Show that there are infinitely many primes $p$ of the form $p=a^2+b^2+c^2+1$

Question

I know that any prime can be written as the sum of four squares. But I don't know how to know one of these squares is $0$.

Answer 1

The key point is that any integer that is not of the form $4^m(8k+7)$ can be written as the sum of three squares (see Legendre's $3$-squares theorem). For sure $p\neq 1+4^m(8k+7)$ holds for infinitely many primes, for instance for any prime of the form $8k+3$.