Lucky number of Euler

Definition

A lucky number of Euler is a prime number $p$ such that the polynomial:

$n^{2}-n+p$

takes prime number values for $n\in \{1,2,\dots ,p-1\}$.

This condition is equivalent to the condition that the ring of integers in $\mathbb {Q} ({\sqrt {1-4p}})$ be a unique factorization domain, or equivalently, the class number of the field be equal to one.

Occurrence

There are only six lucky numbers of Euler:

2, 3, 5, 11, 17, 41 View list of OEIS