Editing Number theory (section)

=== Elementary number theory ===
[[File:Paul Erdos with Terence Tao.jpg|thumb|upright=1.22|Number theorists [[Paul Erdős]] and [[Terence Tao]] in 1985, when Erdős was 72 and Tao was 10]]

Elementary number theory deals with the topics in number theory by means of basic methods in arithmetic.<ref name=":1">{{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |publisher=Facts On File |year=2005 |isbn=0-8160-5124-0 |location=New York |pages=359-60 |language=en |chapter=Number theory}}</ref> Its primary subjects of study are [[divisibility]], [[factorization]], and [[primality]], as well as [[Modular arithmetic|congruences]] in [[modular arithmetic]].<ref>{{Cite book |last=Nathanson |first=Melvyn B. |title=Elementary Methods in Number Theory |publisher=Springer |year=2000 |isbn=0-387-98912-9 |language=en |chapter=Preface}}</ref><ref name=":3" /> Other topic in elementary number theory are [[Diophantine equation|Diophantine equations]], [[continued fraction]], [[Integer partition|integer partitions]], and [[Diophantine approximation|Diophantine approximations]].<ref name=":2">{{Cite web |last=Bukhshtab |first=A.A. |date=2014 |title=Elementary number theory |url=https://encyclopediaofmath.org/wiki/Elementary_number_theory |url-status=live |access-date=2025-05-03 |website=Encyclopedia of Mathematics |publisher=Springer}}</ref>

Arithmetic is the study of numerical operations and investigates how numbers are combined and transformed using the arithmetic operations of [[addition]], [[subtraction]], [[multiplication]], [[Division (mathematics)|division]], [[exponentiation]], extraction of [[Nth root|roots]], and [[logarithm]]s. Multiplication, for instance, is an operation that combines two numbers, referred to as factors, to form a single number, termed the [[Product (mathematics)|product]], such as <math>2 \times 3 = 6</math>.<ref>{{multiref|{{harvnb|Romanowski|2008|p=303}}|{{harvnb|Musser|Peterson|Burger|2013|pp=[https://books.google.com/books?id=8jh7DwAAQBAJ&pg=PA101 101–102]}}}}</ref>

Divisibility is a property between two nonzero integers related to division. An integer <math>a</math> is said to be divisible by a nonzero integer <math>b</math> if <math>a</math> is a multiple of <math>b</math>; that is, if there exists an integer <math>q</math> such that <math>a = bq</math>. An equivalent formulation is that <math>b</math> divides <math>a</math> and is denoted by a vertical bar, which in this case is <math>b | a</math>. Conversely, if this were not the case, then <math>a</math> would not be divided evenly by <math>b</math>, resulting in a remainder. [[Euclid's division lemma]] asserts that <math>a</math> and <math>b</math> can generally be written as <math>a = bq + r</math>, where the remainder <math>r < b</math> accounts for the leftover quantity. Elementary number theory studies [[divisibility rules]] in order to quickly identify if a given integer is divisible by a fixed divisor. For instance, it is known that any integer is divisible by 3 if its decimal [[digit sum]] is divisible by 3.<ref name="Richmond-Richmond-2009">Richmond & Richmond (2009), [{{Google books|plainurl=y|id=HucyKYx0_WwC|page=102|text=divisible by}} Section 3.4 (Divisibility Tests), p. 102–108]</ref><ref name=":4" />

A common divisor of several nonzero integers is an integer that divides all of them. The [[greatest common divisor]] (gcd) is the largest of such divisors. Two integers are said to be coprime or relatively prime to one another if their greatest common divisor, and simultaneously their only divisor, is 1. The [[Euclidean algorithm]] computes the greatest common divisor of two integers <math>a,b</math> by means of repeatedly applying the division lemma and shifting the divisor and remainder after every step. The algorithm [[Extended Euclidean algorithm|can be extended]] to solve a special case of [[Linear Diophantine equation|linear Diophantine equations]] <math>ax + by = 1</math>. A Diophantine equation is an equation with several unknowns and integer coefficients. Another kind of Diophantine equation is described in the [[Pythagorean theorem]], <math>x^2 + y^2 = z^2</math>, whose solutions are called Pythagorean triples if they are all integers.<ref name=":4" /><ref name=":6" />

Elementary number theory studies the divisibility properties of integers such as [[Parity (mathematics)|parity]] (even and odd numbers), [[prime numbers]], and [[perfect numbers]]. Important number-theoric functions include the [[Divisor function|divisor-counting function]], the [[divisor summatory function]] and its modifications, and [[Euler's totient function]]. A [[prime number]] is an integer greater than 1 whose only positive divisors are 1 and the prime itself. A positive integer greater than 1 that is not prime is called a composite number. [[Euclid's theorem]] demonstrates that there are infinitely many prime numbers that comprise the set {2, 3, 5, 7, 11, ...}. The [[sieve of Eratosthenes]] was devised as an efficient algorithm for identifying all primes up to a given natural number by eliminating all composite numbers. The [[distribution of primes]] is unpredictable and is a major subject of study in number theory. Formulas for a partial sequence of primes, including [[Lucky numbers of Euler|Euler's prime-generating polynomials]] have been developed. However, these cease to function as the primes become too large.<ref>{{Cite book |last=Nathanson |first=Melvyn B. |title=Elementary Methods in Number Theory |publisher=Springer |year=2000 |isbn=0-387-98912-9 |language=en |chapter=Divisibility and Primes}}</ref><ref name=":5" />

[[Factorization]] is a method of expressing a number as a [[Product (mathematics)|product]]. Specifically in number theory, [[integer factorization]] is the decomposition of an integer into a product of integers. The process of repeatedly applying this procedure until all factors are prime is known as [[prime factorization]]. A fundamental property of primes is shown in [[Euclid's lemma]]. It is a consequence of the lemma that if a prime divides a product of integers, then that prime divides at least one of the factors in the product. The [[Fundamental theorem of arithmetic|unique factorization theorem]] is the fundamental theorem of arithmetic that relates to prime factorization. The theorem states that every integer greater than 1 can be factorised into a product of prime numbers and that this factorisation is unique up to the order of the factors. For example, <math>120</math> is expressed uniquely as <math>2 \times 2 \times 2 \times 3 \times 5</math> or simply <math>2^3 \times 3 \times 5</math>.<ref>{{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |publisher=Facts On File |year=2005 |isbn=0-8160-5124-0 |location=New York |language=en |chapter=Fundamental theorem of arithmetic}}</ref><ref name=":4" />

[[Modular arithmetic]] works with finite sets of integers and introduces the concepts of congruence and residue classes. A congruence of two integers <math>a, b</math> modulo <math>n</math> (a positive integer called the modulus) is an [[equivalence relation]] whereby <math>n | (a - b)</math> is true. Performing [[Euclidean division]] on both <math>a</math> and <math>n</math>, and on <math>b</math> and <math>n</math>, yields the same remainder. This written as <math display="inline">a \equiv b \pmod{n}</math>. In a manner analogous to the 12-hour clock, the sum of 4 and 9 is equal to 13, yet congruent to 1. A residue class modulo <math>n</math> is a set that contains all integers congruent to a specified <math>r</math> modulo <math>n</math>. For example, <math>6\Z + 1</math> contains all multiples of 6 incremented by 1. Modular arithmetic provides a range of formulas for rapidly solving congruences of very large powers. An influential theorem is [[Fermat's little theorem]], which states that if a prime <math>p</math> is coprime to some integer <math>a</math>, then <math display="inline">a^{p - 1} \equiv 1 \pmod{p}</math> is true. Euler's theorem extends this to assert that every integer <math>n</math> satisfies the congruence<math display="block">a^{\varphi(n)} \equiv 1 \pmod{n},</math>where Euler's totient function <math>\varphi</math> counts all positive integers up to <math>n</math> that are coprime to <math>n</math>. Modular arithmetic also provides formulas that are used to solve congruences with unknowns in a similar vein to equation solving in algebra, such as the [[Chinese remainder theorem]].<ref>{{Cite book |last=Shoup |first=Victor |title=A Computational Introduction to Number Theory and Algebra |publisher=Cambridge University Press |year=2005 |isbn=978-0-511-11363-5 |language=en}}</ref>

More specifically, elementary number theory works with ''[[elementary proof|elementary proofs]]'', a term that excludes the use of [[complex numbers]] but may include basic analysis.<ref name=":2" /> For example, the [[prime number theorem]] was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by [[Paul Erdős|Erdős]] and [[Atle Selberg|Selberg]].{{sfn|Goldfeld|2003}} The term is somewhat ambiguous. For example, proofs based on complex [[Tauberian theorem]]s, such as [[Wiener–Ikehara theorem|Wiener–Ikehara]], are often seen as quite enlightening but not elementary despite using [[Fourier analysis]], not complex analysis. Here as elsewhere, an ''elementary'' proof may be longer and more difficult for most readers than a more advanced proof.

Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, many of the proofs of these results are not particularly accessible, in part because the range of tools they use is, if anything, unusually broad within mathematics.<ref>See, for example, the initial comment in {{harvnb|Iwaniec|Kowalski|2004|p=1}}.</ref>