A395608 a(n) = least prime p such that Mordell's equation y^2 = x^3 + p has exactly n integral solutions with y >= 0.

7, 2, 101, 37, 89, 1009, 73, 58049, 17, 23417, 15193, 7057, 4481, 92563129, 2089, 76033889, 14704649, 7150249, 4251817, 3543193, 11228689, 3193361

Offset

0,1

Comments

a(31) = 3120110801, a(32) = 10438333993.

Links

Table of n, a(n) for n=0..21.

Helmut Richter, Solutions of Mordell's equation y^2 = x^3 + k.

XiaoYang Zhang, k is a prime number. The Mordell equation y^2=x^3-K has 11 sets of integer solutions for k=7057. For what value of k will there be 15 to 30 sets of solutions? (in Chinese).

Example

a(4) = 89 because 89 is the least prime p such that y^2 = x^3 + p has 4 integral solutions with nonnegative y: {{-4, 5}, {-2, 9}, {10, 33}, {55, 408}}.
a(5) = 1009 because 1009 is the least prime p such that y^2 = x^3 + p has 5 integral solutions with nonnegative y: {{-10, 3}, {6, 35}, {8, 39}, {1355, 49878}, {2520, 126503}}.

Prog

(Magma)
SetClassGroupBounds ("GRH");
sols := [**];
primes := [p : p in [2..2000] | IsPrime(p)];
for k in primes do
    Append(~sols, [k, #IntegralPoints(EllipticCurve([0, 0, 0, 0, k]))]);
end for;
for n in [0..6] 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. A081121, A179162, A392144.

Sequence in context: A038268 A100983 A350621 * A103237 A395478 A338305

Adjacent sequences: A395605 A395606 A395607 * A395609 A395610 A395611

Keyword

nonn,hard,more

Author

Zhining Yang, Jun 02 2026

Status

approved