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

Author

Ctibor O. Zizka, Sep 01 2023

Keywords

• nonn

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.

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

Crossrefs

Cf. A000040, A000096, A000217, A002061, A005563, A006254, A054504, A081119, A364421.

Programs

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

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.