mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, an elementary proof is a

mathematical proof A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use othe ...

that only uses basic techniques. More specifically, the term is used in

number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

to refer to proofs that make no use of

complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic ...

. Historically, it was once thought that certain theorems, like the

prime number theorem In mathematics, the prime number theorem (PNT) describes the asymptotic analysis, 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 p ...

, 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. 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 Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning wa ...

using complex analysis. Many mathematicians then attempted to construct elementary proofs of the theorem, without success.

G. H. Hardy Godfrey Harold Hardy (7 February 1877 – 1 December 1947) was an English mathematician, known for his achievements in number theory and mathematical analysis. In biology, he is known for the Hardy–Weinberg principle, a basic principle of pop ...

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 to find elementary proofs of the prime number theorem.

Friedman's conjecture

Harvey Friedman conjectured, "Every theorem published in the ''

Annals of Mathematics The ''Annals of Mathematics'' is a mathematical journal published every two months by Princeton University and the Institute for Advanced Study. History The journal was established as ''The Analyst'' in 1874 and with Joel E. Hendricks as t ...

'' 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 (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

arithmetic and

mathematical induction Mathematical induction is a method for mathematical proof, proving that a statement P(n) is true for every natural number n, that is, that the infinitely many cases P(0), P(1), P(2), P(3), \dots all hold. This is done by first proving a ...

. 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 number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...

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