A364248 For n >= 3, a(n) is the least r >= 0 such that the elliptic equation y^2 = n^3 + n^2 + 2*r*n + r^2 has an integer solution.
0, 2, 5, 9, 14, 0, 27, 5, 44, 12, 65, 21, 0, 32, 119, 9, 152, 35, 21, 77, 230, 0, 275, 117, 54, 14, 377, 41, 434, 32, 55, 221, 0, 27, 629, 285, 52, 20, 779, 49, 860, 11, 21, 437, 1034, 0, 1127, 75, 34, 182, 1325, 27, 110, 159, 19, 725, 1652, 10, 1769, 837, 0, 320, 195, 99, 2144, 374
Offset: 3
Keywords
Examples
n = 6: y^2 = 6^3 + (6 + r)^2 is valid for r = 9, 19, 47. Least r is 9 thus a(6) = 9 and [y, n, n+r] is [21, 6, 15]. n = 7: y^2 = 7^3 + (7 + r)^2 is valid for r = 14, 164. Least r is 14 thus a(7) = 14 and [y, n, n+r] is [28, 7, 21].
Links
- Robin Visser, Table of n, a(n) for n = 3..10000
- Josef Gebel, Attila Pethő, and Horst G. Zimmer, Computing S-integral points on elliptic curves, Rend. Circ. Matem. Palermo 30 (1910), 377-407. In: Cohen, H. (eds) Algorithmic Number Theory. ANTS 1996. Lecture Notes in Computer Science, vol 1122. Springer, Berlin, Heidelberg.
- The L-functions and modular forms database, LMFDB
- Joseph H. Silverman, The Arithmetic of Elliptic Curves
- Eric Weisstein's World of Mathematics, Mordell Curve
Programs
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PARI
a(n)=vecmin(((select((x)->x[1]>=0&&x[2]>=n,thue(thueinit(x^2-1,1),n^3)))~[1]))-n \\ Thomas Scheuerle, Sep 03 2023
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Sage
def a(n): for d in Integer(n^3).divisors(): if ((d-n^3/d)%2 == 0) and ((d-n^3/d)/2 >= n): return (d-n^3/d)/2 - n # Robin Visser, Sep 30 2023
Formula
a(p) = p*(p - 3)/2, p >= 5 a prime number.
a(n) = 0 for n from A005563.
Comments