mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, an elementary proof is a mathematical proof that only uses basic techniques. More specifically, the term is used in

number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Math ...

to refer to proofs that make no use of complex analysis. Historically, it was once thought that certain theorems, like the

prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic distribution of the prime numbers among the positive integers. It formalizes the intuitive idea that primes become less common as they become larger by precisely quantifying ...

, could only be proved by invoking "higher" mathematical theorems or techniques. However, as time progresses, many of these results have also been subsequently reproven using only elementary techniques. While there is generally no consensus as to what counts as elementary, the term is nevertheless a common part of the

mathematical jargon The language of mathematics has a vast vocabulary of specialist and technical terms. It also has a certain amount of jargon: commonly used phrases which are part of the culture of mathematics, rather than of the subject. Jargon often appears in l ...

. An elementary proof is not necessarily simple, in the sense of being easy to understand or trivial. In fact, some elementary proofs can be quite complicated — and this is especially true when a statement of notable importance is involved..

Prime number theorem

The distinction between elementary and non-elementary proofs has been considered especially important in regard to the

. 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: However, in 1948, Atle Selberg produced new methods which led him and

Paul Erdős Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in ...

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 In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...

. Also in that sense, these proofs are elementary.

Friedman's conjecture

Harvey Friedman

conjecture In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1 ...

d, "Every theorem published in the '' Annals of Mathematics'' whose statement involves only finitary mathematical objects (i.e., what logicians call an arithmetical statement) can be proved in elementary arithmetic.". The form of elementary arithmetic referred to in this conjecture can be formalized by a small set of axioms concerning

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

arithmetic and mathematical induction. For instance, according to this conjecture,

Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers , , and satisfy the equation for any integer value of greater than 2. The cases and have bee ...

should have an elementary proof; Wiles's proof of Fermat's Last Theorem is not elementary. However, there are other simple statements about arithmetic such as the existence of iterated exponential functions that cannot be proven in this theory.

References

{{DEFAULTSORT:Elementary Proof Elementary mathematics Mathematical proofs