单墫 余红兵 冯志刚 刘培杰

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.

$(x^2+xy+y^2)(z^2+zw+w^2)=(xz-yw)^2+(xz-yw)[wx+y(z+w)]+[wx+y(z+w)]^2$

\begin{vmatrix}
x& y\cr
-y & x+y
\end{vmatrix}

Let $$f(x_1,x_2,\dotsc,x_n)$$ be a homogeneous polynomial. Let

$S=\{f(a_1,a_2,\dotsc,a_n)\mid a_1,a_2,\dotsc,a_n \in\Bbb Z\}.$

If $$S$$ satisfies the following condition: for all $$m,n\in S$$, we have $$mn\in S$$. Can we determine all the homogeneous polynomials $$f$$?

For example, $$x^n(n\in\Bbb N),x^2+n y^2(n\in\Bbb Z), x^2+xy+y^2,x^3+y^3+z^3-3xyz$$, and $$x^2+y^2+z^2+w^2$$ are all appropriate examples.

Richard Taylor(就是协助 Andrew Wiles 完成了Fermat’s Last Theorem 的证明的那位) 写了一篇很有趣的文章 Modular Arithmetic: Driven by Inherent Beauty and Human Curiosity(The Institute Letter, 2012, Summer, 6-8). 这文章指出: Euclid 在他的几何原本 已经得到方程

$$x^2+y^2=z^2$$

$$x^2+y^2=2z^2$$

$$x^2+y^2=nz^2,$$

Taylor 就说了这么多. 那么, 我们来尝试找出这方程的所有有理解, 以及所有整数解.

$$x^2+y^2=(a^2+b^2)z^2,$$

证明 1

• a=1. 此时 $$1+(d+2)d=(d+1)^2$$ 是合数;
• $$a\geqslant2$$. 此时 $$a+ad=a(d+1)$$ 是合数.

证明 2

$$(m+1)!+2,(m+1)!+3,\dotsc,(m+1)!+m+1$$ 是 $$m$$ 个连续合数.

证明 3

$a, a+d, a+2d, \dotsc, a+(m-1)d$

证明 4

$f(x)=\sum\limits_{i=0}^ma_ix^i,$

证明 6

$m_i|\left(a+i\right), i=1, 2, \dotsc, n.$

[证明 6 更新于 北京时间 2015 年 6 月 24 日]

1.21 唯一因式分解定理的证明

2.5  五边形与五角数

$p_m(k)=\dfrac{mk(k-1)}2+k.$

2.8  一个不平凡的结论

2.9 什么数恰好有 $$60$$ 个因数?

$$kn = x^2+y^2+1$$

$$n$$ is a odd number, then there exists positive integer $$k\gt0$$ such that $$kn = x^2+y^2+1$$ for some integers $$x,y$$.

with use of the Chinese remainder theorem we have to solve this problem only for power of primes:

suppose that $$n=p_1^{a_1}p_2^{a_2}\dotsm p_k^{a_k}$$, then we know that for each $$i$$, there exist $$x_i, y_i$$ such that $$p_i^{a_i}$$ divides  $$x_i^2+y_i^2+1$$. Now consider these equations:

$X\equiv x_i\pmod {p_i^{a_i}}, i= 1,2,\dotsc,k.$

these equations have solution because of  Chinese remainder theorem.

similarly these equation have solution:

$Y\equiv y_i\pmod {p_i^{a_i}}, i= 1,2,\dotsc,k.$

now $$n$$ divides  $$X^2+Y^2+1.$$

then we can apply hansel’s lemma. Actually we want to show that if for some $$\alpha$$, there exist $$x,y$$ such that $$p^\alpha$$ divides  $$x^2+y^2+1$$, then for  $$\alpha +1$$ such $$x$$  and $$y$$ exist. For this because in case $$\alpha$$, $$p$$ cannot divide both $$x$$  and $$y$$, then we can use hansel for improve $$\alpha$$ to $$\alpha+1.$$

References

1. 华罗庚, 数论导引.
2. Hardy, An introducton to the theory of numbers. 有中文本
3. Tom M. Apostol, Introduction to analytic number therory. 有中文本

Number theory will be understood, not as a collection of tricks and isolated results, but as a coherent and interconnected theory.

Green(与 Tao 合作 Green–Tao theorem 的那位)的(博士)导师 Gowers, 写了两篇关于此定理的文章: Why isn’t the fundamental theorem of arithmetic obvious?  和 Proving the fundamental theorem of arithmetic.

$H=\{1,5,9,13,17,\dotsc\}$

Sey Y.Kim 的证明也收录在 Biscuits of Number Theory 一书.