Is there a method for determining if a system of quadratic diophantine equations has any solutions?
My specific example (which comes from this question) is:
$$\frac{4}{3}x^2 + \frac{4}{3}x + 1 = y^2$$
$$\frac{8}{3}x^2 + \frac{8}{3}x + 1 = z^2$$
I want to know if there are any positive integer triples $(x,y,z)$ which satisfy both equations.
Best Answer
Answered at the source question. Briefly: There are no such $(x,y,z)$, because then we'd have a nonconstant arithmetic progression of four squares: $$ 1 = 1^2, \ \frac43(x^2+x)+1 = y^2, \ \frac83(x^2+x)+1 = z^2, \ 4(x^2+x)+1 = (2x+1)^2. $$ The impossibility of such a progression is a theorem of Euler (1780, answering a question "first raised by Fermat in 1640" according to Keith Conrad's exposition). This even proves that there are no rational solutions $(x,y,z)$ other than the obvious ones with $x=0$ or $x=-1$.