1. 设 $$p$$ 为大于 $$3$$ 的素数, 证明 $$\dfrac{p^p-1}{p-1}$$ 和 $$\dfrac{p^p+1}{p+1}$$ 不能都是素数幂;
2. 设 $$n\gt5$$, 证明 $$n!$$ 不能整除它的正约数之和;
3. 设 $$A$$, $$B$$ 划分正整数集, 如果$$A+A$$ 和 $$B+B$$ 都只含有有限个素数, 证明$$A$$ 或 $$B$$ 是全体奇数的集合;
4. 设 $$M$$ 是给定正整数, 证明对每个充分大的素数 $$p$$, 存在$$M$$个连续的 $$\bmod p$$ 的二次非剩余;
5. 设 $$q$$ 是一个不大于$$\dfrac{\pi^2}6 -1$$ 的正有理数, 证明 $$q$$ 可写为若干互异单位分数的平方和;
6. 对每个充分大的正整数 $$k$$, 存在若干互异正整数, 其和为 $$k$$, 其倒数和为 $$1$$;
7. 在 $$n^2$$ 和 $$(n+1)^2$$ 间总有一些正整数的积是一个平方数的两倍;
8. 若一些单位根之和在单位圆上, 则必亦为单位根;
9. 设 $$f(x)=a_0+a_1x+a_2x^2+\dotsb$$ 是一个整系数的形式幂级数, 假定 $$\dfrac{f^\prime(x)}{f(x)}$$ 也是一个整系数的形式幂级数, 证明对任意下标 $$k$$, $$a_k$$ 能被 $$a_0$$ 整除.

$$\dfrac pq$$ can expressed as the finite sum of reciprocals of distinct squares if and only if

$\frac pq\in[0, \frac{\pi^2}6-1)\cup[1,\frac{\pi^2}6).$

If  $$n$$ is an integer exceeding $$77$$ then there exist positive integers $$k$$, $$a_1$$, $$a_2$$, $$\dotsc$$, $$a_k$$ such that:

1. $$1\lt a_1\lt a_2\lt \dotsc \lt a_k;$$
2.  $$a_1+ a_2+ \dotsb + a_k=n;$$
3.  $$\frac1{a_1}+ \frac1{a_2}+ \dotsb + \frac1{a_k}=1.$$

His proof is constructive and fairly short, but it does require a long table of decompositions for relatively small values of $$n$$. It would be interesting to see a non-constructive proof that doesn’t require such a long list.

Granville and Selfridge, Product of integers in an interval, modulo squares: “We prove a conjecture of Irving Kaplansky which asserts that between any pair of consecutive positive squares there is a set of distinct integers whose product is twice a square.”

The details are Electronic Journal of Combinatorics, Volume 8(1), 2001.

$|\sum_{i=1}^k n_i\zeta_i|= 1,$

where $$n_i\in\mathbb Z$$, then $$\sum\limits_{i=1}^k n_i \zeta_i$$ is also an $$n$$-th root of unit.

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