A396598 a(n) = least prime p such that Mordell's equation y^2 = x^3 - p has exactly n integral solutions with y >= 0.

3, 2, 7, 47, 307, 3967, 1999, 4799, 503, 431, 2351, 720703, 1727639, 522127, 344719, 1406591

Offset

0,1

Comments

5*10^8 < a(16) <= 42979358951, a(21) = 28279, a(22)=419040143, a(23) = 245862719, a(24) = 249007807, a(25) = 141073183.

Links

Table of n, a(n) for n=0..15.

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) = 307 because 307 is the least prime p such that y^2 = x^3 - p has 4 integral solutions with nonnegative y: {{7, 6}, {11, 32}, {71, 598}, {939787, 911054064}}.
a(6) = 1999 because 1999 is the least prime p such that y^2 = x^3 - p has 6 integral solutions with nonnegative y: {{22, 93}, {40, 249}, {74, 635}, {100, 999}, {299, 5170}, {562, 13323}}.

Prog

(Magma)
SetClassGroupBounds ("GRH");
sols := [**];
primes := [p : p in [2..1000] | IsPrime(p)];
for k in primes do
    Append(~sols, [k, #IntegralPoints(EllipticCurve([0, 0, 0, 0, -k]))]);
end for;
for n in [0..10] 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. A179175, A392395, A392548.

Sequence in context: A363400 A230710 A265009 * A358656 A021757 A143312

Adjacent sequences: A396593 A396594 A396597 * A396599 A396600 A396601

Keyword

nonn,hard,more

Author

Zhining Yang, May 30 2026

Status

approved