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2024/03/01 13:26

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References for:

Liubo (ancient Chinese board game)

The Littlewood conjectureplugin-autotooltip__plain plugin-autotooltip_bigThe Littlewood conjecture

The Littlewood Conjecture, proposed by UK mathematician John Littewood in 1930, states that for any two real numbers α and β,

$${\displaystyle \liminf _{n\to \infty }\ n\,\Vert n\alpha \Vert \,\Vert n\beta \Vert =0,}$$

where $${\displaystyle \Vert x\Vert :=\min(|x-\lfloor x\rfloor |,|x-\lceil x\rceil |)}$$ is the distance to the nearest integer.

In plain language :

Any two real numbers α and β can be simultaneously approximated at least moderately well by rationals…

The Newman conjectureplugin-autotooltip__plain plugin-autotooltip_bigThe 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

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