A364248 - OEIS

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

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OFFSET

3,2

COMMENTS

The equation y^2 = n^3 + A*n^2 + B*n + C, where A = 1, B = 2*r, C = r^2 is a minimal model of an elliptic curve with integral coefficients, for details see the Links section.

For n >= 4, n*(n - 3)/2 >= a(n) >= 2.

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

FORMULA

a(p) = p*(p - 3)/2, p >= 5 a prime number.

a(n) = 0 for n from A005563.

EXAMPLE

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].

PROG

(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

(Sage)