Sophie Germain's theorem for prime pairs \(p, 6p+1\) (Q1090696)

Let \(p\) be an odd prime and suppose that there exist integers \(X, Y, Z\) prime to \(p\) satisfying the Fermat equation \(X^p+Y^p+Z^p=0\). Suppose that \(q=6p+1\) is a prime. Working in the quadratic field \(\mathbb Q(\sqrt{-3})\), the author deduces from this certain rational divisibility conditions depending on \(q\). His computations show that among the \(14443\) prime pairs \((p,q)\) with \(p<10^6\) there are \(1615\) pairs satisfying these conditions.