A193531 Number of integer solutions to the quartic elliptic curve y^2 = 5*x^4 - 4*n.

2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 0

Offset

1,1

Comments

The quintic x^5+n*x+m is reducible into cubic and quadratic factors if and only a(n) != 0.

Links

Table of n, a(n) for n=1..86.

Example

We have following parametrization: (X^3 - d*X^2 + (d^2 - e)*X + (2*d*e - d^3))*(X^2 + d*X + e) =  -d^3*e + 2*d*e^2 + (-d^4 + 3*d^2*e - e^2)*X + X^5.
Solving the equation (-d^4 + 3*d^2*e - e^2) = n for e we have e=(3*d^2 +/-sqrt(5*d^4 - 4*n))/2. So 5*d^4 - 4*n must be a perfect square (then y^2=5*x^4-4*n has at least one integer solution).

Prog

(Magma) [IntegralQuarticPoints([5, 0, 0, 0, -4*n]) : n in [1..55]];

Crossrefs

Cf. A193524, A193528.

Sequence in context: A193033 A318253 A249772 * A093492 A139380 A128771

Adjacent sequences: A193528 A193529 A193530 * A193532 A193533 A193534

Keyword

nonn

Author

Artur Jasinski, Jul 29 2011

Status

approved