Prove that the equation $a^2+b^2=c^2+3$ has infinitely many integer solutions $(a,b,c)$. Prove that the equation $a^2+b^2=c^2+3$ has infinitely many integer solutions $(a,b,c)$.
My attempt:
$(a+1)(a-1)+(b+1)(b-1)=c^2+1$
This form didn't help so I thought of $\mod 3$, but that didn't help either. Please help. Thank you.

Note that the requirement is simply to demonstrate infinitely many solutions - we are not required to find all solutions.
Setting $c=b+1$, we see that $a^2 = c^2-b^2+3 = 2b+4$
Therefore for any even $a>2$,  we can choose $b=\frac{a^2-4}{2}$ and $c=b+1$. This gives infinitely many solutions as required.
