arXiv subjects
Publications and source records attributed to Erin Bevilacqua.
We establish general criteria for a countable group $Γ$ to have fixed price 1 depending on a choice of left-invariant proper metric on $Γ$. We apply this criterion to show that if $Γ_1,Γ_2$ are two countable groups satisfying a certain growth condition then $Γ_1\times Γ_2$ has fixed price 1. For example, $Γ\times Γ$ has fixed price 1 for any countable group $Γ$.
For rational $α$, the fractional partition functions $p_α(n)$ are given by the coefficients of the generating function $(q;q)^α_\infty$. When $α=-1$, one obtains the usual partition function. Congruences of the form $p(\ell n + c)\equiv 0 \pmod{\ell}$ for a prime $\ell$ and integer $c$ were studied by Ramanujan. Such congruences exist only for $\ell\in\{5,7,11\}.$ Chan and Wang [4] recently studied congruences for the fractional partition functions and gave several infinite families of congruences using identities of the Dedekind eta-function. Following their work, we use the theory of non-ordinary primes to find a general framework that characterizes congruences modulo any integer. This allows us to prove new congruences such as $p_\frac{57}{61}(17^2n-3)\equiv 0 \pmod{17^2}$.
In this work, we investigate the fewest number of colors needed to guarantee a rainbow solution to the equation $x_1 + x_2 = k x_3$ in $\mathbb{Z}_n$. This value is called the Rainbow number and is denoted by $rb(\mathbb{Z}_n, k)$ for positive integer values of $n$ and $k$. We find that $rb(\mathbb{Z}_p, 1) = 4$ for all primes greater than $3$ and that $rb(\mathbb{Z}_n, 1)$ can be deterimined from the prime factorization of $n$. Furthermore, when $k$ is prime, $rb(\mathbb{Z}_n, k)$ can be determined from the prime factorization of $n$.