[FOM] Dirichlet's theorem; boiling down proofs
joeshipman@aol.com
joeshipman at aol.com
Fri Aug 17 20:02:18 EDT 2007
I propose the thesis "any mathematics result more than a century old is
suitable for undergraduate math majors".
Note that the original proofs may be too difficult for undergraduates,
I am only requiring that today a "boiled-down" proof (which may be
embedded in a much larger theory than existed at the time of the
original proof) could be taught.
So far I have only found one significant counterexample, Dirichlet's
theorem (which, in its logically simplest form, states that if a is
prime to b, there exists a prime congruent to a mod b).
Can anyone think of better counterexamples? Does anyone know of a proof
of Dirichlet's theorem that does not require prerequisites beyond the
standard undergraduate curriculum?
(Two other possible counterexamples, the Prime Number Theorem and the
Transcendence of Pi, are proven sufficiently easily at the following
links that they would, in my opinion, be appropriate for a senior
seminar:
http://www.ma.utexas.edu/users/dafr/M375T/Newman.pdf
http://sixthform.info/maths/files/pitrans.pdf
).
Another version of the thesis is "any mathematics result more than 200
years old is suitable for freshmen" (note that most high schools offer
a full year of Calculus). Results that were merely conjectured more
than 200 years ago but not really proved until later don't count.
-- JS
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