R. Burns

activity

20162022

most citedStructure and asymptotics for Catalan numbers modulo primes using automata

3 citations · 13 across the 13 of their papers we have counts for

collaborators

13 papers

math.NT2022

The appearance function for paper-folding words

Rob Burns

We provide a complete characterisation of the appearance function for paper-folding sequences for factors of any length. We make use of the software package {\tt Walnut} to establi…

math.NT2022

Factorials and Legendre's three-square theorem: II

Rob Burns

Let $S ˉ$ denote the set of integers $n$ such that $n!$ cannot be written as a sum of three squares. Let $S ˉ (n)$ denote $S ˉ \cap [1, n]$ . We establish an exact formul…

math.NT2021★ 1 cited

Factorials and Legendre's three-square theorem

Rob Burns

We provide a necessary and sufficient condition for $n!$ to be a sum of three squares. The condition is based on the binary representation of $n$ and can be expressed by the operat…

math.NT2020

Extremely symmetric primes

Rob Burns

We introduce extremely symmetric primes and provide some elementary properties of these.

math.NT2020

The Sum of Squares function in the ring $Z_{n}$

Rob Burns

We consider the sum of squares function in the ring $Z_{n}$ . We determine formulae in a number of cases when $n$ is a power of a prime.

math.CO2019

Counting birthday collisions using partitions

Rob Burns, Jen McKenzie

We use partitions to provide some formulae for counting s-collisions and other events in various forms of the Birthday Problem.

math.NT2017★ 1 cited

Representing numbers as the sum of squares and powers in the ring $Z_{n}$

Rob Burns

We examine the representation of numbers as the sum of two squares in $Z_{n}$ for a general positive integer $n$ . Using this information we make some comments about the dens…

math.NT2017

The Catalan numbers have no forbidden residue modulo primes

Rob Burns

Let $C_{n}$ be the $n$ th Catalan number. For any prime $p \geq 5$ we show that the set ${C_{n} : n \in N}$ contains all residues mod $p$ . In addition all residues are attai…

math.NT2017★ 1 cited

Structure and asymptotics for Motzkin numbers modulo primes using automata

Rob Burns

We establish a lower bound of 2/p(p-1) for the asymptotic density of the Motzkin numbers divisible by a general prime number p > 3. We provide a criteria for when this asymptotic d…

math.NT2017★ 3 cited

Structure and asymptotics for Catalan numbers modulo primes using automata

Rob Burns

Let $C_{n}$ be the $n$ th Catalan number. We show that the asymptotic density of the set ${n : C_{n} \equiv 0 mod p}$ is $1$ for all primes $p$ , We also show that if $n = p^{k} - 1$ the…