# 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 |