Elementary treatment of $p^a \pm p^b + 1 = x^2$

Abstract

We give a shorter simpler proof of a result of Szalay on the equation $2^a + 2^b + 1 = z^2$. We give an elementary proof of a result of Luca on the equation of the title for prime $p > 2$. The elementary treatment is made possible by a lemma which is also useful for other Diophantine equations.