Department of Mathematics
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Sir Michael Atyah remarked during an interview: any good theorem should have several proofs, more the better. Ever since Euclid proved the infinitude of primes (INFP), there has been a plethora of proofs of INFP including the famous theorem of Euler which was the beginning of analytic number theory. Patterned after Euler's proof of INFP, Dirichlet proved the INFP in every arithmetic progression of the form an+b with gcd(a, b) = 1. Dirichlet's proof employs very powerful analytic techniques (e.g. Dirichlet series). However, there are some well-known special cases of Dirichlet's theorem for which "pure algebraic" proofs exist. In this presentation, we analyze such algebraic proofs, define what ought to be an algebraic proof, and give several new simple, polynomial-based algebraic proofs of special cases of Dirichlet Theorem. However, it is well-known that not all cases of Dirichlet theorem can have algebraic proofs (see [1]).

1. M. Ram Murty, Primes in arithmetic progressions, J. Madras Univ., Section B 51 (1988), 161Ð169.
2. M. Raussen and C. Skau, Interview with Michael Atiyah and Isadore Singer, Notices Amer. Math. Soc. 52 (2005), no.2, 225Ð233.