[FOM] Only one proof
joeshipman at aol.com
Mon Aug 31 17:12:19 EDT 2009
I've researched this further -- there is a proof by Kochiba and
Uchiyama from 1966 "On the Existence of Prime Numbers in Arithmetic
Progressions" (Proc. japan Acad. 42/7, 696-701) which uses Selberg's
famous inequality from his proof of PNT to reduce the general case to
the "1 mod n" case which can be handled by more algebraic techniques,
though it is still all about the vanishing of Dirichlet characters.
The proof by Zassenhaus in Commentarii Mathematici Helvetici (Dec.
1949, v22, #1, pp. 232-259) is allegedly purely algebraic, but I'd like
to see an English version before pronouncing on that.
So it looks like even Dirichlet's theorem was eventually derived by
other methods, though apart from possibly the 1949 Zassenhaus proof
they all owe a huge debt to Dirichlet's work.
That leaves Feit/Thompson as my best example of an important theorem
with essentially only one proof. Can anyone think of another? (It's not
enough that there is one idea common to all proofs unless that idea is
the only difficulty -- I'm looking for a theorem where all proofs are
essentially similar both globally and locally.)
From: joeshipman at aol.com
I've now found a 1948 paper by Selberg (published in the April 1949
Annals of Mathematics 50/2 pp.297-304) which has an "elementary" proof
which avoids compex numbers, though it still involves nonvanishing of =0
real "characters" and also depends on Selberg's "
elementary" proof of the Prime Number Theorem). So it still depends on
Dirichlet to some extent and it took 111 years to find!
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