Infinitude of Primes
A one-line proof

$\displaystyle0\lt\prod_{p}\sin \left(\frac{\pi}{p}\right)=\prod_{p}\sin \left(\pi\frac{1+2\displaystyle\prod_{p'}p'}{p}\right)=0.$

(There is a great video, explaining the above formula.)

Reference

1. S. Northshield, A One-Line Proof of the Infinitude of Primes, Am Math Monthly Volume 122, Number 5, May 2015, pp. 466-466(1)

• Infinitude of Primes
  1. Infinitude of Primes - A Topological Proof
  2. Infinitude of Primes - A Topological Proof without Topology
  3. Infinitude of Primes Via *-Sets
  4. Infinitude of Primes Via Coprime Pairs
  5. Infinitude of Primes Via Fermat Numbers
  6. Infinitude of Primes Via Harmonic Series
  7. Infinitude of Primes Via Lower Bounds
  8. Infinitude of Primes - via Fibonacci Numbers
  9. New Proof of Euclid's Theorem
  10. Infinitude Of Primes From Legendre's Formula
  11. Infinitude of Primes - via Bertrand's Postulate
  12. Infinitude of Primes - An Impossible Injection
  13. Infinitude of Primes - from Infinitude of Mutual Primes
  14. Infinitude of Primes Via Euler's Product Formula
  15. Infinitude of Primes Via Euler's Product Formula for Pi
  16. Infinitude of Primes Via Powers of 2
  17. Infinitude of Primes As a Quickie
  18. A One-Line Proof of the Infinitude of Primes
  19. Infinitude of Primes - A. Thue's Proof
  20. Why The Number of Primes Could Not Be Finite?

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