A393931 a(n) is the least positive integer k such that the elliptic curve y^2 = x^3 - k*x + 1 has exactly n integer solutions with y >= 0.

3, 2, 17, 10, 8, 1, 74, 58, 9, 88, 4, 28, 25, 36, 16, 400, 49, 784, 169, 196, 217, 1240, 4900, 1681, 47128, 14392, 625, 1296, 844, 45889, 99148, 21592, 15376, 31684, 21904, 289, 11449, 196249, 7609, 21316, 3721, 167104, 573049

Offset

1,1

Comments

a(44) > 10^6, a(45) = 16384, a(52) = 120409, a(66)=550624.

Conjecture: For all n > 3, a(n) is not prime.

Links

Table of n, a(n) for n=1..43.

Example

a(7) = 74 because 74 is the least k such that the elliptic curve y^2 = x^3 - 74*x + 1 has 7 integral solutions with nonnegative y: {{-8,9}, {-3,14}, {0,1}, {9,8}, {12,29}, {28,141}, {1369,50652}}.

Prog

(Magma)
SetClassGroupBounds("GRH");
max_n := 15;
max_k := 100;
sols := [**];
for k in [1..max_k] do
    Append(~sols, [k, #IntegralPoints(EllipticCurve([0, 0, 0, -k, 1]))]);
end for;
for n in [1..max_n] do
    for sol in sols do
        if sol[2] eq n then
            printf "a(%o) = %o\n", n, sol[1];
            break;
        end if;
    end for;
end for;

Crossrefs

Cf. A392144, A392395, A394164.

Sequence in context: A072045 A189731 A342140 * A126354 A361084 A360416

Adjacent sequences: A393927 A393928 A393930 * A393932 A393933 A393934

Keyword

nonn,more

Author

Zhining Yang, Mar 03 2026

Status

approved