Commentary on Euclid IX.20: infinitude of primes (JC)
- The statement of the theorem makes use of an elegant notion of
infinity: "greater than any assigned multitude."
- 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.
- 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.
- 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
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.