====== The Newman conjecture ======

The Newman conjecture was created by Morris Newman in 1960. It remains unsolved.

It's stated formally as :

{{:spacer.png?40|}}For any integers// m// and// r// such that  $${\displaystyle 0\leq r\leq m-1}$$ $${\displaystyle 0\leq r\leq m-1}$${{:spacer.png?40|}}the value of the partition function  $${\displaystyle p(n)}$$ {{:spacer.png?40|}}satisfies the congruence $${\displaystyle p(n)\equiv r{\pmod {m}}}$$ {{:spacer.png?40|}}for infinitely many non-negative integers //n//.

In plain language :

Given arbitrary //m//, //r//, are there infinitely values of //n// such that the partition function at //n// is congruent to //r// mod //m//?

See : [[https://www.jstor.org/stable/1993300?origin=crossref|Periodicity Modulo m and Divisibility Properties of the Partition Function]]  //Transactions of the American Mathematical Society, // Vol. 97, No. 2 pp. 225-236

And [[https://en.wikipedia.org/wiki/Newman%27s_conjecture|Wikipedia]]