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

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

2006-08-31

arxiv.org/abs/math/0608796v1

Mathematics

Number Theory

Scientific paper

We give a shorter simpler proof of a result of Szalay on the equation $2^a + 2^b + 1 = x^2$. 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 used to obtain a bound on $n$ in the equation $x^2 + C = y^n$, where $x$ and $y$ are primes or prime powers, and $2 | C$; the bound depends only on the primes dividing $C$.

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