Sam Northshield 用一句话就说明了质数的无穷性. 这个很精彩的证明是这样的:

Proof.  If the set of primes is finite, then

$0\lt \prod_p \sin\left(\frac \pi p\right)= \prod_p \sin\left(\frac{\pi\Big(1+2\prod\limits_{p’}p’\Big)}p\right)=0. \qquad \Box$

$\sin\left(\frac{\pi\Big(1+2\prod\limits_{p’}p’\Big)}q\right)=\sin k\pi=0.$

### References

1. Sam Northshield, A One-Line Proof of the Infinitude of Primes, The American Mathematical Monthly, Vol. 122, No. 5 (May 2015), p. 466
2. Michael Hardy and Catherine Woodgold, Prime Simplicity,  the Mathematical Intelligencer, Volume 31, Issue 4, December 2009,  44-52
3. Harold Edwards, Contradict or Construct?,  the Mathematical Intelligencer, Volume 32, Issue 1, March 2010, p.3
4. Harold Edwards, Essays in Constructive Mathematics,  Springer, 2010

### One Response to “A one sentence Proof of the Infinitude of Primes”

1. ; Euclid 的那个质数无穷性的著名证明是通过反证法. 事实不是如此.

为什么不是反证法呢？假设素数有限，找到一个不在其中的有一个素数，与假设矛盾啊。能否解释一下：）

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