# 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 Thirteen 2013-03-22 17:24:58 2013-03-22 17:24:58 PrimeFan (13766) PrimeFan (13766) 6 PrimeFan (13766) Feature msc 11A99