Elementary proof

Elementary proof

In mathematics a proof is said to be elementary if it avoids difficult ideas from distant areas of mathematics. For example, the term is used in number theory to refer to proofs that make no use of complex analysis. An elementary proof in combinatorics, using methods such as direct enumeration, is similarly called a "combinatorial proof".

The distinction between elementary and non-elementary proofs has been considered important in regard to the prime number theorem. It 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. G. H. Hardy in 1921 expressed strong reservations; he considered that the essential 'depth' of the result ruled out elementary proofs. In 1948, Selberg produced new methods which led him and Paul Erdős to find elementary proofs of this result. [ cite paper | last = Goldfeld | first = Dorian M. | authorlink = Dorian M. Goldfeld | title = The Elementary Proof of the Prime Number Theorem: An Historical Perspective | date = 2003
url = http://www.math.columbia.edu/~goldfeld/ErdosSelbergDispute.pdf | format = PDF
accessdate = 2007-11-14 ]

Note that under this definition of an elementary proof, there is no logical distinction. Sometimes people use the word "elementary" to refer to a proof that can be carried out in Peano Arithmetic. However this does not tie in with the above meaning of "elementary". If we take "elementary" to have the meaning from the first paragraph in this article, elementary and complex analytic proofs of the prime number theorem can both be carried out in Peano Arithmetic. The meaning from the first paragraph is the standard one.

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