Restricted Integer Partition Sums
- Brandt Kronholm, Assistant Professor of Mathematics, Juniata College
- Mark Kozek, Assistant Professor of Mathematics, Whittier College
The partition function of an integer $n$, denoted $p(n)$, counts how many ways positive integers can add to $n$. For example, $p(4) = 5$ since $4$ can be expressed as: (1+1+1+1), (1+1+2), (2+2), (3+1), (4). The restricted partition function of an integer $n$ into exactly $m$ parts, denoted $p(n,m)$, is relatively unexplored. If $r$ is an integer $- 5 \leq r < 5$, we observe $p(55+r,5) + p(55-r,5) \equiv p(r) \pmod 5$. It begs the question, when do similar statements hold true? By using modern computational tools, investigation quickly shows that similar statements are true if $m$ is an odd prime. We attempt to prove when statements like those above are true and what patterns might dictate their occurrence.