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Peter J. C. Moses

Publications and source records attributed to Peter J. C. Moses.

6 recordsLinked to original sources
arXiv2015

The ménage problem with a known mathematician

We give a solution of the following combinatorial problem: "Let one from $n$ married couples in the ménage problem (see Problem 1) be a couple of a known mathematician $M$ and his wife. After the ladies are seated at every other chair, $M$ (in token of respect) is the first man allowed to choose one of the remaining chairs. To find the number of ways of seating the other men, with no man seated next to his wife, if $M$ chooses the chair that is $d$ seats clockwise from his wife's chair."

math.CO↗
arXiv2014

Tangent power sums and their applications

For integer $m, p,$ we study tangent power sum $\sum^m_{k=1}\tan^{2p}\frac{πk}{2m+1}.$ We prove that, for every $m, p,$ it is integer, and, for a fixed p, it is a polynomial in $m$ of degree $2p.$ We give recurrent, asymptotical and explicit formulas for these polynomials and indicate their connections with Newman's digit sums in base $2m.$

math.NT↗
arXiv2012

On intervals (kn,(k+1)n) containing a prime for all n>1

We study values of k for which the interval (kn,(k+1)n) contains a prime for every n>1. We prove that the list of such integers k includes k=1,2,3,5,9,14, and no others, at least for k<=50,000,000. For every known k of this list, we give a good upper estimate of the smallest N_k(m), such that, if n>=N_k(m), then the interval (kn,(k+1)n) contains at least m primes.

math.NT↗
arXiv2012

A family of digit functions with large periods

For odd n>=3, we consider a general hypothetical identity for the differences S_{n,0}(x) of multiples of n with even and odd digit sums in the base n-1 in interval [0,x), which we prove in the cases n=3 and n=5 and empirically confirm for some other n. We give a verification algorithm for this identity for any odd n. The hypothetical identity allows to give a general recursion for S_{n,0}(x) for every integer x depending on the residue of x modulo p(n)=2n(n-1)^{n-1}, such that p(3)=24, p(5)=2560, p(7)=653184, etc.

math.NT↗