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

0, 4, 5, 720

Offset: 1

Author

Seiichi Manyama, Jul 10 2019

Keywords

• nonn
• fini
• full

Comments

(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.

Examples

			  0^3 + 20*  0 =         0 =     0^2.
  4^3 + 20*  4 =       144 =    12^2.
  5^3 + 20*  5 =       225 =    15^2.
720^3 + 20*720 = 373262400 = 19320^2.
		

Crossrefs

Cf. A029728, A116108, A126203.

Programs

PARI

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

SageMath

[i[0] for i in EllipticCurve([20, 0]).integral_points()] # Seiichi Manyama, Aug 25 2019