Elementary proof
Encyclopedia
In mathematics
, an elementary proof is a mathematical proof
that only uses basic techniques. More specifically, the term is used in number theory
to refer to proofs that make no use of complex analysis
. For some time it was thought that certain theorems, like the prime number theorem
, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
While the meaning has not always been defined precisely, the term is commonly used in mathematical jargon
. An elementary proof is not necessarily simple, in the sense of being easy to understand: some elementary proofs can be quite complicated.
. 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
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.
conjectured, "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 arithmetic and mathematical induction. For instance, according to this conjecture, Fermat's Last Theorem
should have an elementary proof; Wiles' 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an elementary proof is a mathematical proof
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
that only uses basic techniques. More specifically, the term is used in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
to refer to proofs that make no use of complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
. For some time it was thought that certain theorems, like the prime number theorem
Prime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
, could only be proved using "higher" mathematics. However, over time, many of these results have been reproved using only elementary techniques.
While the meaning has not always been defined precisely, the term is commonly used in mathematical jargon
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 lectures, and sometimes in print, as informal...
. An elementary proof is not necessarily simple, in the sense of being easy to understand: some elementary proofs can be quite complicated.
Prime number theorem
The distinction between elementary and non-elementary proofs has been considered especially important in regard to the prime number theoremPrime number theorem
In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....
. This theorem was first proved in 1896 by Jacques Hadamard
Jacques Hadamard
Jacques Salomon Hadamard FRS was a French mathematician who made major contributions in number theory, complex function theory, differential geometry and partial differential equations.-Biography:...
and Charles Jean de la Vallée-Poussin
Charles Jean de la Vallée-Poussin
Charles-Jean Étienne Gustave Nicolas de la Vallée Poussin was a Belgian mathematician. He is most well known for proving the Prime number theorem.The king of Belgium ennobled him with the title of baron.-Biography:...
using complex analysis. Many mathematicians then attempted to construct elementary proofs of the theorem, without success. G. H. Hardy
G. H. Hardy
Godfrey Harold “G. H.” Hardy FRS was a prominent English mathematician, known for his achievements in number theory and mathematical analysis....
expressed strong reservations; he considered that the essential "depth" of the result ruled out elementary proofs:
However, in 1948, Atle Selberg
Atle Selberg
Atle Selberg was a Norwegian mathematician known for his work in analytic number theory, and in the theory of automorphic forms, in particular bringing them into relation with spectral theory...
produced new methods which led him and Paul Erdős
Paul Erdos
Paul Erdős was a Hungarian mathematician. Erdős published more papers than any other mathematician in history, working with hundreds of collaborators. He worked on problems in combinatorics, graph theory, number theory, classical analysis, approximation theory, set theory, and probability theory...
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.
Friedman's conjecture
Harvey FriedmanHarvey Friedman
Harvey Friedman is a mathematical logician at Ohio State University in Columbus, Ohio. He is noted especially for his work on reverse mathematics, a project intended to derive the axioms of mathematics from the theorems considered to be necessary...
conjectured, "Every theorem published in the Annals of Mathematics
Annals of Mathematics
The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study. It ranks amongst the most prestigious mathematics journals in the world by criteria such as impact factor.-History:The journal began as The Analyst in 1874 and was...
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 arithmetic and mathematical induction. For instance, according to this conjecture, Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
should have an elementary proof; Wiles' proof of Fermat's Last Theorem
Wiles' proof of Fermat's Last Theorem
Wiles's proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves released by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Wiles first announced his proof in June 1993 in a version that was soon recognized...
is not elementary. However, there are other simple statements about arithmetic such as the existence of iterated exponential
Tetration
In mathematics, tetration is an iterated exponential and is the next hyper operator after exponentiation. The word tetration was coined by English mathematician Reuben Louis Goodstein from tetra- and iteration. Tetration is used for the notation of very large numbers...
functions that cannot be proven in this theory.