Witness identities for three Ramanujan congruences

Abstract

For the unrestricted partition function $p(n)$ for integers $n\ge 0$, it is known that $p(49n+19)\equiv 0(\mathrm{mod}49)$, $p(49n+33)\equiv 0(\mathrm{mod}49)$, and $p(49n+40)\equiv 0(\mathrm{mod}49)$ for all $n\ge 0$. We find witness identities for these Ramanujan congruences.