Mordell's equation y^2 = x^3 ± k^2

northwolves

$ y^2 = x^3 ± k^2$ 与  $y^2 = x^3 ± k$ 相比，计算$n$个解的最小$k$值有什么优化策略 ？


A392548  a(n) is the least positive integer k such that Mordell's equation $y^2 = x^3 - k^2$ has exactly n integer solutions with y >= 0.

DATA
3, 1, 2, 10, 80, 530, 26, 1160, 208, 1664, 730, 1090, 5840, 8720, 46720, 69760, 214370

OFFSET
0,1

COMMENTS
Conjecture： a(n) is even for all n>1.

a(30) = 293410.

LINKS
Xiaoyang Zhang, <a href="https://www.zhihu.com/question/1993841306041071339">The Mordell's equation y² = x³ - k² (k is an integer and 0 < k < 1000).When k takes what value, does it have the most sets of positive integer solutions?(in Chinese)</a>

EXAMPLE
a(6) = 26 because 26 is the least k such that y^2 = x^3 - k^2 has 6 integral solutions with nonnegative y: {{10,18},{13,39},{26,130},{130,1482},{338,6214},{901,27045}}.


A392549  a(n) is the least positive integer k such that Mordell's equation $y^2 = x^3 + k^2$ has exactly n integer solutions with y >= 0.

DATA
2, 11, 1, 6, 3, 10, 80, 62, 63, 210, 55, 840, 15, 440, 120, 960, 3240, 561, 2415, 510, 665, 19320, 1155, 5320

OFFSET
1,1

COMMENTS
a(25) > 80000 if it exists. a(26) = 31185, a(27) = 9240, a(29) = 73920, a(33) = 54285.

EXAMPLE
a(6) = 10 because 10 is the least k such that y^2 = x^3 + k^2 has 6 integral solutions with nonnegative y: {{-4, 6},{0, 10},{5, 15},{20, 90},{24, 118},{2660, 137190}}.