Solving x² + y² + z² = xyz - 1 for positive integers

[mtayab1994]

Homework Statement

Solve the following equation in N^3:

x^2+y^2+z^2=xyz-1

The Attempt at a Solution

I moved all the terms to one side, but I don't know what to do from there on.

 

[HallsofIvy]

That's a cubic Diophantine equation and those are notoriously difficult!

 

[mtayab1994]

Do you have any idea on how i can start it? I think i should make x=t and x=s as my two parameters and i'll get an equation in z. Is that a good idea?

 

[Joffan]

Consider the possibilities mod 2, then mod 4, on the two sides of the equation as originally given.

 

[mtayab1994]

Consider the possibilities mod 2, then mod 4, on the two sides of the equation as originally given.

yea that's what i did i got lhs=1mod4 and rhs is -1mod4 thus this equation has no solution.