FOM: Test Case for "Elementary" Proofs JSHIPMAN at
Thu Dec 11 09:36:59 EST 1997

   Dirichlet's theorem (If n,k>1 have no common factor then the sequence [n+k,
n+2k,n+3k,n+4k, ... ] contains a prime) is much simpler to state than the Prime
Number Theorem, and unlike the P.N.T. no proof has ever been found that does
not go through complex analysis.  Number theorists regarded Erdos and Selberg's
"elementary" proof of P.N.T. as a big deal even though it came out in 1949, 53
years after the original proofs of Hadamard and de la Vallee Poussin (they were
both around to see it, Hadamard lived to 98 and de la Vallee Poussin to 96).  I
am sure an "elementary proof" of Dirichlet's theorem would also be a big deal.
Here's great chance to put Reverse Mathematics on the map, Steve! -- Joe Shipman
P.S. Regarding CH--the whole point of the independence results is that there is
no "elementary" proof.  But does the lack of an elementary proof *necessarily*
devalue the meaning or significance of a proposition?   Dirichlet's theorem's
meaning is so obviously clear that my 8-year old son conjectured it last year!
(He was bitterly disappointed I couldn't show him the proof, he enjoyed the
proofs of the infinitude of primes and the irrationality of sqrt(2) so much!)-JS

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