Prime Number Theorem
The distinction between elementary and non-elementary proofs has been considered especially important in regard to the prime number theorem. This theorem was first proved in 1896 by Jacques Hadamard and Charles Jean de la Vallée-Poussin using complex analysis. Many mathematicians then attempted to construct elementary proofs of the theorem, without success. G. H. Hardy expressed strong reservations; he considered that the essential "depth" of the result ruled out elementary proofs:
No elementary proof of the prime number theorem is known, and one may ask whether it is reasonable to expect one. Now we know that the theorem is roughly equivalent to a theorem about an analytic function, the theorem that Riemann's zeta function has no roots on a certain line. A proof of such a theorem, not fundamentally dependent on the theory of functions, seems to me extraordinarily unlikely. It is rash to assert that a mathematical theorem cannot be proved in a particular way; but one thing seems quite clear. We have certain views about the logic of the theory; we think that some theorems, as we say "lie deep" and others nearer to the surface. If anyone produces an elementary proof of the prime number theorem, he will show that these views are wrong, that the subject does not hang together in the way we have supposed, and that it is time for the books to be cast aside and for the theory to be rewritten. —G. H. Hardy (1921). Lecture to Mathematical Society of Copenhagen. Quoted in Goldfeld (2003), p. 3However, in 1948, Atle Selberg produced new methods which led him and Paul Erdős to find elementary proofs of the prime number theorem.
A possible formalization of the notion of "elementary" in connection to a proof of a number-theoretical result is the restriction that the proof can be carried out in Peano arithmetic. Also in that sense, these proofs are elementary.
Read more about this topic: Elementary Proof
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