[FOM] Dirichlet's theorem; boiling down proofs

[joeshipman@aol.com]

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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