More specifically, the term is used in number theory to refer to proofs that make no use of complex analysis.
While there is generally no consensus as to what counts as elementary, the term is nevertheless a common part of the mathematical jargon.
[1] 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.
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.However, in 1948, Atle Selberg produced new methods which led him and Paul Erdős to find elementary proofs of the prime number theorem.