

A193531


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


5



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
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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^44*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



