Solutions of q=xy+yz+zxq=\frac{x}{y} +\frac{y}{z} + \frac{z}{x} s.t. q≥3q \geq 3

Is it true that for every rational q3 , the following equation has a solution over N ?

q=xy+yz+zx

Answer

The problem

N=xy+yz+zx

with N,x,y,zZ was considered by Andrew Bremner and Richard Guy in “Two more representation problems” published in the Proceedings of the Edinburgh Mathematical Society, vol. 40 pp.1-17 in 1997. An online copy is available here. They showed solutions only occurred for those N where the elliptic curve

t2=u3+N2u2+8Nu+16

has rank at least 1.

For small N>0, the first solution is for N=6, with x=18, y=4 and z=3.

Attribution
Source : Link , Question Author : Mahan , Answer Author : Aryabhata

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