Alexandru Gica: Some strange primes, p.213-217

Abstract:

The aim of this paper is to analyze the set of prime numbers $p>3$ for which $p-a^2$ is four times a prime for any positive odd integer $a$ such that $a^2<p$ (we consider that 1 is a prime number). We show that for such prime numbers $p$ we have $p=x^2+4$, where $x$ is a prime number. We compute also the class number $h(-4p)$ for the quadratic imaginary field $\mathbb{Q} (i\sqrt {p})$ using a famous formula of Gauss. There are only six prime numbers with the above property.

Key Words: Class number, sum of squares and primes.

2000 Mathematics Subject Classification: Primary: 11R29,
Secondary: 11P99.

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