# thirteen

Thirteen is in many ways a rather ordinary number, yet some people are afraid of it (see triskaidekaphobia). For a mathematician there is nothing particularly unlucky about 13; in fact, 13 is a lucky prime (since it is not crossed off in the well-defined sieving process for lucky numbers).

13 is a Fibonacci number^{} (being the sum of 5 and 8), and like all other odd-indexed Fibonacci numbers, it is also a Markov number^{}. It appears in the following solutions to ${13}^{2}+{y}^{2}+{z}^{2}=39yz$: (1, 5, 13), (1, 13, 34), (5, 13, 194), (13, 34, 1325), (13, 194, 7561), (13, 1325, 51641), (13, 7561, 294685), (13, 51641, 2012674), (13, 294685, 11485154), (13, 2012674, 78442645), (13, 11485154, 447626321), etc. Starting with 1, the number 13 represents a new low for the Mertens function^{}, but this is neither unlucky nor special (A051401 in Sloane’s OEIS lists other lows of the Mertens function).

This is not to say that there isn’t anything special about 13: for example, it is the only integer solution to ${x}^{4}={n}^{2}+{(n+1)}^{2}$ (besides the obvious $x=1$ and $n=0$).

Title | thirteen |
---|---|

Canonical name | Thirteen |

Date of creation | 2013-03-22 17:24:58 |

Last modified on | 2013-03-22 17:24:58 |

Owner | PrimeFan (13766) |

Last modified by | PrimeFan (13766) |

Numerical id | 6 |

Author | PrimeFan (13766) |

Entry type | Feature |

Classification | msc 11A99 |