A394624 - OEIS

A394624

Largest x for which x^3 - n = y^2 is an integer solution to this Mordell's equation, 0 if no such x exists.

1, 3, 0, 5, 0, 0, 32, 2, 0, 0, 15, 0, 17, 0, 4, 0, 0, 3, 7, 6, 0, 0, 3, 0, 5, 35, 3, 37, 0, 0, 0, 0, 0, 0, 11, 0, 0, 0, 22, 14, 0, 0, 0, 5, 21, 0, 63, 28, 65, 0, 0, 0, 29, 7, 56, 18, 0, 0, 0, 136, 5, 0, 568, 4, 0, 0, 23, 0, 0, 0, 8, 6, 0, 99, 0, 101, 0, 0, 20, 0, 13, 0, 27, 0, 0, 0, 7, 0, 5, 0

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

LINKS

Table of n, a(n) for n=1..90.

EXAMPLE

For n = 1, a(n) = 1 is the largest x for which x^3 - 1 = 0 is a square.

For n = 2, a(n) = 3 is the largest x for which x^3 - 2 = 25 is a square.

For n = 3, 5, 6, ... (cf. A081121) Mordell's equation y^2 = x^3 - n has no integer solution, so a(n) = 0.

For n = 4, a(n) = 5 is the largest x for which x^3 - 4 = 121 = y^2 for y = +-11, the other inequivalent solution being x = 2, y = +-2.

PROG

(PARI) apply( {A394624(n, L=20*n^2)=if(L=ellratpoints(ellinit([0, 0, 0, 0, -n]), [L, 1]), L[#L][1])}, [1..99]) \\ Search limit L corresponds to Hall's conjecture with over-estimate of the largest known ratio sqrt(x)/n ~ 4.26.

CROSSREFS

Cf. A081120 (number of integral solutions to y^2 = x^3 - n), A134109 (only those with y >= 0), A081121 (n for which there are no integral solutions), A081119 (number of integral solutions to y^2 = x^3 + n).

Sequence in context: A393746 A002656 A234434 * A234020 A348259 A276833

Adjacent sequences: A394621 A394622 A394623 * A394625 A394626 A394627

KEYWORD

nonn

AUTHOR

M. F. Hasler, Mar 26 2026

STATUS

approved