What is the condition for ax2+by2=cax^2 + by^2 = cax2+by2=c to have solutions if a,b,c>0a, b, c > 0a,b,c>0?
c≥abc \ge \sqrt{ab}c≥ab
There must exist a prime ppp such that (a/p)=−1(a/p) = -1(a/p)=−1
Legendre's theorem on ternary quadratic forms
It always has solutions for c>0c > 0c>0