Is it true that for every rational q≥3 , the following equation has a solution over N ?

q=xy+yz+zx

**Answer**

The problem

N=xy+yz+zx

with N,x,y,z∈Z 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*