Lucky number of Euler: Difference between revisions

Revision as of 21:41, 15 January 2012

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

@@ Line 7: / Line 7: @@
 takes [[prime number]] values for <math>n \in \{ 1,2,\dots,p-1 \}</math>.
-This condition is equivalent to the condition that the ring of integers in <math>\mathbb{Q}(\sqrt{-1 - 4p})</math> be a unique factorization domain, or equivalently, the class number of the field be equal to one.
+This condition is equivalent to the condition that the ring of integers in <math>\mathbb{Q}(\sqrt{1 - 4p})</math> be a unique factorization domain, or equivalently, the class number of the field be equal to one.
 ==Occurrence==