Commentary on Euclid IX.20: infinitude of primes (JC)
  1. The statement of the theorem makes use of an elegant notion of infinity: "greater than any assigned multitude."
  2. The notation is quite different than that current in the 20th century. Where we would say "Let p1, p1, ... pn be the assigned prime numbers", Euclid says "A, B, C." It was clear to Euclid and his readers that he referred to a list of numbers of arbitrary length. The literary style differs, but not the substance.
  3. Consider the statement "For let the least number measured by A, B, C be taken, and let it be DE; let the unit DF be added to DE. Then EF is either prime or not." Here Euclid clearly uses geometric language. Numbers are intervals on the line. Adding the interval EF to DE results in EF. A number can be measured by another if one can lay out the first number N times along the second: as a carpenter would lay out a measuring stick N times along the baseline of house to be constructed. If nothing is left over, the stick "measures" the given length.
  4. Proposition IX.20 is the earliest answer to the many deep questions about prime numbers. Other questions are (a) are there infinitely many twin primes such as 11, 13 and 17, 19? There is a great body of experimental evidence. But there is no proof, hence neither certainty nor understanding. (b) Goldbach's conjecture: can every even integer greater than two be written as a sum of two prime numbers? Experimental evidence is easy to come by 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, etc. (c) how many prime numbers are there less than a given number? (d) are there arbitrarily long arithmetic progressions in the primes? (e) the Riemann hypothesis; The answers to (a), (b), and (e) are unknown. (c) an answer - that there are roughly x/log(x) primes less than x - was conjectured in the late 1700's and was proved by independently by Hadamard and de la Valee Poussin in 1896. Regarding (e), this was a byproduct Riemann's effort to answer (c), and was was formulated in 1859. The Riemann hypothesis would give important statistical information on the distribution of prime numbers. It is one of the 23 problems David Hilbert announced in 1900 as important for the future of mathematics. Indeed it has been so (consider the Weil conjectures and their influence on number theory and algebraic geometry). The Riemann hypothesis is also one of the Clay Mathematics Institute's Millennium Prize Problems . (d) An arithmetic progression of primes is a sequence like 199, 409, 619, 829, 1039, 1249, 1459, 1669, 1879, 2089. The conjecture was proved in 2004 by Ben Green and Terence Tao.