title: The Prime Number Theorem (London Mathematical Society Student Texts)
author:  G. J. O. Jameson 
description: "...strongly recommended to those wishing to teach some analytic number theory at the undergraduate level." Mathematical Reviews
zlib_text: 
  
  
  
  
  
  # ALSO BY MARIO LIVIO
  
  
  
  
  
  _The Accelerating Universe: Infinite Expansion,  
  
  
  the Cosmological Constant, and the Beauty of the Cosmos_
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  
  # APPENDIX 10
  
  
  
  
  
  Euclid's proof that infinitely many primes exist is based on the method of
  reductio ad absurdum. He began by assuming the contradictory—that only a
  finite number of primes exist. If that is true, however, then one of them must
  be the largest prime. Let us denote that prime by _P._ Euclid then constructed
  a new number by the following process: He multiplied together all the primes
  from 2 up to (and including) _P_ , and then he added 1 to the product. The new
  number is therefore
  
  
  
  
  
  ![image](../Images/Livi_9780307485526_epub_258_r1.jpg)
  
  
  
  
  
  By the original assumption, this number must be composite (not a prime),
  because it is obviously larger than _P_ , which was assumed to be the largest
  prime. Consequently, this number must be divisible by at least one of the
  existing primes. However, from its construction