Mathematics Colloquia and Seminars

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For odd primes $p$ let $\chi_p(r) := (\frac{r}{p})$ denote the Legendre symbol. With this, the Legendre-signed partition numbers, denoted $\mathfrak{p}(n,\chi_{p})$, are then defined to be the coefficients appearing in the series expansion $$\prod_{r=1}^{p-1}\prod_{m=0}^{\infty}\frac{1}{1-\chi_{p}(r)q^{mp+r}} = 1 + \sum_{n=1}^\infty \mathfrak{p}(n,\chi_{p})q^n.$$ It is known that: 1) one has $\mathfrak{p}(n,\chi_{5}) = 0$ for all $n \equiv 2 \,(\mathrm{mod}\,10)$; and 2) the sequences $(\mathfrak{p}(n,\chi_{p}))_{n \geq 1}$ do not have such a periodic vanishing whenever $p \not\equiv 1 \,(\mathrm{mod}\,8)$ and $p \neq 5$. In this talk we discuss the recent result that $\mathfrak{p}(n,\chi_{17})$ vanishes only when the input $n$ is odd and $1-24n$ is congruent to a quartic residue $(\mathrm{mod}\,17)$, as well as a similar vanishing in the sequence $(\mathfrak{p}(n,-\chi_{17}))_{n\geq 1}$.