16 problems

Let $p(n)$ denote the number of partitions of the positive integer $n$. Parkin–Shanks conjecture. The values $p(n)$ that are even and the values $p(n)$ that are odd each have natur…

Let $p(n)$ denote the partition function, and for integers $d\ge 1$ and $n\ge 0$ define the Jensen polynomial … A polynomial is hyperbolic if all of its roots are real. Chen–Jia–Wa…

Let $M$ be a positive integer, and let $p(n)$ denote the number of partitions of the positive integer $n$. For an integer $r$, consider the residue class of $p(n)$ modulo $M$. Newm…

Let $p(n)$ denote the partition function, let $\Delta$ be the forward difference operator $\Delta(a)(n)=a(n)-a(n-1)$, and let $n_0(r)$ be a threshold such that $\Delta^r(p)(n)>0$ f…

Quadratic-progression parity conjecture. The theorem that both parities occur infinitely often among these values, with $(m,6)=1$, should hold for all square-free $D$ satisfying…

Partition Parity Conjecture. The values $p(n)$ are distributed equally between odd and even values, namely

Let $p(n)$ denote the number of partitions of the positive integer $n$, with $p(0)=1$. For a modulus $m$, consider the natural density of indices at which $p(n)$ is divisible by…

Let $p(n)$ denote the number of nonincreasing sequences of positive integers whose sum is $n$. Parkin's conjecture. Half of the partition numbers are odd and the other half are eve…

Let $p(n)$ be the partition function, and let $R(n)$ denote the first term of the Hardy–Ramanujan–Rademacher series for $p(n)$. For a sequence $a_n$, write … for its degree-$d$ Jen…

Partition-function Jensen-polynomial conjecture. For every positive integer $m\geq 4$, there exists a positive integer $N(m)$ such that, for every $n\geq N(m)$, the polynomial…

Higher-order partition-function inequality. For $n\geq 2$,

Let $p(n)$ denote the number of partitions of $n$. Desalvo–Pak's conjecture. For every integer $n\geq45$, … The coefficient $\frac{\pi}{\sqrt{24}}$ is asymptotically best possible,…

Let $p(n)$ denote the number of partitions of the nonnegative integer $n$, and let $\Delta$ be the difference operator with respect to $n$. For a fixed integer $r$, let $n(r)$ be a…

Let $p(n)$ denote the number of partitions of the nonnegative integer $n$, and let $\Delta$ be the difference operator with respect to $n$. For a fixed integer $r\geq 1$, write…

Let $p(n)$ denote the number of partitions of the positive integer $n$. For integers $n\geq 45$, consider the consecutive ratio inequality involving $p(n-1)$, $p(n)$, and $p(n+1)$.…

Let $p(n)$ denote the partition function, and let $r$ and $t$ be integers satisfying $0\leq r<t$. Subbarao's conjecture. The values $p(tn+r)$ should take each parity value infinite…