Lucky number of Euler

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Definition

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

takes prime number values for .

This condition is equivalent to the condition that the ring of integers in 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 . 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.

Polynomial value Polynomial value Polynomial value Polynomial value Polynomial value Polynomial value
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