Ramanujan の没後百周年の命日
Ken Ono & Sarah Trebat-Leder の
The 1729 K3 surface
に載っている恒等式に具体的数値を入れてみた。
出力結果
20^3 + (-17)^3 = 3087
14^3 + 7^3 = 3087
46^3 + (-37)^3 = 46683
30^3 + 27^3 = 46683
84^3 + (-63)^3 = 342657
54^3 + 57^3 = 342657
134^3 + (-95)^3 = 1548729
86^3 + 97^3 = 1548729
196^3 + (-133)^3 = 5176899
126^3 + 147^3 = 5176899
270^3 + (-177)^3 = 14137767
174^3 + 207^3 = 14137767
356^3 + (-227)^3 = 33420933
230^3 + 277^3 = 33420933
454^3 + (-283)^3 = 70911477
294^3 + 357^3 = 70911477
564^3 + (-345)^3 = 138342519
366^3 + 447^3 = 138342519
686^3 + (-413)^3 = 252383859
446^3 + 547^3 = 252383859
820^3 + (-487)^3 = 435866697
534^3 + 657^3 = 435866697
966^3 + (-567)^3 = 719144433
630^3 + 777^3 = 719144433
1124^3 + (-653)^3 = 1141589547
734^3 + 907^3 = 1141589547
1294^3 + (-745)^3 = 1753226559
846^3 + 1047^3 = 1753226559
1476^3 + (-843)^3 = 2616501069
966^3 + 1197^3 = 2616501069
1670^3 + (-947)^3 = 3808184877
1094^3 + 1357^3 = 3808184877
1876^3 + (-1057)^3 = 5421417183
1230^3 + 1527^3 = 5421417183
2094^3 + (-1173)^3 = 7567881867
1374^3 + 1707^3 = 7567881867
2324^3 + (-1295)^3 = 10380120849
1526^3 + 1897^3 = 10380120849
2566^3 + (-1423)^3 = 14013983529
1686^3 + 2097^3 = 14013983529
2820^3 + (-1557)^3 = 18651212307
1854^3 + 2307^3 = 18651212307
3086^3 + (-1697)^3 = 24502164183
2030^3 + 2527^3 = 24502164183
3364^3 + (-1843)^3 = 31808668437
2214^3 + 2757^3 = 31808668437
3654^3 + (-1995)^3 = 40847020389
2406^3 + 2997^3 = 40847020389
3956^3 + (-2153)^3 = 51931111239
2606^3 + 3247^3 = 51931111239
4270^3 + (-2317)^3 = 65415693987
2814^3 + 3507^3 = 65415693987
4596^3 + (-2487)^3 = 81699785433
3030^3 + 3777^3 = 81699785433
4934^3 + (-2663)^3 = 101230204257
3254^3 + 4057^3 = 101230204257
5284^3 + (-2845)^3 = 124505245179
3486^3 + 4347^3 = 124505245179
5646^3 + (-3033)^3 = 152078489199
3726^3 + 4647^3 = 152078489199
6020^3 + (-3227)^3 = 184562749917
3974^3 + 4957^3 = 184562749917
6406^3 + (-3427)^3 = 222634155933
4230^3 + 5277^3 = 222634155933
6804^3 + (-3633)^3 = 267036369327
4494^3 + 5607^3 = 267036369327
7214^3 + (-3845)^3 = 318584940219
4766^3 + 5947^3 = 318584940219
7636^3 + (-4063)^3 = 378171797409
5046^3 + 6297^3 = 378171797409
8070^3 + (-4287)^3 = 446769875097
5334^3 + 6657^3 = 446769875097
8516^3 + (-4517)^3 = 525437875683
5630^3 + 7027^3 = 525437875683
8974^3 + (-4753)^3 = 615325168647
5934^3 + 7407^3 = 615325168647
9444^3 + (-4995)^3 = 717676825509
6246^3 + 7797^3 = 717676825509
9926^3 + (-5243)^3 = 833838790869
6566^3 + 8197^3 = 833838790869
10420^3 + (-5497)^3 = 965263189527
6894^3 + 8607^3 = 965263189527
10926^3 + (-5757)^3 = 1113513769683
7230^3 + 9027^3 = 1113513769683
11444^3 + (-6023)^3 = 1280271482217
7574^3 + 9457^3 = 1280271482217
11974^3 + (-6295)^3 = 1467340196049
7926^3 + 9897^3 = 1467340196049
12516^3 + (-6573)^3 = 1676652549579
8286^3 + 10347^3 = 1676652549579
13070^3 + (-6857)^3 = 1910275938207
8654^3 + 10807^3 = 1910275938207
13636^3 + (-7147)^3 = 2170418637933
9030^3 + 11277^3 = 2170418637933
14214^3 + (-7443)^3 = 2459436065037
9414^3 + 11757^3 = 2459436065037
14804^3 + (-7745)^3 = 2779837171839
9806^3 + 12247^3 = 2779837171839
15406^3 + (-8053)^3 = 3134290978539
10206^3 + 12747^3 = 3134290978539
16020^3 + (-8367)^3 = 3525633241137
10614^3 + 13257^3 = 3525633241137
16646^3 + (-8687)^3 = 3956873255433
11030^3 + 13777^3 = 3956873255433
17284^3 + (-9013)^3 = 4431200797107
11454^3 + 14307^3 = 4431200797107
17934^3 + (-9345)^3 = 4951993197879
11886^3 + 14847^3 = 4951993197879
18596^3 + (-9683)^3 = 5522822557749
12326^3 + 15397^3 = 5522822557749
19270^3 + (-10027)^3 = 6147463093317
12774^3 + 15957^3 = 6147463093317
19956^3 + (-10377)^3 = 6829898622183
13230^3 + 16527^3 = 6829898622183
20654^3 + (-10733)^3 = 7574330183427
13694^3 + 17107^3 = 7574330183427
21364^3 + (-11095)^3 = 8385183794169
14166^3 + 17697^3 = 8385183794169
22086^3 + (-11463)^3 = 9267118342209
14646^3 + 18297^3 = 9267118342209
22820^3 + (-11837)^3 = 10225033614747
15134^3 + 18907^3 = 10225033614747
23566^3 + (-12217)^3 = 11264078463183
15630^3 + 19527^3 = 11264078463183
24324^3 + (-12603)^3 = 12389659103997
16134^3 + 20157^3 = 12389659103997
25094^3 + (-12995)^3 = 13607447555709
16646^3 + 20797^3 = 13607447555709
25876^3 + (-13393)^3 = 14923390211919
17166^3 + 21447^3 = 14923390211919
26670^3 + (-13797)^3 = 16343716550427
17694^3 + 22107^3 = 16343716550427
27476^3 + (-14207)^3 = 17874947978433
18230^3 + 22777^3 = 17874947978433
28294^3 + (-14623)^3 = 19523906813817
18774^3 + 23457^3 = 19523906813817
29124^3 + (-15045)^3 = 21297725402499
19326^3 + 24147^3 = 21297725402499
29966^3 + (-15473)^3 = 23203855371879
19886^3 + 24847^3 = 23203855371879
30820^3 + (-15907)^3 = 25250077020357
20454^3 + 25557^3 = 25250077020357
31686^3 + (-16347)^3 = 27444508842933
21030^3 + 26277^3 = 27444508842933
32564^3 + (-16793)^3 = 29795617192887
21614^3 + 27007^3 = 29795617192887
33454^3 + (-17245)^3 = 32312226079539
22206^3 + 27747^3 = 32312226079539
34356^3 + (-17703)^3 = 35003527102089
22806^3 + 28497^3 = 35003527102089
35270^3 + (-18167)^3 = 37879089519537
23414^3 + 29257^3 = 37879089519537
36196^3 + (-18637)^3 = 40948870456683
24030^3 + 30027^3 = 40948870456683
37134^3 + (-19113)^3 = 44223225246207
24654^3 + 30807^3 = 44223225246207
38084^3 + (-19595)^3 = 47712917906829
25286^3 + 31597^3 = 47712917906829
39046^3 + (-20083)^3 = 51429131757549
25926^3 + 32397^3 = 51429131757549
40020^3 + (-20577)^3 = 55383480167967
26574^3 + 33207^3 = 55383480167967
41006^3 + (-21077)^3 = 59588017444683
27230^3 + 34027^3 = 59588017444683
42004^3 + (-21583)^3 = 64055249853777
27894^3 + 34857^3 = 64055249853777
43014^3 + (-22095)^3 = 68798146779369
28566^3 + 35697^3 = 68798146779369
44036^3 + (-22613)^3 = 73830152018259
29246^3 + 36547^3 = 73830152018259
45070^3 + (-23137)^3 = 79165195210647
29934^3 + 37407^3 = 79165195210647
46116^3 + (-23667)^3 = 84817703406933
30630^3 + 38277^3 = 84817703406933
47174^3 + (-24203)^3 = 90802612770597
31334^3 + 39157^3 = 90802612770597
48244^3 + (-24745)^3 = 97135380417159
32046^3 + 40047^3 = 97135380417159
49326^3 + (-25293)^3 = 103831996389219
32766^3 + 40947^3 = 103831996389219
50420^3 + (-25847)^3 = 110908995767577
33494^3 + 41857^3 = 110908995767577
51526^3 + (-26407)^3 = 118383470918433
34230^3 + 42777^3 = 118383470918433
52644^3 + (-26973)^3 = 126273083876667
34974^3 + 43707^3 = 126273083876667
53774^3 + (-27545)^3 = 134596078865199
35726^3 + 44647^3 = 134596078865199
54916^3 + (-28123)^3 = 143371294950429
36486^3 + 45597^3 = 143371294950429
56070^3 + (-28707)^3 = 152618178833757
37254^3 + 46557^3 = 152618178833757
57236^3 + (-29297)^3 = 162356797779183
38030^3 + 47527^3 = 162356797779183
58414^3 + (-29893)^3 = 172607852676987
38814^3 + 48507^3 = 172607852676987
59604^3 + (-30495)^3 = 183392691243489
39606^3 + 49497^3 = 183392691243489
[189, 3087, 46683, 342657, 1548729, 5176899, 14137767, 33420933, 70911477, 138342519, 252383859, 435866697, 719144433, 1141589547, 1753226559, 2616501069, 3808184877, 5421417183, 7567881867, 10380120849, 14013983529, 18651212307, 24502164183, 31808668437, 40847020389, 51931111239, 65415693987, 81699785433, 101230204257, 124505245179, 152078489199, 184562749917, 222634155933, 267036369327, 318584940219, 378171797409, 446769875097, 525437875683, 615325168647, 717676825509, 833838790869, 965263189527, 1113513769683, 1280271482217, 1467340196049, 1676652549579, 1910275938207, 2170418637933, 2459436065037, 2779837171839, 3134290978539, 3525633241137, 3956873255433, 4431200797107, 4951993197879, 5522822557749, 6147463093317, 6829898622183, 7574330183427, 8385183794169, 9267118342209, 10225033614747, 11264078463183, 12389659103997, 13607447555709, 14923390211919, 16343716550427, 17874947978433, 19523906813817, 21297725402499, 23203855371879, 25250077020357, 27444508842933, 29795617192887, 32312226079539, 35003527102089, 37879089519537, 40948870456683, 44223225246207, 47712917906829, 51429131757549, 55383480167967, 59588017444683, 64055249853777, 68798146779369, 73830152018259, 79165195210647, 84817703406933, 90802612770597, 97135380417159, 103831996389219, 110908995767577, 118383470918433, 126273083876667, 134596078865199, 143371294950429, 152618178833757, 162356797779183, 172607852676987, 183392691243489]
に載っている恒等式に具体的数値を入れてみた。
# n=1のとき(x,y)=(z,w)=(6,-3)
def A(n)
x, y = 6 * n * n - 4 * n + 4, -3 * n * n - 5 * n + 5
z, w = 4 * n * n - 4 * n + 6, 5 * n * n - 5 * n - 3
puts "#{x}^3 + (#{y})^3 = #{x ** 3 + y ** 3}"
puts "#{z}^3 + #{w}^3 = #{z ** 3 + w ** 3}"
end
def B(n)
63 * (3 * n * n - 3 * n + 1) * (n * n + n + 1) * (n * n - 3 * n + 3)
end
n = 100
(2..n).each{|i| A(i)}
p (1..n).map{|i| B(i)}
出力結果
20^3 + (-17)^3 = 3087
14^3 + 7^3 = 3087
46^3 + (-37)^3 = 46683
30^3 + 27^3 = 46683
84^3 + (-63)^3 = 342657
54^3 + 57^3 = 342657
134^3 + (-95)^3 = 1548729
86^3 + 97^3 = 1548729
196^3 + (-133)^3 = 5176899
126^3 + 147^3 = 5176899
270^3 + (-177)^3 = 14137767
174^3 + 207^3 = 14137767
356^3 + (-227)^3 = 33420933
230^3 + 277^3 = 33420933
454^3 + (-283)^3 = 70911477
294^3 + 357^3 = 70911477
564^3 + (-345)^3 = 138342519
366^3 + 447^3 = 138342519
686^3 + (-413)^3 = 252383859
446^3 + 547^3 = 252383859
820^3 + (-487)^3 = 435866697
534^3 + 657^3 = 435866697
966^3 + (-567)^3 = 719144433
630^3 + 777^3 = 719144433
1124^3 + (-653)^3 = 1141589547
734^3 + 907^3 = 1141589547
1294^3 + (-745)^3 = 1753226559
846^3 + 1047^3 = 1753226559
1476^3 + (-843)^3 = 2616501069
966^3 + 1197^3 = 2616501069
1670^3 + (-947)^3 = 3808184877
1094^3 + 1357^3 = 3808184877
1876^3 + (-1057)^3 = 5421417183
1230^3 + 1527^3 = 5421417183
2094^3 + (-1173)^3 = 7567881867
1374^3 + 1707^3 = 7567881867
2324^3 + (-1295)^3 = 10380120849
1526^3 + 1897^3 = 10380120849
2566^3 + (-1423)^3 = 14013983529
1686^3 + 2097^3 = 14013983529
2820^3 + (-1557)^3 = 18651212307
1854^3 + 2307^3 = 18651212307
3086^3 + (-1697)^3 = 24502164183
2030^3 + 2527^3 = 24502164183
3364^3 + (-1843)^3 = 31808668437
2214^3 + 2757^3 = 31808668437
3654^3 + (-1995)^3 = 40847020389
2406^3 + 2997^3 = 40847020389
3956^3 + (-2153)^3 = 51931111239
2606^3 + 3247^3 = 51931111239
4270^3 + (-2317)^3 = 65415693987
2814^3 + 3507^3 = 65415693987
4596^3 + (-2487)^3 = 81699785433
3030^3 + 3777^3 = 81699785433
4934^3 + (-2663)^3 = 101230204257
3254^3 + 4057^3 = 101230204257
5284^3 + (-2845)^3 = 124505245179
3486^3 + 4347^3 = 124505245179
5646^3 + (-3033)^3 = 152078489199
3726^3 + 4647^3 = 152078489199
6020^3 + (-3227)^3 = 184562749917
3974^3 + 4957^3 = 184562749917
6406^3 + (-3427)^3 = 222634155933
4230^3 + 5277^3 = 222634155933
6804^3 + (-3633)^3 = 267036369327
4494^3 + 5607^3 = 267036369327
7214^3 + (-3845)^3 = 318584940219
4766^3 + 5947^3 = 318584940219
7636^3 + (-4063)^3 = 378171797409
5046^3 + 6297^3 = 378171797409
8070^3 + (-4287)^3 = 446769875097
5334^3 + 6657^3 = 446769875097
8516^3 + (-4517)^3 = 525437875683
5630^3 + 7027^3 = 525437875683
8974^3 + (-4753)^3 = 615325168647
5934^3 + 7407^3 = 615325168647
9444^3 + (-4995)^3 = 717676825509
6246^3 + 7797^3 = 717676825509
9926^3 + (-5243)^3 = 833838790869
6566^3 + 8197^3 = 833838790869
10420^3 + (-5497)^3 = 965263189527
6894^3 + 8607^3 = 965263189527
10926^3 + (-5757)^3 = 1113513769683
7230^3 + 9027^3 = 1113513769683
11444^3 + (-6023)^3 = 1280271482217
7574^3 + 9457^3 = 1280271482217
11974^3 + (-6295)^3 = 1467340196049
7926^3 + 9897^3 = 1467340196049
12516^3 + (-6573)^3 = 1676652549579
8286^3 + 10347^3 = 1676652549579
13070^3 + (-6857)^3 = 1910275938207
8654^3 + 10807^3 = 1910275938207
13636^3 + (-7147)^3 = 2170418637933
9030^3 + 11277^3 = 2170418637933
14214^3 + (-7443)^3 = 2459436065037
9414^3 + 11757^3 = 2459436065037
14804^3 + (-7745)^3 = 2779837171839
9806^3 + 12247^3 = 2779837171839
15406^3 + (-8053)^3 = 3134290978539
10206^3 + 12747^3 = 3134290978539
16020^3 + (-8367)^3 = 3525633241137
10614^3 + 13257^3 = 3525633241137
16646^3 + (-8687)^3 = 3956873255433
11030^3 + 13777^3 = 3956873255433
17284^3 + (-9013)^3 = 4431200797107
11454^3 + 14307^3 = 4431200797107
17934^3 + (-9345)^3 = 4951993197879
11886^3 + 14847^3 = 4951993197879
18596^3 + (-9683)^3 = 5522822557749
12326^3 + 15397^3 = 5522822557749
19270^3 + (-10027)^3 = 6147463093317
12774^3 + 15957^3 = 6147463093317
19956^3 + (-10377)^3 = 6829898622183
13230^3 + 16527^3 = 6829898622183
20654^3 + (-10733)^3 = 7574330183427
13694^3 + 17107^3 = 7574330183427
21364^3 + (-11095)^3 = 8385183794169
14166^3 + 17697^3 = 8385183794169
22086^3 + (-11463)^3 = 9267118342209
14646^3 + 18297^3 = 9267118342209
22820^3 + (-11837)^3 = 10225033614747
15134^3 + 18907^3 = 10225033614747
23566^3 + (-12217)^3 = 11264078463183
15630^3 + 19527^3 = 11264078463183
24324^3 + (-12603)^3 = 12389659103997
16134^3 + 20157^3 = 12389659103997
25094^3 + (-12995)^3 = 13607447555709
16646^3 + 20797^3 = 13607447555709
25876^3 + (-13393)^3 = 14923390211919
17166^3 + 21447^3 = 14923390211919
26670^3 + (-13797)^3 = 16343716550427
17694^3 + 22107^3 = 16343716550427
27476^3 + (-14207)^3 = 17874947978433
18230^3 + 22777^3 = 17874947978433
28294^3 + (-14623)^3 = 19523906813817
18774^3 + 23457^3 = 19523906813817
29124^3 + (-15045)^3 = 21297725402499
19326^3 + 24147^3 = 21297725402499
29966^3 + (-15473)^3 = 23203855371879
19886^3 + 24847^3 = 23203855371879
30820^3 + (-15907)^3 = 25250077020357
20454^3 + 25557^3 = 25250077020357
31686^3 + (-16347)^3 = 27444508842933
21030^3 + 26277^3 = 27444508842933
32564^3 + (-16793)^3 = 29795617192887
21614^3 + 27007^3 = 29795617192887
33454^3 + (-17245)^3 = 32312226079539
22206^3 + 27747^3 = 32312226079539
34356^3 + (-17703)^3 = 35003527102089
22806^3 + 28497^3 = 35003527102089
35270^3 + (-18167)^3 = 37879089519537
23414^3 + 29257^3 = 37879089519537
36196^3 + (-18637)^3 = 40948870456683
24030^3 + 30027^3 = 40948870456683
37134^3 + (-19113)^3 = 44223225246207
24654^3 + 30807^3 = 44223225246207
38084^3 + (-19595)^3 = 47712917906829
25286^3 + 31597^3 = 47712917906829
39046^3 + (-20083)^3 = 51429131757549
25926^3 + 32397^3 = 51429131757549
40020^3 + (-20577)^3 = 55383480167967
26574^3 + 33207^3 = 55383480167967
41006^3 + (-21077)^3 = 59588017444683
27230^3 + 34027^3 = 59588017444683
42004^3 + (-21583)^3 = 64055249853777
27894^3 + 34857^3 = 64055249853777
43014^3 + (-22095)^3 = 68798146779369
28566^3 + 35697^3 = 68798146779369
44036^3 + (-22613)^3 = 73830152018259
29246^3 + 36547^3 = 73830152018259
45070^3 + (-23137)^3 = 79165195210647
29934^3 + 37407^3 = 79165195210647
46116^3 + (-23667)^3 = 84817703406933
30630^3 + 38277^3 = 84817703406933
47174^3 + (-24203)^3 = 90802612770597
31334^3 + 39157^3 = 90802612770597
48244^3 + (-24745)^3 = 97135380417159
32046^3 + 40047^3 = 97135380417159
49326^3 + (-25293)^3 = 103831996389219
32766^3 + 40947^3 = 103831996389219
50420^3 + (-25847)^3 = 110908995767577
33494^3 + 41857^3 = 110908995767577
51526^3 + (-26407)^3 = 118383470918433
34230^3 + 42777^3 = 118383470918433
52644^3 + (-26973)^3 = 126273083876667
34974^3 + 43707^3 = 126273083876667
53774^3 + (-27545)^3 = 134596078865199
35726^3 + 44647^3 = 134596078865199
54916^3 + (-28123)^3 = 143371294950429
36486^3 + 45597^3 = 143371294950429
56070^3 + (-28707)^3 = 152618178833757
37254^3 + 46557^3 = 152618178833757
57236^3 + (-29297)^3 = 162356797779183
38030^3 + 47527^3 = 162356797779183
58414^3 + (-29893)^3 = 172607852676987
38814^3 + 48507^3 = 172607852676987
59604^3 + (-30495)^3 = 183392691243489
39606^3 + 49497^3 = 183392691243489
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