Analogues to Fermat primes related to Pell's equation
In this work the author studies the Fermat primes related to Pell equations. He considers the primes of the form \[ p=2x^{2}-1=2dy^{2}+1. \] So it is known that if \((x_{1},y_{1})\) is the positive solution of \(x^{2}-dy^{2}=1\), then the other solutions are \((x_{n},y_{n})\), where \[ x_{n}+y_{n}\sqrt{d}=(x_{1}+y_{1}\sqrt{d})^{n} \] for \(n\geq 1\). Therefore, if \(p_{n}=2x_{n}^{2}-1\) is prime, then \(n=2^{m}\) is a power of \(2\). So there are analogues to the Fermat numbers of the form \(2^{n}+1\).
scientific article; zbMATH DE number 5675007