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A193531
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Number of integer solutions to the quartic elliptic curve y^2 = 5*x^4 - 4*n.
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5
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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
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1,1
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COMMENTS
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The quintic x^5+n*x+m is reducible into cubic and quadratic factors if and only a(n) != 0.
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LINKS
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EXAMPLE
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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).
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PROG
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(Magma) [IntegralQuarticPoints([5, 0, 0, 0, -4*n]) : n in [1..55]];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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