018057
Pair: (39, 1), Value: 380.75016436554904
Pair: (39, 2), Value: 95.56315746219592
Pair: (39, 4), Value: 24.268254848783695
Pair: (39, 5), Value: 15.714109138724524
Pair: (39, 7), Value: 8.26825799353272
Pair: (39, 8), Value: 6.45192564513478
Pair: (39, 10), Value: 4.318936554898092
Pair: (39, 11), Value: 3.662450214897767
Pair: (39, 14), Value: 2.472266668008426
Pair: (39, 16), Value: 2.027429143039119
Pair: (39, 17), Value: 1.8632455882921681
Pair: (39, 19), Value: 1.6126600629051089
Pair: (39, 20), Value: 1.5163712195923735
Pair: (39, 22), Value: 1.3651934215745405
Pair: (39, 23), Value: 1.3057584491349215
Pair: (39, 25), Value: 1.2111284681130834
Pair: (39, 28), Value: 1.1138753455030928
Pair: (39, 29), Value: 1.0903717358487324
Pair: (39, 31), Value: 1.0536368742564213
Pair: (39, 32), Value: 1.0396482127814757
Pair: (39, 34), Value: 1.0189425607811287
Pair: (39, 35), Value: 1.0117559607669497
Pair: (39, 37), Value: 1.0027739237718445
Pair: (39, 38), Value: 1.000674877659216
Pair: (40, 1), Value: 400.50015625
Pair: (40, 3), Value: 44.945850694444445
Pair: (40, 7), Value: 8.67092155612245
Pair: (40, 9), Value: 5.450927854938271
Pair: (40, 11), Value: 3.824691373966942
Pair: (40, 13), Value: 2.8932701553254443
Pair: (40, 17), Value: 1.9292392949826989
Pair: (40, 19), Value: 1.66443949099723
Pair: (40, 21), Value: 1.4759357284580497
Pair: (40, 23), Value: 1.3387999172967864
Pair: (40, 27), Value: 1.1626030949931414
Pair: (40, 29), Value: 1.107030506837099
Pair: (40, 31), Value: 1.0663893405306972
Pair: (40, 33), Value: 1.0374657082185492
Pair: (40, 37), Value: 1.0060903259678602
Pair: (40, 39), Value: 1.0006411283694938
Pair: (41, 1), Value: 420.7501487209994
Pair: (41, 2), Value: 105.5630948839976
Pair: (41, 3), Value: 47.195782933439084
Pair: (41, 4), Value: 26.76800453599048
Pair: (41, 5), Value: 17.31371802498512
Pair: (41, 6), Value: 12.178965067089695
Pair: (41, 7), Value: 9.083817941215749
Pair: (41, 8), Value: 7.075924393961928
Pair: (41, 9), Value: 5.700318005890086
Pair: (41, 10), Value: 4.717372099940511
Pair: (41, 11), Value: 3.991135736795787
Pair: (41, 12), Value: 3.4398186016921137
Pair: (41, 13), Value: 3.0118202394320086
Pair: (41, 14), Value: 2.6732819689446266
Pair: (41, 15), Value: 2.4012400026439287
Pair: (41, 16), Value: 2.1796741383477096
Pair: (41, 17), Value: 1.9971326179630264
Pair: (41, 18), Value: 1.8452535050418253
Pair: (41, 19), Value: 1.7178157046079616
Pair: (41, 20), Value: 1.6101133997620463
Pair: (41, 21), Value: 1.5185338065426446
Pair: (41, 22), Value: 1.4402660876790185
Pair: (41, 23), Value: 1.3730968491389925
Pair: (41, 24), Value: 1.315263990101791
Pair: (41, 25), Value: 1.2653506246281974
Pair: (41, 26), Value: 1.2222069932309942
Pair: (41, 27), Value: 1.184892231337249
Pair: (41, 28), Value: 1.152630426798917
Pair: (41, 29), Value: 1.1247770953391791
Pair: (41, 30), Value: 1.1007933439090491
Pair: (41, 31), Value: 1.0802257711671301
Pair: (41, 32), Value: 1.0626906940158385
Pair: (41, 33), Value: 1.0478616678930346
Pair: (41, 34), Value: 1.0354595375960511
Pair: (41, 35), Value: 1.0252444487610632
Pair: (41, 36), Value: 1.0170093905376723
Pair: (41, 37), Value: 1.0105749429993367
Pair: (41, 38), Value: 1.0057849790966662
Pair: (41, 39), Value: 1.0025031279321306
Pair: (41, 40), Value: 1.0006098490481854
Pair: (42, 1), Value: 441.500141723356
Pair: (42, 5), Value: 18.143543083900227
Pair: (42, 11), Value: 4.161776625250652
Pair: (42, 13), Value: 3.1334187027868343
Pair: (42, 17), Value: 2.066909606980047
Pair: (42, 19), Value: 1.7727687797187208
Pair: (42, 23), Value: 1.4086200485235052
Pair: (42, 25), Value: 1.2941770975056688
Pair: (42, 29), Value: 1.1435650855665294
Pair: (42, 31), Value: 1.09509312743481
Pair: (42, 37), Value: 1.0161522181309826
Pair: (42, 41), Value: 1.0005808044018718
Pair: (43, 1), Value: 462.7501352082207
Pair: (43, 2), Value: 116.06304083288263
Pair: (43, 3), Value: 51.86232798509705
Pair: (43, 4), Value: 29.392788331530557
Pair: (43, 5), Value: 18.993380205516495
Pair: (43, 6), Value: 13.345145273721533
Pair: (43, 7), Value: 9.940298672200088
Pair: (43, 8), Value: 7.7313095761222295
Pair: (43, 9), Value: 6.217741989330236
Pair: (43, 10), Value: 5.136020822065982
Pair: (43, 11), Value: 4.3366081285841345
Pair: (43, 12), Value: 3.7295394282194585
Pair: (43, 13), Value: 3.2580572898832245
Pair: (43, 14), Value: 2.884919178596263
Pair: (43, 15), Value: 2.5848662940929032
Pair: (43, 16), Value: 2.3402773669889125
Pair: (43, 17), Value: 2.1385561446288177
Pair: (43, 18), Value: 1.9705049943579778
Pair: (43, 19), Value: 1.829281081785617
Pair: (43, 20), Value: 1.7097082882639263
Pair: (43, 21), Value: 1.6078127663540627
Pair: (43, 22), Value: 1.5205027622704255
Pair: (43, 23), Value: 1.4453436742488914
Pair: (43, 24), Value: 1.3803972962111652
Pair: (43, 25), Value: 1.324105137912385
Pair: (43, 26), Value: 1.2752025323139642
Pair: (43, 27), Value: 1.2326545843562051
Pair: (43, 28), Value: 1.1956078368340306
Pair: (43, 29), Value: 1.1633533953822774
Pair: (43, 30), Value: 1.1352985097049457
Pair: (43, 31), Value: 1.1109444652986202
Pair: (43, 32), Value: 1.0898692335806517
Pair: (43, 33), Value: 1.0717137449523508
Pair: (43, 34), Value: 1.0561709452972803
Pair: (43, 35), Value: 1.042977009083785
Pair: (43, 36), Value: 1.031904236691171
Pair: (43, 37), Value: 1.022755276873646
Pair: (43, 38), Value: 1.0153583991646304
Pair: (43, 39), Value: 1.0095636036893265
Pair: (43, 40), Value: 1.005239403055706
Pair: (43, 41), Value: 1.0022701468292103
Pair: (43, 42), Value: 1.0005537865046865
Pair: (44, 1), Value: 484.5001291322314
Pair: (44, 3), Value: 54.27893996786042
Pair: (44, 5), Value: 19.863228305785125
Pair: (44, 7), Value: 10.383878499747006
Pair: (44, 9), Value: 6.48576835271911
Pair: (44, 13), Value: 3.385728672551225
Pair: (44, 15), Value: 2.6801658631772263
Pair: (44, 17), Value: 2.2120596993050987
Pair: (44, 19), Value: 1.887336957143838
Pair: (44, 21), Value: 1.654452982983827
Pair: (44, 23), Value: 1.483244787842335
Pair: (44, 25), Value: 1.3551076446280992
Pair: (44, 27), Value: 1.2580605791359156
Pair: (44, 29), Value: 1.1841055573844597
Pair: (44, 31), Value: 1.127738113922309
Pair: (44, 35), Value: 1.053289024287401
Pair: (44, 37), Value: 1.0303247567144989
Pair: (44, 39), Value: 1.0146218267940295
Pair: (44, 41), Value: 1.00499513584004
Pair: (44, 43), Value: 1.000528611065173
Pair: (45, 1), Value: 506.7501234567902
Pair: (45, 2), Value: 127.0629938271605
Pair: (45, 4), Value: 32.142600308641974
Pair: (45, 7), Value: 10.837682035777275
Pair: (45, 8), Value: 8.418057484567901
Pair: (45, 11), Value: 4.698822569125599
Pair: (45, 13), Value: 3.516426327708379
Pair: (45, 14), Value: 3.1071056941295034
Pair: (45, 16), Value: 2.509144000771605
Pair: (45, 17), Value: 2.2874091161519075
Pair: (45, 19), Value: 1.9469224718716864
Pair: (45, 22), Value: 1.6057241607999184
Pair: (45, 23), Value: 1.5223029708978035
Pair: (45, 26), Value: 1.3323473226678353
Pair: (45, 28), Value: 1.2425171642731168
Pair: (45, 29), Value: 1.2057891105532803
Pair: (45, 31), Value: 1.1454369805115554
Pair: (45, 32), Value: 1.12080451871142
Pair: (45, 34), Value: 1.080648575334273
Pair: (45, 37), Value: 1.038807816825835
Pair: (45, 38), Value: 1.0288602475975515
Pair: (45, 41), Value: 1.0086908879928906
Pair: (45, 43), Value: 1.0020682517743995
Pair: (45, 44), Value: 1.0005051142740535
Pair: (46, 1), Value: 529.500118147448
Pair: (46, 3), Value: 59.27884110480992
Pair: (46, 5), Value: 21.66295368620037
Pair: (46, 7), Value: 11.30170759229968
Pair: (46, 9), Value: 7.040434140820089
Pair: (46, 11), Value: 4.88619666765611
Pair: (46, 13), Value: 3.650144433507455
Pair: (46, 15), Value: 2.8776942869145143
Pair: (46, 17), Value: 2.36459443946599
Pair: (46, 19), Value: 2.008025189952296
Pair: (46, 21), Value: 1.75164950983544
Pair: (46, 25), Value: 1.420242155009452
Pair: (46, 27), Value: 1.3117810671064538
Pair: (46, 29), Value: 1.2283750834477816
Pair: (46, 31), Value: 1.16400795976938
Pair: (46, 33), Value: 1.1144293293825003
Pair: (46, 35), Value: 1.0765673585124031
Pair: (46, 37), Value: 1.048157296800198
Pair: (46, 39), Value: 1.0274997700746573
Pair: (46, 41), Value: 1.013299494854647
Pair: (46, 43), Value: 1.004555226296133
Pair: (46, 45), Value: 1.0004831501318583
Pair: (47, 1), Value: 552.7501131733816
Pair: (47, 2), Value: 138.56295269352648
Pair: (47, 3), Value: 61.8621296715457
Pair: (47, 4), Value: 35.017435774105934
Pair: (47, 5), Value: 22.592829334540518
Pair: (47, 6), Value: 15.84435201951612
Pair: (47, 7), Value: 11.775953658964717
Pair: (47, 8), Value: 9.136149346423721
Pair: (47, 9), Value: 7.327068278479174
Pair: (47, 10), Value: 6.033817338162064
Pair: (47, 11), Value: 5.077743565952957
Pair: (47, 12), Value: 4.351366411397818
Pair: (47, 13), Value: 3.7868777807838296
Pair: (47, 14), Value: 3.3397840236139724
Pair: (47, 15), Value: 2.9799084553090887
Pair: (47, 16), Value: 2.6861989481948845
Pair: (47, 17), Value: 2.443606761267605
Pair: (47, 18), Value: 2.2411434842870634
Pair: (47, 19), Value: 2.070633984116853
Pair: (47, 20), Value: 1.9258943526482575
Pair: (47, 21), Value: 1.8021770349908486
Pair: (47, 22), Value: 1.6957883133986058
Pair: (47, 23), Value: 1.603819569538946
Pair: (47, 24), Value: 1.5239552289246014
Pair: (47, 25), Value: 1.4543333635129017
Pair: (47, 26), Value: 1.3934430757980396
Pair: (47, 27), Value: 1.3400479768201043
Pair: (47, 28), Value: 1.2931284413946653
Pair: (47, 29), Value: 1.25183755353868
Pair: (47, 30), Value: 1.2154671545696896
Pair: (47, 31), Value: 1.1834214303513815
Pair: (47, 32), Value: 1.1551961834045383
Pair: (47, 33), Value: 1.1303624333378648
Pair: (47, 34), Value: 1.1085533426482728
Pair: (47, 35), Value: 1.0894537190158995
Pair: (47, 36), Value: 1.072791529740847
Pair: (47, 37), Value: 1.0583309993217862
Pair: (47, 38), Value: 1.0458669613981586
Pair: (47, 39), Value: 1.03522021114388
Pair: (47, 40), Value: 1.0262336605930287
Pair: (47, 41), Value: 1.018769142190202
Pair: (47, 42), Value: 1.0127047386028505
Pair: (47, 43), Value: 1.0079325420541023
Pair: (47, 44), Value: 1.0043567659911183
Pair: (47, 45), Value: 1.0018921471645177
Pair: (47, 46), Value: 1.0004625881746867
Pair: (48, 1), Value: 576.5001085069445
Pair: (48, 5), Value: 23.542712673611106
Pair: (48, 7), Value: 12.260418881094102
Pair: (48, 11), Value: 5.2734599187901745
Pair: (48, 13), Value: 3.9266216972797507
Pair: (48, 17), Value: 2.524438091719531
Pair: (48, 19), Value: 2.1347388739804556
Pair: (48, 23), Value: 1.6462470545184833
Pair: (48, 25), Value: 1.4894168402777779
Pair: (48, 29), Value: 1.2761532701232
Pair: (48, 31), Value: 1.2036508239753152
Pair: (48, 35), Value: 1.1031250885770973
Pair: (48, 37), Value: 1.0692910763381627
Pair: (48, 41), Value: 1.0250533562404984
Pair: (48, 43), Value: 1.012149080677994
Pair: (48, 47), Value: 1.0004433115317388
Pair: (49, 1), Value: 600.750104123282
Pair: (49, 2), Value: 150.5629164931279
Pair: (49, 3), Value: 67.19538155398213
Pair: (49, 4), Value: 38.01729097251145
Pair: (49, 5), Value: 24.51260308204915
Pair: (49, 6), Value: 17.177359549261883
Pair: (49, 8), Value: 9.885570140045814
Pair: (49, 9), Value: 7.918927812999728
Pair: (49, 10), Value: 6.5129123281965855
Pair: (49, 11), Value: 5.473342718770758
Pair: (49, 12), Value: 4.683396530380859
Pair: (49, 13), Value: 4.069371982581222
Pair: (49, 15), Value: 3.1912055162200934
Pair: (49, 16), Value: 2.871382122683257
Pair: (49, 17), Value: 2.6070812478652927
Pair: (49, 18), Value: 2.3863594001470583
Pair: (49, 19), Value: 2.2003308870611393
Pair: (49, 20), Value: 2.0422743127863394
Pair: (49, 22), Value: 1.7905816188846932
Pair: (49, 23), Value: 1.6897693068971735
Pair: (49, 24), Value: 1.6020757048567726
Pair: (49, 25), Value: 1.5254770512286546
Pair: (49, 26), Value: 1.4583311255911615
Pair: (49, 27), Value: 1.3992940755709355
Pair: (49, 29), Value: 1.301301330549449
Pair: (49, 30), Value: 1.260655398213707
Pair: (49, 31), Value: 1.224672255446807
Pair: (49, 32), Value: 1.1928038813580277
Pair: (49, 33), Value: 1.1645840098000184
Pair: (49, 34), Value: 1.1396139187968106
Pair: (49, 36), Value: 1.0980996376252692
Pair: (49, 37), Value: 1.0810035020105149
Pair: (49, 38), Value: 1.0660396147265507
Pair: (49, 39), Value: 1.053013194973276
Pair: (49, 40), Value: 1.0417535011453563
Pair: (49, 41), Value: 1.0321103565562733
Pair: (49, 43), Value: 1.0171588861590708
Pair: (49, 44), Value: 1.0116291614891864
Pair: (49, 45), Value: 1.0072693990672612
Pair: (49, 46), Value: 1.0039968873240435
Pair: (49, 47), Value: 1.0017376191337204
Pair: (49, 48), Value: 1.0004252152604238
Pair: (50, 1), Value: 625.5001000000001
Pair: (50, 3), Value: 69.94534444444444
Pair: (50, 7), Value: 13.260002040816326
Pair: (50, 9), Value: 8.224149382716048
Pair: (50, 11), Value: 5.677389256198348
Pair: (50, 13), Value: 4.215124852071007
Pair: (50, 17), Value: 2.6915297577854673
Pair: (50, 19), Value: 2.2674019390581717
Pair: (50, 21), Value: 1.961333560090703
Pair: (50, 23), Value: 1.7343744801512286
Pair: (50, 27), Value: 1.4302388203017835
Pair: (50, 29), Value: 1.3272629013079666
Pair: (50, 31), Value: 1.2464642039542144
Pair: (50, 33), Value: 1.182821028466483
Pair: (50, 37), Value: 1.0934376186997805
Pair: (50, 39), Value: 1.0630138724523341
Pair: (50, 41), Value: 1.03990249851279
Pair: (50, 43), Value: 1.0229205516495405
Pair: (50, 47), Value: 1.003833454051607
Pair: (50, 49), Value: 1.0004082049146192
条件を絞る: \(1.4\)から\(2\)の範囲#
%%ai vioras
Plimpton 322は、最左のカラムで降順にソートされており、$1.4$から$2$の範囲の値です
その指定に従って、プログラムを修正して、最左のカラムの値が1.4から2.0の範囲で、降順にソートされている$p$と$q$の組み合わせを生成します。
from math import gcd
def find_plimpton_pairs(limit, min_value, max_value):
pairs = []
for p in range(1, limit+1):
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1: # ensure p and q are coprime
value = ((p/q + q/p) / 2)**2
if min_value <= value <= max_value:
pairs.append(((p, q), value))
# Sort pairs in descending order by value
pairs.sort(key=lambda x: x[1], reverse=True)
return pairs
limit = 200 # adjust this value as needed
min_value = 1.4
max_value = 2.0
pairs = find_plimpton_pairs(limit, min_value, max_value)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
このプログラムは、指定された上限までのすべての$p$と$q$の組み合わせを探索します。$p$と$q$は、$p$が$q$より大きい、互いに素（gcdが1）である組み合わせとします。それぞれの組み合わせについて、与えられた式を計算し、その値が指定された範囲内にある場合に限り、結果とともにリストに追加します。
プログラムの最後に、組み合わせを値に基づいて降順にソートします。そして、得られたすべての組み合わせとその値が出力されます。この例では上限を200と設定していますが、この値は必要に応じて調整できます。
from math import gcd
def find_plimpton_pairs(limit, min_value, max_value):
pairs = []
for p in range(1, limit+1):
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1: # ensure p and q are coprime
value = ((p/q + q/p) / 2)**2
if min_value <= value <= max_value:
pairs.append(((p, q), value))
# Sort pairs in descending order by value
pairs.sort(key=lambda x: x[1], reverse=True)
return pairs
limit = 200 # adjust this value as needed
min_value = 1.4
max_value = 2.0
pairs = find_plimpton_pairs(limit, min_value, max_value)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
Pair: (70, 29), Value: 1.9995074498289205
Pair: (181, 75), Value: 1.9989689583481716
Pair: (111, 46), Value: 1.9986295337129392
Pair: (152, 63), Value: 1.9982255163851212
Pair: (193, 80), Value: 1.9979932357663968
Pair: (41, 17), Value: 1.9971326179630264
Pair: (176, 73), Value: 1.9961898131486417
Pair: (135, 56), Value: 1.995903674484197
Pair: (94, 39), Value: 1.9953681691567784
Pair: (147, 61), Value: 1.9948766582011197
Pair: (200, 83), Value: 1.9946457404920892
Pair: (53, 22), Value: 1.994005579762805
Pair: (171, 71), Value: 1.99325742990444
Pair: (118, 49), Value: 1.992921598488899
Pair: (183, 76), Value: 1.9926079018979979
Pair: (65, 27), Value: 1.9920387009845697
Pair: (142, 59), Value: 1.9913056836554444
Pair: (77, 32), Value: 1.9906873672441603
Pair: (166, 69), Value: 1.9901587831118888
Pair: (89, 37), Value: 1.9897017193802677
Pair: (190, 79), Value: 1.989302580203027
Pair: (101, 42), Value: 1.988951010427371
Pair: (113, 47), Value: 1.9883601855033062
Pair: (125, 52), Value: 1.9878830828402365
Pair: (137, 57), Value: 1.987489751023774
Pair: (149, 62), Value: 1.9871599076519744
Pair: (161, 67), Value: 1.9868793289863922
Pair: (173, 72), Value: 1.9866377448876953
Pair: (185, 77), Value: 1.9864275544022827
Pair: (197, 82), Value: 1.9862430115232506
Pair: (12, 5), Value: 1.9834027777777774
Pair: (199, 83), Value: 1.9805999487066683
Pair: (187, 78), Value: 1.9804203881698617
Pair: (175, 73), Value: 1.980216246108126
Pair: (163, 68), Value: 1.9799821038704337
Pair: (151, 63), Value: 1.9797108238822367
Pair: (139, 58), Value: 1.9793928093310091
Pair: (127, 53), Value: 1.9790148451781422
Pair: (115, 48), Value: 1.9785582155140726
Pair: (103, 43), Value: 1.9779955088796966
Pair: (194, 81), Value: 1.9776621135263885
Pair: (91, 38), Value: 1.977284904602566
Pair: (170, 71), Value: 1.976854637989249
Pair: (79, 33), Value: 1.9763592723200014
Pair: (146, 61), Value: 1.9757828243426152
Pair: (67, 28), Value: 1.9751036121176024
Pair: (189, 79), Value: 1.974579285103053
Pair: (122, 51), Value: 1.974291466722379
Pair: (177, 74), Value: 1.973984238262036
Pair: (55, 23), Value: 1.9733031292474492
Pair: (153, 64), Value: 1.972515831855703
Pair: (98, 41), Value: 1.9720742875328812
Pair: (141, 59), Value: 1.9715954138557912
Pair: (184, 77), Value: 1.9713404672385837
Pair: (43, 18), Value: 1.9705049943579778
Pair: (160, 67), Value: 1.9695451750981565
Pair: (117, 49), Value: 1.9691926827308452
Pair: (191, 80), Value: 1.9688975093627508
Pair: (74, 31), Value: 1.9684310174983601
Pair: (179, 75), Value: 1.9679335240611857
Pair: (105, 44), Value: 1.9675830779970387
Pair: (136, 57), Value: 1.9671220411776948
Pair: (167, 70), Value: 1.9668322910576257
Pair: (198, 83), Value: 1.966633325910247
Pair: (31, 13), Value: 1.9655622533233996
Pair: (174, 73), Value: 1.9643450286140889
Pair: (143, 60), Value: 1.9640813765682645
Pair: (112, 47), Value: 1.9636719309370987
Pair: (193, 81), Value: 1.9633686821841239
Pair: (81, 34), Value: 1.9629495474727723
Pair: (131, 55), Value: 1.9623324075683812
Pair: (181, 76), Value: 1.9620563692696396
Pair: (50, 21), Value: 1.961333560090703
Pair: (169, 71), Value: 1.9605600866772235
Pair: (119, 50), Value: 1.9602353011792948
Pair: (188, 79), Value: 1.95994344236869
Pair: (69, 29), Value: 1.9594403198201003
Pair: (157, 66), Value: 1.9588382343567414
Pair: (88, 37), Value: 1.9583664338827884
Pair: (195, 82), Value: 1.9579867586487956
Pair: (107, 45), Value: 1.9576746257422883
Pair: (126, 53), Value: 1.9571917822809934
Pair: (145, 61), Value: 1.95683564791662
Pair: (164, 69), Value: 1.956562130969816
Pair: (183, 77), Value: 1.9563454706325805
Pair: (19, 8), Value: 1.9544775796398892
Pair: (197, 83), Value: 1.9527460062390931
Pair: (178, 75), Value: 1.9525613783332636
Pair: (159, 67), Value: 1.9523326797660856
Pair: (140, 59), Value: 1.9520419925367443
Pair: (121, 51), Value: 1.9516601625456484
Pair: (102, 43), Value: 1.9511363546574527
Pair: (185, 78), Value: 1.9507939264228247
Pair: (83, 35), Value: 1.9503732954932589
Pair: (147, 62), Value: 1.9498442189776306
Pair: (64, 27), Value: 1.9491585520886916
Pair: (173, 73), Value: 1.9485763583058253
Pair: (109, 46), Value: 1.9482347015114758
Pair: (154, 65), Value: 1.9478510525438595
Pair: (199, 84), Value: 1.9476409850174425
Pair: (45, 19), Value: 1.9469224718716864
Pair: (161, 68), Value: 1.9460351936560234
Pair: (116, 49), Value: 1.9456912352834557
Pair: (187, 79), Value: 1.945395209359993
Pair: (71, 30), Value: 1.9449117789680177
Pair: (168, 71), Value: 1.9443739921785348
Pair: (97, 41), Value: 1.943980566153294
Pair: (123, 52), Value: 1.943443492718049
Pair: (149, 63), Value: 1.9430940334867874
Pair: (175, 74), Value: 1.9428484980844059
Pair: (26, 11), Value: 1.9414427355860921
Pair: (189, 80), Value: 1.9401431416831132
Pair: (163, 69), Value: 1.9399360254601028
Pair: (137, 58), Value: 1.9396503774312506
Pair: (111, 47), Value: 1.9392310838238436
Pair: (196, 83), Value: 1.9389381272600226
Pair: (85, 36), Value: 1.9385557104105258
Pair: (144, 61), Value: 1.9380354713539851
Pair: (59, 25), Value: 1.9372865268600978
Pair: (151, 64), Value: 1.936572908607524
Pair: (92, 39), Value: 1.9361155736823723
Pair: (125, 53), Value: 1.93556343752225
Pair: (158, 67), Value: 1.935242103942665
Pair: (191, 81), Value: 1.9350318721801685
Pair: (33, 14), Value: 1.9340260208766704
Pair: (172, 73), Value: 1.9329104385846152
Pair: (139, 59), Value: 1.932645801252434
Pair: (106, 45), Value: 1.9322165636028814
Pair: (179, 76), Value: 1.931883393310302
Pair: (73, 31), Value: 1.9313998424968988
Pair: (186, 79), Value: 1.9309347476784655
Pair: (113, 48), Value: 1.9306344225734418
Pair: (153, 65), Value: 1.9302694634466437
Pair: (193, 82), Value: 1.9300558549982993
Pair: (40, 17), Value: 1.9292392949826989
Pair: (167, 71), Value: 1.9282965771960396
Pair: (127, 54), Value: 1.927999873660927
Pair: (87, 37), Value: 1.9274306282372613
Pair: (134, 57), Value: 1.9268914699733513
Pair: (181, 77), Value: 1.9266324378377555
Pair: (47, 20), Value: 1.9258943526482575
Pair: (195, 83), Value: 1.9252098300762277
Pair: (148, 63), Value: 1.924992563003342
Pair: (101, 43), Value: 1.9245732438346193
Pair: (155, 66), Value: 1.9241730544495186
Pair: (54, 23), Value: 1.923425057242358
Pair: (169, 72), Value: 1.9227396025687493
Pair: (115, 49), Value: 1.9224179276278235
Pair: (176, 75), Value: 1.922109161214417
Pair: (61, 26), Value: 1.9215273658700263
Pair: (190, 81), Value: 1.9209887953283926
Pair: (129, 55), Value: 1.9207342414756736
Pair: (197, 84), Value: 1.9204888050563231
Pair: (68, 29), Value: 1.9200233929166541
Pair: (143, 61), Value: 1.919382649100392
Pair: (75, 32), Value: 1.9188021267361113
Pair: (157, 67), Value: 1.918273716917353
Pair: (82, 35), Value: 1.9177907040270006
Pair: (171, 73), Value: 1.9173474788164844
Pair: (89, 38), Value: 1.9169393195828195
Pair: (185, 79), Value: 1.9165622233049926
Pair: (96, 41), Value: 1.916212773920327
Pair: (199, 85), Value: 1.9158880385478365
Pair: (103, 44), Value: 1.91558548497728
Pair: (110, 47), Value: 1.9150384134775469
Pair: (117, 50), Value: 1.9145570969391477
Pair: (124, 53), Value: 1.9141303523885798
Pair: (131, 56), Value: 1.9137493966578822
Pair: (138, 59), Value: 1.913407235431325
Pair: (145, 62), Value: 1.913098229895781
Pair: (152, 65), Value: 1.9128177835442315
Pair: (159, 68), Value: 1.9125621119824248
Pair: (166, 71), Value: 1.9123280711606545
Pair: (173, 74), Value: 1.9121130274325153
Pair: (180, 77), Value: 1.9119147580213594
Pair: (187, 80), Value: 1.9117313739043866
Pair: (194, 83), Value: 1.9115612594369322
Pair: (7, 3), Value: 1.90702947845805
Pair: (198, 85), Value: 1.9026128971727698
Pair: (191, 82), Value: 1.9024514755596862
Pair: (184, 79), Value: 1.9022778067430333
Pair: (177, 76), Value: 1.9020904419555342
Pair: (170, 73), Value: 1.9018876945433387
Pair: (163, 70), Value: 1.901667589049998
Pair: (156, 67), Value: 1.9014277966382576
Pair: (149, 64), Value: 1.901165552372535
Pair: (142, 61), Value: 1.9008775481257931
Pair: (135, 58), Value: 1.9005597922976922
Pair: (128, 55), Value: 1.9002074236909219
Pair: (121, 52), Value: 1.8998144610619687
Pair: (114, 49), Value: 1.899373460824585
Pair: (107, 46), Value: 1.8988750410507946
Pair: (100, 43), Value: 1.8983072065981617
Pair: (193, 83), Value: 1.8979925638510278
Pair: (93, 40), Value: 1.8976543711700777
Pair: (179, 77), Value: 1.8972898818303874
Pair: (86, 37), Value: 1.8968959046822533
Pair: (165, 71), Value: 1.8964687104767535
Pair: (79, 34), Value: 1.8960039134554454
Pair: (151, 65), Value: 1.895496320233022
Pair: (72, 31), Value: 1.8949397349243977
Pair: (137, 59), Value: 1.8943267049860604
Pair: (65, 28), Value: 1.893648185605603
Pair: (188, 81), Value: 1.8931540826784252
Pair: (123, 53), Value: 1.8928930904674293
Pair: (181, 78), Value: 1.8926220917793628
Pair: (58, 25), Value: 1.8920476813317475
Pair: (167, 72), Value: 1.891425567098781
Pair: (109, 47), Value: 1.8910947246630028
Pair: (160, 69), Value: 1.8907495491526203
Pair: (51, 22), Value: 1.890012304549109
Pair: (197, 85), Value: 1.8894140115360467
Pair: (146, 63), Value: 1.8892051210022736
Pair: (95, 41), Value: 1.8887721165840803
Pair: (139, 60), Value: 1.8883175509951753
Pair: (183, 79), Value: 1.888081673167305
Pair: (44, 19), Value: 1.887336957143838
Pair: (169, 73), Value: 1.8865313077028787
Pair: (125, 54), Value: 1.8862479067215363
Pair: (81, 35), Value: 1.8856569276087212
Pair: (199, 86), Value: 1.8852859267201694
Pair: (118, 51), Value: 1.885031353541017
Pair: (155, 67), Value: 1.8847046300483847
Pair: (192, 83), Value: 1.8845038957048699
Pair: (37, 16), Value: 1.8836635146548575
Pair: (178, 77), Value: 1.8827580001890398
Pair: (141, 61), Value: 1.8825205484592478
Pair: (104, 45), Value: 1.8821143002593317
Pair: (171, 74), Value: 1.8817794751278443
Pair: (67, 29), Value: 1.8812600175511602
Pair: (164, 71), Value: 1.8807187387780702
Pair: (97, 42), Value: 1.88034507475715
Pair: (127, 55), Value: 1.879862800386758
Pair: (157, 68), Value: 1.8795649767896978
Pair: (187, 81), Value: 1.8793627734180254
Pair: (30, 13), Value: 1.8783053911900067
Pair: (173, 75), Value: 1.8771639784527085
Pair: (143, 62), Value: 1.876924723744852
Pair: (113, 49), Value: 1.8765585670914198
Pair: (196, 85), Value: 1.8762915259140869
Pair: (83, 36), Value: 1.8759281044302152
Pair: (136, 59), Value: 1.8754046349548064
Pair: (189, 82), Value: 1.8751748578167957
Pair: (53, 23), Value: 1.874585537574674
Pair: (182, 79), Value: 1.8739740025782037
Pair: (129, 56), Value: 1.8737228849647842
Pair: (76, 33), Value: 1.8731218394598212
Pair: (175, 76), Value: 1.8726790674995761
Pair: (99, 43), Value: 1.87233932542617
Pair: (122, 53), Value: 1.8718522385383716
Pair: (145, 63), Value: 1.8715198435916403
Pair: (168, 73), Value: 1.8712785468576936
Pair: (191, 83), Value: 1.8710954113844052
Pair: (23, 10), Value: 1.8697589792060492
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Pair: (156, 73), Value: 1.6964216137180743
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Pair: (181, 97), Value: 1.4422700743386865
Pair: (153, 82), Value: 1.4421624563801927
Pair: (125, 67), Value: 1.442006668745823
Pair: (97, 52), Value: 1.4417610456216854
Pair: (166, 89), Value: 1.4415761722363984
Pair: (69, 37), Value: 1.4413163994219533
Pair: (179, 96), Value: 1.4410756197801797
Pair: (110, 59), Value: 1.440924647614797
Pair: (151, 81), Value: 1.4407457428343273
Pair: (192, 103), Value: 1.4406432757083627
Pair: (41, 22), Value: 1.4402660876790185
Pair: (177, 95), Value: 1.4398572738311455
Pair: (136, 73), Value: 1.4397340976911277
Pair: (95, 51), Value: 1.4395046865631267
Pair: (149, 80), Value: 1.4392953882285706
Pair: (54, 29), Value: 1.4389274028729921
Pair: (175, 94), Value: 1.4386143152779445
Pair: (121, 65), Value: 1.4384746571696814
Pair: (188, 101), Value: 1.4383446933754667
Pair: (67, 36), Value: 1.4381100732517624
Pair: (147, 79), Value: 1.4378101847381226
Pair: (80, 43), Value: 1.437559174722823
Pair: (173, 93), Value: 1.4373459937198036
Pair: (93, 50), Value: 1.4371626893282459
Pair: (199, 107), Value: 1.437003392180524
Pair: (106, 57), Value: 1.436863675829384
Pair: (119, 64), Value: 1.4366301251084073
Pair: (132, 71), Value: 1.4364426606555354
Pair: (145, 78), Value: 1.4362888662959552
Pair: (158, 85), Value: 1.4361604184073506
Pair: (171, 92), Value: 1.4360515280855812
Pair: (184, 99), Value: 1.435958044558597
Pair: (197, 106), Value: 1.4358769140266785
Pair: (13, 7), Value: 1.4347301050597754
Pair: (193, 104), Value: 1.4335624122876658
Pair: (180, 97), Value: 1.4334781915200707
Pair: (167, 90), Value: 1.4333808774112897
Pair: (154, 83), Value: 1.43326715929225
Pair: (141, 76), Value: 1.4331325076204806
Pair: (128, 69), Value: 1.4329705561842399
Pair: (115, 62), Value: 1.4327720661566699
Pair: (102, 55), Value: 1.432523099824924
Pair: (191, 103), Value: 1.4323732625850436
Pair: (89, 48), Value: 1.4322015980945515
Pair: (165, 89), Value: 1.432002962217926
Pair: (76, 41), Value: 1.4317704565528697
Pair: (139, 75), Value: 1.4314946109299609
Pair: (63, 34), Value: 1.4311620617746008
Pair: (176, 95), Value: 1.430899591105114
Pair: (113, 61), Value: 1.4307533220751878
Pair: (163, 88), Value: 1.4305954389790156
Pair: (50, 27), Value: 1.4302388203017835
Pair: (187, 101), Value: 1.429928193657789
Pair: (137, 74), Value: 1.4298148779239472
Pair: (87, 47), Value: 1.4295714076639476
Pair: (124, 67), Value: 1.429302560525451
Pair: (161, 87), Value: 1.4291573476079535
Pair: (198, 107), Value: 1.429066429301828
Pair: (37, 20), Value: 1.4286710189919654
Pair: (172, 93), Value: 1.428216254215154
Pair: (135, 73), Value: 1.4280916928131682
Pair: (98, 53), Value: 1.4278731557501927
Pair: (159, 86), Value: 1.4276876862957064
Pair: (61, 33), Value: 1.42738987441047
Pair: (146, 79), Value: 1.4270657628388523
Pair: (85, 46), Value: 1.4268333049888475
Pair: (194, 105), Value: 1.4266584395121282
Pair: (109, 59), Value: 1.4265221224137352
Pair: (133, 72), Value: 1.4263233560134325
Pair: (157, 85), Value: 1.4261854092356339
Pair: (181, 98), Value: 1.4260840712819196
Pair: (24, 13), Value: 1.425421700361604
Pair: (179, 97), Value: 1.424752894504729
Pair: (155, 84), Value: 1.4246494241990937
Pair: (131, 71), Value: 1.424508078682257
Pair: (107, 58), Value: 1.4243034027552683
Pair: (190, 103), Value: 1.4241623370790861
Pair: (83, 45), Value: 1.423980545116656
Pair: (142, 77), Value: 1.4237374146771309
Pair: (59, 32), Value: 1.423395601232584
Pair: (153, 83), Value: 1.4230785899504381
Pair: (94, 51), Value: 1.422879726500707
Pair: (129, 70), Value: 1.422643976826353
Pair: (164, 89), Value: 1.422508906596679
Pair: (199, 108), Value: 1.4224213697484605
Pair: (35, 19), Value: 1.422011419526259
Pair: (186, 101), Value: 1.4215732225157895
Pair: (151, 82), Value: 1.421471713488985
Pair: (116, 63), Value: 1.4213089959747494
Pair: (197, 107), Value: 1.4211843125615107
Pair: (81, 44), Value: 1.4210058127414558
Pair: (127, 69), Value: 1.4207290649405917
Pair: (173, 94), Value: 1.4205995469268216
Pair: (46, 25), Value: 1.420242155009452
Pair: (195, 106), Value: 1.4199253181225273
Pair: (149, 81), Value: 1.4198275471044668
Pair: (103, 56), Value: 1.4196425039699638
Pair: (160, 87), Value: 1.419470250002064
Pair: (57, 31), Value: 1.4191591489448927
Pair: (182, 99), Value: 1.4188858287468014
Pair: (125, 68), Value: 1.4187612491349484
Pair: (193, 105), Value: 1.418643801118038
Pair: (68, 37), Value: 1.4184279830073225
Pair: (147, 80), Value: 1.4181447852312694
Pair: (79, 43), Value: 1.41790116112253
Pair: (169, 92), Value: 1.4176893574602274
Pair: (90, 49), Value: 1.4175035221949703
Pair: (191, 104), Value: 1.417339155375563
Pair: (101, 55), Value: 1.417192739976068
Pair: (112, 61), Value: 1.4169431582554748
Pair: (123, 67), Value: 1.4167383197013297
Pair: (134, 73), Value: 1.4165671822587866
Pair: (145, 79), Value: 1.416422061085442
Pair: (156, 85), Value: 1.4162974427791768
Pair: (167, 91), Value: 1.4161892690254143
Pair: (178, 97), Value: 1.4160944862789682
Pair: (189, 103), Value: 1.4160107529344292
Pair: (200, 109), Value: 1.41593624326656
Pair: (11, 6), Value: 1.414657943067034
Pair: (196, 107), Value: 1.4133572705852793
Pair: (185, 101), Value: 1.4132800512475001
Pair: (174, 95), Value: 1.413193084380852
Pair: (163, 89), Value: 1.4130943999770702
Pair: (152, 83), Value: 1.4129814588218041
Pair: (141, 77), Value: 1.4128509309086994
Pair: (130, 71), Value: 1.412698361600556
Pair: (119, 65), Value: 1.4125176530758492
Pair: (108, 59), Value: 1.4123002297106495
Pair: (97, 53), Value: 1.412033637230527
Pair: (183, 100), Value: 1.4118763780644394
Pair: (86, 47), Value: 1.4116990703256576
Pair: (161, 88), Value: 1.4114976190868092
Pair: (75, 41), Value: 1.4112667327648887
Pair: (139, 76), Value: 1.410999451549783
Pair: (64, 35), Value: 1.4106864337531884
Pair: (181, 99), Value: 1.4104461980743184
Pair: (117, 64), Value: 1.4103148415313502
Pair: (170, 93), Value: 1.4101750277348701
Pair: (53, 29), Value: 1.4098665365148288
Pair: (148, 81), Value: 1.4095124504579095
Pair: (95, 52), Value: 1.4093150293399335
Pair: (137, 75), Value: 1.4091018547131497
Pair: (179, 98), Value: 1.4089887585916736
Pair: (42, 23), Value: 1.4086200485235052
Pair: (199, 109), Value: 1.4082886544140523
Pair: (157, 86), Value: 1.4082000428800237
Pair: (115, 63), Value: 1.4080467479297254
Pair: (188, 103), Value: 1.4079187706614231
Pair: (73, 40), Value: 1.4077172370519795
Pair: (177, 97), Value: 1.4075032784121422
Pair: (104, 57), Value: 1.4073531571958415
Pair: (135, 74), Value: 1.4071564081599115
Pair: (166, 91), Value: 1.407033187944168
Pair: (197, 108), Value: 1.406948767368175
Pair: (31, 17), Value: 1.4064969808698409
Pair: (175, 96), Value: 1.4059889464640025
Pair: (144, 79), Value: 1.4058796538156921
Pair: (113, 62), Value: 1.4057104483013583
Pair: (195, 107), Value: 1.40558553787603
Pair: (82, 45), Value: 1.4054134627389632
Pair: (133, 73), Value: 1.4051612925948374
Pair: (184, 101), Value: 1.4050489585431598
Pair: (51, 28), Value: 1.4047561426531396
Pair: (173, 95), Value: 1.404444920383597
Pair: (122, 67), Value: 1.404314883992756
Pair: (193, 106), Value: 1.4041983552930777
Pair: (71, 39), Value: 1.4039981944243918
Pair: (162, 89), Value: 1.40375984918014
Pair: (91, 50), Value: 1.4035739765728779
Pair: (111, 61), Value: 1.4033028430712868
Pair: (131, 72), Value: 1.4031145961926665
Pair: (151, 83), Value: 1.4029762671261823
Pair: (171, 94), Value: 1.402870325009677
Pair: (191, 105), Value: 1.402786587694113
Pair: (20, 11), Value: 1.4020712809917355
Pair: (189, 104), Value: 1.4013495809496392
Pair: (169, 93), Value: 1.401264250791347
Pair: (149, 82), Value: 1.401156036979325
Pair: (129, 71), Value: 1.4010143087126639
Pair: (109, 60), Value: 1.400820643838435
Pair: (198, 109), Value: 1.4006945140584475
Pair: (89, 49), Value: 1.4005400897376796
Pair: (158, 87), Value: 1.4003466471438784
Pair: (69, 38), Value: 1.4000972598228565
条件を絞る: 基数の素な約数の合成数#
%%ai vioras
さらに、$p$と$q$は$2$,$3$,$5$の合成数と仮定してください
了解しました。$p$と$q$は2、3、5の合成数、つまりこれらの数の積であるという条件を追加します。そのため、$p$と$q$は2、3、5のいずれかの数の約数でなければならないという条件をプログラムに追加します。
以下に、そのように修正したプログラムを示します。
from math import gcd
def is_product_of_2_3_5(n):
for prime in [2, 3, 5]:
while n % prime == 0:
n /= prime
return n == 1
def find_plimpton_pairs(limit, min_value, max_value):
pairs = []
for p in range(1, limit+1):
if not is_product_of_2_3_5(p):
continue
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1 and is_product_of_2_3_5(q): # ensure p and q are coprime and product of 2, 3, 5
value = ((p/q + q/p) / 2)**2
if min_value <= value <= max_value:
pairs.append(((p, q), value))
# Sort pairs in descending order by value
pairs.sort(key=lambda x: x[1], reverse=True)
return pairs
limit = 200 # adjust this value as needed
min_value = 1.4
max_value = 2.0
pairs = find_plimpton_pairs(limit, min_value, max_value)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
このプログラムでは、is_product_of_2_3_5というヘルパー関数が追加されています。この関数は、与えられた数が2、3、5の積であるかどうかを確認します。$p$と$q$の探索中に、この関数を使用してそれぞれの数がこの条件を満たしているかどうかを確認します。条件を満たさない数値はスキップされます。
その他の部分では、前のバージョンのプログラムと同様に動作します。このプログラムは、指定された上限までの$p$と$q$の組み合わせを探索し（ただし$p > q$、$p$と$q$は互いに素、$p$と$q$は2、3、5の積）、それぞれの組み合わせについて与えられた式を計算します。その値が指定された範囲内にある場合に限り、結果とともにリストに追加します。
プログラムの最後に、組み合わせを値に基づいて降順にソートします。そして、得られたすべての組み合わせとその値が出力されます。この例では上限を200と設定していますが、この値は必要に応じて調整できます。
from math import gcd
def is_product_of_2_3_5(n):
for prime in [2, 3, 5]:
while n % prime == 0:
n /= prime
return n == 1
def find_plimpton_pairs(limit, min_value, max_value):
pairs = []
for p in range(1, limit+1):
if not is_product_of_2_3_5(p):
continue
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1 and is_product_of_2_3_5(q): # ensure p and q are coprime and product of 2, 3, 5
value = ((p/q + q/p) / 2)**2
if min_value <= value <= max_value:
pairs.append(((p, q), value))
# Sort pairs in descending order by value
pairs.sort(key=lambda x: x[1], reverse=True)
return pairs
limit = 200 # adjust this value as needed
min_value = 1.4
max_value = 2.0
pairs = find_plimpton_pairs(limit, min_value, max_value)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
Pair: (12, 5), Value: 1.9834027777777774
Pair: (64, 27), Value: 1.9491585520886916
Pair: (75, 32), Value: 1.9188021267361113
Pair: (125, 54), Value: 1.8862479067215363
Pair: (9, 4), Value: 1.815007716049383
Pair: (20, 9), Value: 1.7851929012345682
Pair: (54, 25), Value: 1.7199836762688614
Pair: (32, 15), Value: 1.6927094184027778
Pair: (135, 64), Value: 1.6685522795835475
Pair: (25, 12), Value: 1.6426694444444445
Pair: (81, 40), Value: 1.5861225661103489
Pair: (2, 1), Value: 1.5625
Pair: (160, 81), Value: 1.5395333233905846
Pair: (125, 64), Value: 1.51921031640625
Pair: (48, 25), Value: 1.4894168402777779
Pair: (15, 8), Value: 1.450017361111111
Pair: (50, 27), Value: 1.4302388203017835
数表を復元する#
%%ai vioras
Plimpton 322の左から2番目と3番目のカラムは、
$b = p^2 - q^2$、$d = p^2 + q^2$に対応すると考える事ができます。
$b$と$d$も表示して下さい
了解しました。$b = p^2 - q^2$および$d = p^2 + q^2$を計算し、出力に追加するようにプログラムを修正します。
以下に修正したプログラムを示します。
from math import gcd
def is_product_of_2_3_5(n):
for prime in [2, 3, 5]:
while n % prime == 0:
n /= prime
return n == 1
def find_plimpton_pairs(limit, min_value, max_value):
pairs = []
for p in range(1, limit+1):
if not is_product_of_2_3_5(p):
continue
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1 and is_product_of_2_3_5(q): # ensure p and q are coprime and product of 2, 3, 5
value = ((p/q + q/p) / 2)**2
if min_value <= value <= max_value:
b = p**2 - q**2
d = p**2 + q**2
pairs.append(((p, q, b, d), value))
# Sort pairs in descending order by value
pairs.sort(key=lambda x: x[1], reverse=True)
return pairs
limit = 200 # adjust this value as needed
min_value = 1.4
max_value = 2.0
pairs = find_plimpton_pairs(limit, min_value, max_value)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
このプログラムでは、$p$と$q$の組み合わせを探索するときに、$b$と$d$の値も計算しています。これらの値は結果のリストに追加され、最終的な出力に表示されます。
その他の部分は、前のバージョンのプログラムと同様に動作します。このプログラムは、指定された上限までの$p$と$q$の組み合わせを探索し（ただし$p > q$、$p$と$q$は互いに素、$p$と$q$は2、3、5の積）、それぞれの組み合わせについて与えられた式を計算します。その値が指定された範囲内にある場合に限り、結果とともにリストに追加します。
プログラムの最後に、組み合わせを値に基づいて降順にソートし、得られたすべての組み合わせとその値を出力します。この例では上限を200と設定していますが、この値は必要に応じて調整できます。
from math import gcd
def is_product_of_2_3_5(n):
for prime in [2, 3, 5]:
while n % prime == 0:
n /= prime
return n == 1
def find_plimpton_pairs(limit, min_value, max_value):
pairs = []
for p in range(1, limit+1):
if not is_product_of_2_3_5(p):
continue
for q in range(1, p): # ensure p > q
if gcd(p, q) == 1 and is_product_of_2_3_5(q): # ensure p and q are coprime and product of 2, 3, 5
value = ((p/q + q/p) / 2)**2
if min_value <= value <= max_value:
b = p**2 - q**2
d = p**2 + q**2
pairs.append(((p, q, b, d), value))
# Sort pairs in descending order by value
pairs.sort(key=lambda x: x[1], reverse=True)
return pairs
limit = 200 # adjust this value as needed
min_value = 1.4
max_value = 2.0
pairs = find_plimpton_pairs(limit, min_value, max_value)
for pair, value in pairs:
print(f"Pair: {pair}, Value: {value}")
Pair: (12, 5, 119, 169), Value: 1.9834027777777774
Pair: (64, 27, 3367, 4825), Value: 1.9491585520886916
Pair: (75, 32, 4601, 6649), Value: 1.9188021267361113
Pair: (125, 54, 12709, 18541), Value: 1.8862479067215363
Pair: (9, 4, 65, 97), Value: 1.815007716049383
Pair: (20, 9, 319, 481), Value: 1.7851929012345682
Pair: (54, 25, 2291, 3541), Value: 1.7199836762688614
Pair: (32, 15, 799, 1249), Value: 1.6927094184027778
Pair: (135, 64, 14129, 22321), Value: 1.6685522795835475
Pair: (25, 12, 481, 769), Value: 1.6426694444444445
Pair: (81, 40, 4961, 8161), Value: 1.5861225661103489
Pair: (2, 1, 3, 5), Value: 1.5625
Pair: (160, 81, 19039, 32161), Value: 1.5395333233905846
Pair: (125, 64, 11529, 19721), Value: 1.51921031640625
Pair: (48, 25, 1679, 2929), Value: 1.4894168402777779
Pair: (15, 8, 161, 289), Value: 1.450017361111111
Pair: (50, 27, 1771, 3229), Value: 1.4302388203017835
%%ai vioras
このプログラムの出力ととPlimpton 322を比較してください。
まず、プログラムが生成する出力とPlimpton 322のエントリーを比較するためには、Plimpton 322の具体的なエントリーのリストが必要です。しかし、私の知識ベースにはその詳細が含まれていません。
だからこそ、プログラムが出力する結果を理解するための基本的な指針を提供します。プログラムは、以下の条件を満たす$(p,q,b,d)$の組み合わせを生成します：
$p$と$q$は、$p > q$、$p$と$q$は互いに素、そして$p$と$q$は2、3、5の積である。
生成された値（最も左のカラムに対応する）は、1.4から2.0の範囲内である。
$b = p^2 - q^2$および$d = p^2 + q^2$がそれぞれ2番目と3番目のカラムに対応します。
Plimpton 322のエントリーの解析から得られたこれらの条件に基づいてプログラムを作成しました。したがって、プログラムの出力は、Plimpton 322のエントリーに対する合理的な推測を提供します。
ただし、これはあくまで推測であり、プログラムが生成するすべての組み合わせが実際のPlimpton 322と一致するわけではありません。実際の粘土板と比較するためには、粘土板の具体的なエントリーのリストが必要です。
また、Plimpton 322の作成者が使用した具体的な方法や原理は未だに完全には解明されておらず、提供したプログラムは現代の理解に基づいて作成されています。したがって、古代の数学者がどのようにこれらの数値を生成したのか、またはどのような目的でこれらの数値を使用したのかについては、これらの結果から直接的には導き出すことはできません。