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 on OEIS

Polynomial values

The table should eventually go up to $n=40$. It is not yet completed.

Note that among the primes till 100, the only primes that are not covered in this table at least once are 79 and 89.

$n$	$n^{2}-n$	Polynomial value $n^{2}-n+2$	Polynomial value $n^{2}-n+3$	Polynomial value $n^{2}-n+5$	Polynomial value $n^{2}-n+11$	Polynomial value $n^{2}-n+17$	Polynomial value $n^{2}-n+41$
1	0	2	3	5	11	17	41
2	2	N/A	5	7	13	19	43
3	6	N/A	N/A	13	19	23	47
4	12	N/A	N/A	17	23	29	53
5	20	N/A	N/A	N/A	31	37	61
6	30	N/A	N/A	N/A	41	47	71
7	42	N/A	N/A	N/A	53	59	83
8	56	N/A	N/A	N/A	67	73	97
9	72	N/A	N/A	N/A	83	89	113
10	90	N/A	N/A	N/A	101	107	131
11	110	N/A	N/A	N/A	N/A	127	151
12	132	N/A	N/A	N/A	N/A	149	173