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The Newman conjecture

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

It's stated formally as :

For any integers m and r such that $${\displaystyle 0\leq r\leq m-1}$$ $${\displaystyle 0\leq r\leq m-1}$$the value of the partition function $${\displaystyle p(n)}$$ satisfies the congruence $${\displaystyle p(n)\equiv r{\pmod {m}}}$$ 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 : Periodicity Modulo m and Divisibility Properties of the Partition Function Transactions of the American Mathematical Society, Vol. 97, No. 2 pp. 225-236

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