# Euler’s lucky number

A prime number^{} $p$ is one of Euler’s lucky numbers if ${n}^{2}-n+p$ for each $$ is also a prime. Put another way, a lucky number of Euler^{}’s plus the $n$th oblong number produces a list of primes $p$-long. There are only six of them: 2, 3, 5, 11, 17 and 41, these are listed in A014556 of Sloane’s OEIS.

41 is perhaps the most famous of these. We can verify that 2 + 41 is 43, a prime, that 47 is also prime, so are 53, 61, 71, 83, 97, and so on to 1601, giving a list of 41 primes. Predictably, 1681 is divisible by 41, being its square. For $n>p$ the formula does not consistently give only composites or only primes.

Title | Euler’s lucky number |
---|---|

Canonical name | EulersLuckyNumber |

Date of creation | 2013-03-22 16:55:33 |

Last modified on | 2013-03-22 16:55:33 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 4 |

Author | PrimeFan (13766) |

Entry type | Definition |

Classification | msc 11A41 |

Synonym | lucky number of Euler |

Synonym | Eulerian lucky number |