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A197548
Rank of quartic elliptic curve y^2 = 5*x^4 + 4*n.
0
1, 1, 0, 2, 0, 0, 1, 1, 2, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 2, 1, 1, 2, 1, 2, 2, 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 2, 1, 2, 1, 0, 0, 2, 1, 1, 0, 1, 2, 0, 1, 1, 1, 1, 1, 1, 2, 0, 0, 0, 0, 2, 0, 1, 1, 2, 2, 0, 0, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 2, 0, 1, 2
OFFSET
1,4
COMMENTS
If a(n)=0, it means that the number of rational points on curve y^2=5*x^4+4*n is finite; if a(n)>0, the number of rational points is infinite. The value of the rank tells how many points of infinite order is necessary to generate complete infinite set of rational points of given curve.
The quintic trinomial of the form x^5-n*x+m has only finitely many cases such that is factorizable on quadratic and cubic factor with different Elkies coefficient n^5/m^4 if and only a(n)=0; if a(n)>0, then there are infinitely many solutions.
PROG
(Magma) for n := 1 to 100 do print([n, Rank(EllipticCurve([5, 0, 0, 0, 4*n]))]); end for; // Max Alekseyev
CROSSREFS
KEYWORD
nonn
AUTHOR
Artur Jasinski, Oct 16 2011
STATUS
approved