# Polygonal numbers modulo odd primes

Marc Renault's masters thesis "Properties of the Fibonacci Sequence Under Various Moduli" is well known for its investigation of Fibonacci numbers with focus on the distribution of residues, peiods of Fibonacci numbers modulo primes and so on.

Does there exist similar work about Polygonal numbers? More specifically, have the properties of polygonal number sequence under various moduli been investigated at any level of detail? Either for a specific s-gonal number sequence or for all polygonal numbers in general?

I looked up on the net, but could not find such works, hence the request.

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Polygonal numbers are just values of a certain quadratic polynomial with coefficients that are either integers or half-integers. This immediately implies that $2n$ is a period for these sequences modulo $n$. If the polygon has an even number of sides, then the coefficients of the polynomial are integers, and $n$ is itself a period. – Jyrki Lahtonen Jan 30 '12 at 18:58
Fine, I also deduced the thing about period modulo $n$. But what about distribution etc? – NikBels Jan 30 '12 at 19:03
Hmm. If $n=p$ is a prime, then we can complete the square, and deduce that the distribution is an affine mapping applied to the quadratic residues modulo $p$. IOW, typically one value appears once per period, one half of the remaining $p-1$ residues appears twice per period, and the others won't appear at all. There are some degenerate cases, when the quadratic polynomial reduces to a linear one. Can't say much more about the general case. A plan would be to first figure out what happens modulo a prime power, and then apply CRT. – Jyrki Lahtonen Jan 30 '12 at 19:11
I see. Thanks for the clarification, I will try going along the lines suggested by you. – NikBels Jan 31 '12 at 19:26
Just a comment though: Except for 2, triangular numbers for all odd primes p seem to have a period of p. $T_{n+p}-T_{n} = frac{(n+p)^2+(n+p)}{2} - \frac{n^2+n}{2}=\frac{n^2+2np+p^2+n+p-n^2-n}{2}=\frac{p(2n+1)+p^2}{2}\equiv 0\pmod p$ So, $T_{n+p}\equiv T_n \pmod p$. Therefore, $p$ is a period of triangular numbers modulo a given odd prime $p$. Hope I am not wrong. – NikBels Jul 11 '12 at 15:08