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A309071 Complete list of solutions to y^2 = x^3 + 20*x; sequence gives x values. 1

%I #31 Dec 28 2021 23:10:44

%S 0,4,5,720

%N Complete list of solutions to y^2 = x^3 + 20*x; sequence gives x values.

%C (x,y) = (0,0) is this solution. Consider x > 0. If x is square, x^2 + 20 is square and we get (x,y) = (4,12). If x is not square, x = i^2*j where j is squarefree. j | x^2 + 20, so j is 2,5 or 10. If j = 2 or 10, there is no such (x,y). If j = 5, (y/5)^2 = i^2*(5*i^4 + 4). So 5*i^4 + 4 = k^2. That is k^2 - 5*i^4 = 4. i^2 is a square Fibonacci number. i^2 = 1 or 144, so x = 5 or 720.

%e 0^3 + 20* 0 = 0 = 0^2.

%e 4^3 + 20* 4 = 144 = 12^2.

%e 5^3 + 20* 5 = 225 = 15^2.

%e 720^3 + 20*720 = 373262400 = 19320^2.

%o (PARI) for(k=0, 1e5, if(issquare(k*(k^2+20)), print1(k", ")))

%o (SageMath) [i[0] for i in EllipticCurve([20, 0]).integral_points()] # _Seiichi Manyama_, Aug 25 2019

%Y Cf. A029728, A116108, A126203.

%K nonn,fini,full

%O 1,2

%A _Seiichi Manyama_, Jul 10 2019

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