可表示为连续正整数平方和的四次方数

数论爱好者

wayne 发表于 2025-5-8 21:15
我发现.咱们可以直面这个四次方程 $12 m^4-3 n q^2=n^3-n$, 对于给定的n, 用sagemath发现,总是 有理域上的 ...

对此问题,我一窍不通,交给软件去解决,下面这两个答案,必有一个是错的

wayne

mathe 发表于 2025-5-2 10:13
你怎么证明n=176,178等无解呢？

因为先不说四次方, 就算是平方的时候, 刚好 n=176,178 都没解, 也就是pell方程 $m^2-n q^2=\frac{1}{3} (n^3-n)$无解. https://oeis.org/A134419

nyy

如果是立方数等于连续整数的平方和，这个结果是怎么样的？

wayne

用maple的辅助，对于方程 $m^2-n q^2=\frac{1}{3} (n^3-n)$在$n$为平方数的情况下，得到所有解的参数表达${n\to t^2,m\to \frac{t \left(3 s^2+t^4-1\right)}{6 s},q\to \frac{3 s^2-t^4+1}{6 s}}$
当n不为平方数的时候，也存在参数解，$\{m\to \frac{x (s^2+n)-2 n s y}{s^2-n},q\to \frac{y (s^2+n)-2 s x}{s^2-n}\}$， 其中${x,y}$是方程$x^2-n y^2=\frac{1}{3} (n^3-n)$的整数解。
让$s\to\frac{a}{b}$,得到，$(m,q)=(\frac{a^2 x-2 a b n y+b^2 n x}{a^2-b^2 n},\frac{a^2 y-2 a b x+b^2 n y}{a^2-b^2 n})$

1. with(algcurves);
   
2. arr := [2, 11, 23, 24, 26, 33, 47, 50, 52, 59, 73, 74, 88, 96, 97, 107, 122, 146, 148, 177, 184, 191, 193, 194, 218, 239, 241, 242, 244, 249, 276, 292, 297, 299];
   
3. map(n -> {op(parametrization(a^2 - n*b^2 - 1/3*n^3 + 1/3*n, a, b, s)), n}, arr);
   

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wayne

wayne 发表于 2025-5-8 21:15
我发现.咱们可以直面这个四次方程 $12 m^4-3 n q^2=n^3-n$, 对于给定的n, 用sagemath发现,总是 等价于有理 ...

方程$12 m^4-3 n q^2=n^3-n$，给定$n$,得知其亏格为 1 ， j-不变量为 1728，那么等价于椭圆曲线$y^2=x^3+Ax$，要找出其双有理变换转化成Weierstrass 形式，需要在K域有一个有理数的点，也就是$f_4(x)=12 x^4-(n^3-n)=0$，而这个显然失效了。 https://mathoverflow.net/questio ... rtic-elliptic-curve
所以，咱们唯一的依靠就是转化成其Jacobian形式。

现在$12 m^4-3 n q^2=n^3-n$ 转化成Jacobian形式的椭圆曲线表达式是 $y^2 = x^3+\frac{16}{3} \left(n^5-n^3\right) x$, Jacobian坐标是$(X,Y,Z)=(\frac{1}{3} (-16) (n-1) n^4 (n+1) x^2 y,\frac{1}{9} (-16) (n-1) n^4 (n+1) x (n^3-n+12 x^4),-n^3 y^3)$
其中双有理变换表达式是$x=\frac{16 m^2 \left(n^3-n\right)}{3 q^2}, y =\frac{16 m n \left(n^3-n\right) \left(2 n^2+3 q^2-2\right)}{9 q^3}$

https://www.hyperelliptic.org/EFD/oldefd/quartic.html

1. Block[{n=242},{Factor[4 m^4+1/3 (n-n^3)-n q^2],Factor[-y^2+x^3+16/3 (n^5-n^3)*x/.Thread[{x,y}->{(16 m^2 (n^3-n))/(3 q^2),(16 m  n (n^3 - n) (-2 + 2 n^2 + 3 q^2))/(9 q^3)}]]}]

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wayne

https://oeis.org/A383359
https://oeis.org/A383367
https://oeis.org/A383653
https://oeis.org/A383654

nyy

如果是5次方，表达成连续整数的平方和，
结果怎么样？
6次方呢？
7次方的？

wayne

跑了一下$m<1.3*10^10$, 蹦出了一个解

1. 12807728388,1075848147361,316299050378808,(157611601115724,158687449263084)

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wayne

三次方的记录 咱们也可以打破一下的
https://oeis.org/A212018
这个是10^10以内的所有解, 包括负整数。总共146组.

1. {1, 2, 1, 0, 1}
   
2. {11, 22, 9, -6, 15}
   
3. {26, 39, 36, -1, 37}
   
4. {47, 47, 90, 22, 68}
   
5. {65, 26, 205, 90, 115}
   
6. {66, 121, 68, -26, 94}
   
7. {109, 218, 89, -64, 153}
   
8. {921, 162, 4391, 2115, 2276}
   
9. {935, 1375, 1322, -26, 1348}
   
10. {1079, 2158, 881, -638, 1519}
    
11. {2161, 2161, 4138, 989, 3149}
    
12. {2820, 3807, 4328, 261, 4067}
    
13. {2860, 5577, 2532, -1522, 4054}
    
14. {5029, 11449, 862, -5293, 6155}
    
15. {9105, 20250, 3523, -8363, 11886}
    
16. {10681, 21362, 8721, -6320, 15041}
    
17. {12284, 3071, 49104, 23017, 26087}
    
18. {13156, 5577, 40284, 17354, 22930}
    
19. {16761, 5994, 55949, 24978, 30971}
    
20. {18340, 393, 250572, 125090, 125482}
    
21. {41921, 90354, 23241, -33556, 56797}
    
22. {43500, 10368, 178103, 83868, 94235}
    
23. {61721, 65522, 113679, 24079, 89600}
    
24. {63765, 36842, 166423, 64791, 101632}
    
25. {64605, 4307, 500422, 248058, 252364}
    
26. {66317, 7802, 386665, 189432, 197233}
    
27. {75130, 166375, 31124, -67625, 98749}
    
28. {99359, 99359, 190258, 45450, 144808}
    
29. {105731, 211462, 86329, -62566, 148895}
    
30. {116180, 24649, 504260, 239806, 264454}
    
31. {122009, 21531, 580746, 279608, 301138}
    
32. {146821, 137842, 292419, 77289, 215130}
    
33. {159371, 148877, 318386, 84755, 233631}
    
34. {218205, 369603, 258658, -55472, 314130}
    
35. {253393, 468538, 256365, -106086, 362451}
    
36. {260165, 156099, 665670, 254786, 410884}
    
37. {264680, 519168, 230253, -144457, 374710}
    
38. {269588, 170368, 671075, 250354, 420721}
    
39. {314919, 34991, 1889406, 927208, 962198}
    
40. {403130, 57967, 2125952, 1033993, 1091959}
    
41. {404326, 606489, 559764, -23362, 583126}
    
42. {420365, 924803, 190258, -367272, 557530}
    
43. {524095, 123877, 2154818, 1015471, 1139347}
    
44. {601150, 1194649, 501660, -346494, 848154}
    
45. {690381, 942841, 1048662, 52911, 995751}
    
46. {813340, 528000, 1995783, 733892, 1261891}
    
47. {827209, 101306, 4727179, 2312937, 2414242}
    
48. {869241, 572427, 2116654, 772114, 1344540}
    
49. {985864, 674816, 2351151, 838168, 1512983}
    
50. {1046629, 2093258, 854569, -619344, 1473913}
    
51. {1203015, 225423, 5556718, 2665648, 2891070}
    
52. {1348761, 351122, 5283041, 2465960, 2817081}
    
53. {1583274, 3606768, 255609, -1675579, 1931188}
    
54. {1942084, 2048383, 3592384, 772001, 2820383}
    
55. {3109926, 5513104, 3419289, -1046907, 4466196}
    
56. {3145712, 196607, 25165440, 12484417, 12681023}
    
57. {3632486, 797511, 15498012, 7350251, 8147761}
    
58. {3662142, 6713927, 3772980, -1470473, 5243453}
    
59. {3867745, 40474, 75618605, 37789066, 37829539}
    
60. {3871937, 354482, 25592425, 12618972, 12973453}
    
61. {4568353, 4568353, 8747730, 2089689, 6658041}
    
62. {4866290, 6488625, 7550208, 530792, 7019416}
    
63. {5776229, 2991458, 15959781, 6484162, 9475619}
    
64. {7099170, 3786224, 19318595, 7766186, 11552409}
    
65. {7251849, 14720738, 5603081, -4558828, 10161909}
    
66. {8414889, 1694763, 37488686, 17896962, 19591724}
    
67. {10133869, 1370386, 55109531, 26869573, 28239958}
    
68. {10360559, 20721118, 8459361, -6130878, 14590239}
    
69. {10406230, 23762000, 1218491, -11271754, 12490245}
    
70. {13258984, 16477184, 21802733, 2662775, 19139958}
    
71. {13439435, 19876750, 18889075, -493837, 19382912}
    
72. {13674371, 26351229, 12516834, -6917197, 19434031}
    
73. {16667143, 27303838, 20731545, -3286146, 24017691}
    
74. {17613609, 4652843, 68487214, 31917186, 36570028}
    
75. {18749975, 749999, 187499250, 93374626, 94124624}
    
76. {19286215, 44051878, 2127733, -20962072, 23089805}
    
77. {21744866, 75504, 738040371, 368982434, 369057937}
    
78. {21928269, 14111834, 54058937, 19973552, 34085385}
    
79. {27510150, 60250000, 13123031, -23563484, 36686515}
    
80. {29526032, 49242112, 35814295, -6713908, 42528203}
    
81. {32779435, 13111774, 103380885, 45134556, 58246329}
    
82. {36557739, 5988006, 180625199, 87318597, 93306602}
    
83. {37711876, 9427969, 150749264, 70660648, 80088616}
    
84. {38867794, 17654896, 114889331, 48617218, 66272113}
    
85. {39174927, 21171942, 105873315, 42350687, 63522628}
    
86. {40852075, 93468750, 2352141, -45558304, 47910445}
    
87. {48520824, 95282176, 42062539, -26609818, 68672357}
    
88. {52121135, 90306750, 59608785, -15348982, 74957767}
    
89. {57967855, 131745125, 11335186, -60204969, 71540155}
    
90. {61322534, 115569391, 59407704, -28080843, 87488547}
    
91. {80621532, 2239487, 967457520, 482609017, 484848503}
    
92. {102558961, 205117922, 83739041, -60689440, 144428481}
    
93. {112336839, 150176431, 173901474, 11862522, 162038952}
    
94. {113103835, 70394077, 283838218, 106722071, 177116147}
    
95. {114398141, 21144266, 532044367, 255450051, 276594316}
    
96. {140409620, 73839649, 384886440, 155523396, 229363044}
    
97. {145699686, 265936, 6820706363, 3410220214, 3410486149}
    
98. {162279945, 58502250, 539500093, 240498922, 299001171}
    
99. {181848051, 328666734, 192818811, -67923961, 260742772}
    
100. {184130289, 72607441, 584944826, 256168693, 328776133}
     
101. {210044879, 210044879, 402205322, 96080222, 306125100}
     
102. {249782771, 494913671, 210497578, -142208046, 352705624}
     
103. {276710399, 5647151, 3873944214, 1934148532, 1939795682}
     
104. {300259400, 23639903, 2140145192, 1058252645, 1081892547}
     
105. {382361529, 574201386, 528694291, -22753547, 551447838}
     
106. {391415970, 45131823, 2305253168, 1130060673, 1175192495}
     
107. {404474965, 923154626, 51479675, -435837475, 487317150}
     
108. {405791185, 164566082, 1270876753, 553155336, 717721417}
     
109. {441763936, 41415369, 2885490876, 1422037754, 1463453122}
     
110. {460802706, 976628016, 287781287, -344423364, 632204651}
     
111. {462660051, 1056254094, 56116811, -500068641, 556185452}
     
112. {502644461, 1149060841, 44330302, -552365269, 596695571}
     
113. {505803155, 226798, 47773018907, 23886396055, 23886622852}
     
114. {520736449, 1192158906, 5975811, -593091547, 599067358}
     
115. {687035140, 7294848, 13334915757, 6663810455, 6671105302}
     
116. {720640030, 98877328, 3890563083, 1895842878, 1994720205}
     
117. {805306304, 12582911, 12884898816, 6436157953, 6448740863}
     
118. {1015229051, 2030458102, 828931049, -600763526, 1429694575}
     
119. {1087680947, 1294464397, 1848705210, 277120407, 1571584803}
     
120. {1104306049, 176221738, 5527909019, 2675843641, 2852065378}
     
121. {1159387684, 1891150391, 1450551196, -220299597, 1670850793}
     
122. {1497065964, 1385036416, 3008406501, 811685043, 2196721458}
     
123. {1576634475, 688935767, 4753594102, 2032329168, 2721264934}
     
124. {2066242527, 25509167, 37192362570, 18583426702, 18608935868}
     
125. {2670558034, 1124445488, 8205571449, 3540562981, 4665008468}
     
126. {2757505485, 12670627, 81358984318, 40673156846, 40685827472}
     
127. {2804002430, 115024912, 27688560013, 13786767551, 13901792462}
     
128. {2893051557, 5188276962, 3113773595, -1037251683, 4151025278}
     
129. {2919122968, 1094671113, 9512844300, 4209086594, 5303757706}
     
130. {3209893846, 32246736, 64050585117, 32009169191, 32041415926}
     
131. {3592180020, 7663317376, 2149201525, -2757057925, 4906259450}
     
132. {4327782626, 381370544, 29156957111, 14387793284, 14769163827}
     
133. {4619827104, 1172859263, 18325225584, 8576183161, 9749042423}
     
134. {4799999900, 47999999, 95999994000, 47975997001, 48023996999}
     
135. {5009587710, 795640401, 25136340480, 12170350040, 12965990440}
     
136. {5053173099, 5118586343, 9596855934, 2239134796, 7357721138}
     
137. {5379975939, 11620719526, 2930304649, -4345207438, 7275512087}
     
138. {5435885530, 2621526000, 15581854815, 6480164408, 9101690407}
     
139. {5667075271, 1860830686, 19750288633, 8944728974, 10805559659}
     
140. {5860211135, 3713291125, 14566900490, 5426804683, 9140095807}
     
141. {6264891457, 6621933362, 11572117335, 2475091987, 9097025348}
     
142. {6287269274, 9430903911, 8704330164, -363286873, 9067617037}
     
143. {6781937615, 2212123526, 23715217555, 10751547015, 12963670540}
     
144. {7553634975, 8001268533, 13932735798, 2965733633, 10967002165}
     
145. {7596347816, 16558021632, 3808326639, -6374847496, 10183174135}
     
146. {9657496081, 9657496081, 18492697082, 4417600501, 14075096581}
     

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wayne

连续整数的平方和 仍然是平方数，记为$m^2$, $m<10^10$总共有 $107261$组解。合并27组2个解的情况，就是$107234$组。
https://oeis.org/A182379
太多了，我就不贴，放文件里了 https://nestwhile.com/res/A379340/21.1.nb.txt

2个解的我单独新建了一个序列https://oeis.org/A379340

1. {{70, 24, 25, 1, 24}, {70, 25, 24, 0, 24}}
   
2. {{105, 49, 10, -19, 29}, {105, 50, 7, -21, 28}}
   
3. {{143, 11, 86, 38, 48}, {143, 33, 46, 7, 39}}
   
4. {{195, 26, 75, 25, 50}, {195, 50, 47, -1, 48}}
   
5. {{2849, 11, 1718, 854, 864}, {2849, 74, 661, 294, 367}}
   
6. {{3854, 47, 1124, 539, 585}, {3854, 376, 333, -21, 354}}
   
7. {{5681, 299, 634, 168, 466}, {5681, 722, 71, -325, 396}}
   
8. {{8075, 578, 583, 3, 580}, {8075, 722, 433, -144, 577}}
   
9. {{143737, 1969, 6378, 2205, 4173}, {143737, 5329, 2458, -1435, 3893}}
   
10. {{144157, 338, 15681, 7672, 8009}, {144157, 4394, 3533, -430, 3963}}
    
11. {{208395, 50, 58943, 29447, 29496}, {208395, 3025, 7374, 2175, 5199}}
    
12. {{939356, 407, 93124, 46359, 46765}, {939356, 19041, 8032, -5504, 13536}}
    
13. {{1226670, 10552, 23093, 6271, 16822}, {1226670, 24025, 7624, -8200, 15824}}
    
14. {{2259257, 5978, 58339, 26181, 32158}, {2259257, 15873, 34674, 9401, 25273}}
    
15. {{2656724, 11711, 48632, 18461, 30171}, {2656724, 38112, 16019, -11046, 27065}}
    
16. {{2741046, 30712, 25771, -2470, 28241}, {2741046, 34969, 21256, -6856, 28112}}
    
17. {{4598528, 37559, 42212, 2327, 39885}, {4598528, 53889, 24528, -14680, 39208}}
    
18. {{6555549, 5291, 180222, 87466, 92756}, {6555549, 16874, 100461, 41794, 58667}}
    
19. {{7832413, 19729, 110942, 45607, 65335}, {7832413, 86546, 18401, -34072, 52473}}
    
20. {{11818136, 46464, 106321, 29929, 76392}, {11818136, 64009, 85804, 10898, 74906}}
    
21. {{19751043, 6129, 504562, 249217, 255345}, {19751043, 67873, 146474, 39301, 107173}}
    
22. {{32938290, 235224, 2425, -116399, 118824}, {32938290, 235225, 2376, -116424, 118800}}
    
23. {{429323037, 314171, 1521126, 603478, 917648}, {429323037, 976833, 660826, -158003, 818829}}
    
24. {{807759678, 201839, 3594028, 1696095, 1897933}, {807759678, 1545048, 945241, -299903, 1245144}}
    
25. {{1375704770, 535425, 3747428, 1606002, 2141426}, {1375704770, 1097272, 2549083, 725906, 1823177}}
    
26. {{1656510196, 935209, 3383044, 1223918, 2159126}, {1656510196, 3161279, 375256, -1393011, 1768267}}
    
27. {{1981351834, 675672, 4805041, 2064685, 2740356}, {1981351834, 2598959, 1946924, -326017, 2272941}}

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