Editing Number theory (section)

== Main subdivisions ==
=== 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 topics in elementary number theory include [[Diophantine equation|Diophantine equations]], [[continued fractions]], [[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 |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.<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>

[[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>

=== Analytic number theory ===
{{Main|Analytic number theory}}

[[File:Complex zeta.jpg|thumb|[[Riemann zeta function]] ζ(''s'') in the [[complex plane]]. The color of a point ''s'' gives the value of ζ(''s''): dark colors denote values close to zero and hue gives the value's [[Argument (complex analysis)|argument]].]]
[[File:ModularGroup-FundamentalDomain.svg|thumb|The action of the [[modular group]] on the [[upper half plane]]. The region in grey is the standard [[fundamental domain]].]]

Analytic number theory, in contrast to elementary number theory, relies on [[complex numbers]] and techniques from analysis and calculus. Analytic number theory may be defined
* in terms of its tools, as the study of the integers by means of tools from [[Real analysis|real]] and [[Complex analysis|complex]] analysis;{{sfn|Apostol|1976|p=7}} or
* in terms of its concerns, as the study within number theory of estimates on the size and density of certain numbers (e.g., primes), as opposed to identities.<ref>{{harvnb|Granville|2008|loc=section 1}}: "The main difference is that in algebraic number theory [...] one typically considers questions with answers that are given by exact formulas, whereas in analytic number theory [...] one looks for ''good approximations''."</ref>
It studies the distribution of primes, behavior of number-theoric functions, and irrational numbers.<ref>{{Cite web |last=Karatsuba |first=A.A. |date=2014-10-18 |title=Analytic number theory |url=https://encyclopediaofmath.org/wiki/Analytic_number_theory |website=Encyclopedia of Mathematics}}</ref>

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> The following are examples of problems in analytic number theory: the [[prime number theorem]], the [[Goldbach conjecture]], the [[twin prime conjecture]], the [[Hardy–Littlewood conjecture]]s, the [[Waring problem]] and the [[Riemann hypothesis]]. Some of the most important tools of analytic number theory are the [[circle method]], [[sieve theory|sieve methods]] and [[L-functions]] (or, rather, the study of their properties). The theory of [[modular form]]s (and, more generally, [[automorphic forms]]) also occupies an increasingly central place in the toolbox of analytic number theory.<ref>See the remarks in the introduction to {{harvnb|Iwaniec|Kowalski|2004|p=1}}: "However much stronger...".</ref>

[[Mathematical analysis|Analysis]] is the branch of mathematics that studies the [[Limit (mathematics)|limit]], defined as the value to which a sequence or function tends as the argument (or index) approaches a specific value. For example, the limit of the sequence 0.9, 0.99, 0.999, ... is 1. In the context of functions, the limit of <math display="inline">\frac1x</math> as <math>x</math> approaches infinity is 0.<ref>{{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |chapter=Limit}}</ref> The complex numbers extend the real numbers with the imaginary unit <math>i</math> defined as the solution to <math>i^2 = -1</math>. Every complex number can be expressed as <math>x + iy</math>, where <math>x</math> is called the real part and <math>y</math> is called the imaginary part.<ref>{{Cite book |last=Weisstein |first=Eric W. |title=CRC Concise Encyclopedia of Mathematics |year=2002 |chapter=Complex Numbers}}</ref>

The [[distribution of primes]], described by the function <math>\pi</math> that counts all primes up to a given real number, is unpredictable and is a major subject of study in number theory. Elementary 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. The prime number theorem in analytic number theory provides a formalisation of the notion that prime numbers appear less commonly as their numerical value increases. One distribution states, informally, that the function <math>\frac{x}{\log(x)}</math> approximates <math>\pi(x)</math>. Another distribution involves an offset logarithmic integral which converges to <math>\pi(x)</math> more quickly.<ref name=":5" />
[[File:Riemann_Explicit_Formula.gif|thumb|Corrections to an [[Prime-counting function#Exact form|estimate]] of the prime-counting function using zeros of the zeta function.]]
The [[zeta function]] has been demonstrated to be connected to the distribution of primes. It is defined as the series<math display="block"> \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s} = \frac{1}{1^s} + \frac{1}{2^s} + \frac{1}{3^s} + \cdots</math>that converges if <math> s</math> is greater than 1. Euler demonstrated a link involving the infinite product over all prime numbers, expressed as the identity <math display="block">\zeta(s) = \prod_{p \text{ prime}} \left(1 - \frac{1}{p^{s}}\right)^{-1}.</math>Riemann extended the definition to a complex variable and conjectured that all nontrivial cases (<math>0 < \Re(s) < 1</math>) where the function returns a zero are those in which the real part of <math>s</math> is equal to <math display="inline">\frac12</math>. He established a connection between the nontrivial zeroes and the prime-counting function. In what is now recognised as the unsolved [[Riemann hypothesis]], a solution to it would imply direct consequences for understanding the distribution of primes.<ref>{{Cite book |last=Tanton |first=James |title=Encyclopedia of Mathematics |year=2005 |chapter=Zeta function}}</ref>

One may ask analytic questions about [[algebraic number]]s, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define [[prime ideal]]s (generalizations of [[prime number]]s in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question [[Landau prime ideal theorem|can be answered]] by means of an examination of [[Dedekind zeta function]]s, which are generalizations of the [[Riemann zeta function]], a key analytic object at the roots of the subject.<ref>{{harvnb|Granville|2008|loc=section 3}}: "[Riemann] defined what we now call the Riemann zeta function [...] Riemann's deep work gave birth to our subject [...]"</ref> This is an example of a general procedure in analytic number theory: deriving information about the distribution of a [[sequence]] (here, prime ideals or prime numbers) from the analytic behavior of an appropriately constructed complex-valued function.<ref name=":0">See, for example, {{harvnb|Montgomery|Vaughan|2007}}, p. 1.</ref>

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.

Some subjects generally considered to be part of analytic number theory (e.g., [[sieve theory]]) are better covered by the second rather than the first definition.<ref group="note">Sieve theory figures as one of the main subareas of analytic number theory in many standard treatments; see, for instance, {{harvnb|Iwaniec|Kowalski|2004}} or {{harvnb|Montgomery|Vaughan|2007}}</ref> Small sieves, for instance, use little analysis and yet still belong to analytic number theory.<ref group="note">This is the case for some combinatorial sieves such as the [[Brun sieve]], rather than for [[Large sieve|large sieves]]. The study of the latter now includes ideas from [[Harmonic analysis|harmonic]] and [[functional analysis]].</ref>

=== Algebraic number theory ===
{{Main|Algebraic number theory}}

An ''algebraic number'' is any [[complex number]] that is a solution to some polynomial equation <math>f(x)=0</math> with rational coefficients; for example, every solution <math>x</math> of <math>x^5 + (11/2) x^3 - 7 x^2 + 9 = 0 </math> is an algebraic number. Fields of algebraic numbers are also called ''[[algebraic number field]]s'', or shortly ''[[number field]]s''. Algebraic number theory studies algebraic number fields.{{sfn|Milne|2017|p=2}}

It could be argued that the simplest kind of number fields, namely [[Quadratic field|quadratic fields]], were already studied by Gauss, as the discussion of quadratic forms in ''Disquisitiones Arithmeticae'' can be restated in terms of [[ideal (ring theory)|ideals]] and
[[Norm (mathematics)|norms]] in quadratic fields. (A ''quadratic field'' consists of all
numbers of the form <math> a + b \sqrt{d}</math>, where
<math>a</math> and <math>b</math> are rational numbers and <math>d</math>
is a fixed rational number whose square root is not rational.)
For that matter, the eleventh-century [[chakravala method]] amounts—in modern terms—to an algorithm for finding the units of a real quadratic number field. However, neither Bhāskara nor Gauss knew of number fields as such.

The grounds of the subject were set in the late nineteenth century, when ''ideal numbers'', the ''theory of ideals'' and ''valuation theory'' were introduced; these are three complementary ways of dealing with the lack of unique factorization in algebraic number fields. (For example, in the field generated by the rationals
and <math> \sqrt{-5}</math>, the number <math>6</math> can be factorised both as <math> 6 = 2 \cdot 3</math> and
<math> 6 = (1 + \sqrt{-5}) ( 1 - \sqrt{-5})</math>; all of <math>2</math>, <math>3</math>, <math>1 + \sqrt{-5}</math> and
<math> 1 - \sqrt{-5}</math>
are irreducible, and thus, in a naïve sense, analogous to primes among the integers.) The initial impetus for the development of ideal numbers (by [[Ernst Kummer|Kummer]]) seems to have come from the study of higher reciprocity laws,{{sfn|Edwards|2000|p=79}} that is, generalizations of [[quadratic reciprocity]].

Number fields are often studied as extensions of smaller number fields: a field ''L'' is said to be an ''extension'' of a field ''K'' if ''L'' contains ''K''.
(For example, the complex numbers ''C'' are an extension of the reals ''R'', and the reals ''R'' are an extension of the rationals ''Q''.)
Classifying the possible extensions of a given number field is a difficult and partially open problem. Abelian extensions—that is, extensions ''L'' of ''K'' such that the [[Galois group]]<ref group="note">The Galois group of an extension ''L/K'' consists of the operations ([[isomorphisms]]) that send elements of L to other elements of L while leaving all elements of K fixed.
Thus, for instance, ''Gal(C/R)'' consists of two elements: the identity element
(taking every element ''x''&nbsp;+&nbsp;''iy'' of ''C'' to itself) and complex conjugation
(the map taking each element ''x''&nbsp;+&nbsp;''iy'' to ''x''&nbsp;−&nbsp;''iy'').
The Galois group of an extension tells us many of its crucial properties. The study of Galois groups started with [[Évariste Galois]]; in modern language, the main outcome of his work is that an equation ''f''(''x'')&nbsp;=&nbsp;0 can be solved by radicals
(that is, ''x'' can be expressed in terms of the four basic operations together
with square roots, cubic roots, etc.) if and only if the extension of the rationals by the roots of the equation ''f''(''x'')&nbsp;=&nbsp;0 has a Galois group that is [[solvable group|solvable]]
in the sense of group theory. ("Solvable", in the sense of group theory, is a simple property that can be checked easily for finite groups.)</ref> Gal(''L''/''K'') of ''L'' over ''K'' is an [[abelian group]]—are relatively well understood.
Their classification was the object of the programme of [[class field theory]], which was initiated in the late nineteenth century (partly by [[Leopold Kronecker|Kronecker]] and [[Gotthold Eisenstein|Eisenstein]]) and carried out largely in 1900–1950.

An example of an active area of research in algebraic number theory is [[Iwasawa theory]]. The [[Langlands program]], one of the main current large-scale research plans in mathematics, is sometimes described as an attempt to generalise class field theory to non-abelian extensions of number fields.

=== Diophantine geometry ===
{{Main|Diophantine geometry}}

The central problem of Diophantine geometry is to determine when a [[Diophantine equation]] has integer or rational solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.

For example, an equation in two variables defines a curve in the plane. More generally, an equation or system of equations in two or more variables defines a [[algebraic curve|curve]], a [[algebraic surface|surface]], or some other such object in {{math|''n''}}-dimensional space. In Diophantine geometry, one asks whether there are any ''rational points'' (points all of whose coordinates are rationals) or
''integral points'' (points all of whose coordinates are integers) on the curve or surface. If there are any such points, the next step is to ask how many there are and how they are distributed. A basic question in this direction is whether there are finitely
or infinitely many rational points on a given curve or surface.

Consider, for instance, the [[Pythagorean equation]] <math>x^2+y^2 = 1</math>. One would like to know its rational solutions, namely <math>(x,y)</math> such that ''x'' and ''y'' are both rational. This is the same as asking for all integer solutions
to <math>a^2 + b^2 = c^2</math>; any solution to the latter equation gives us a solution <math>x = a/c</math>, <math>y = b/c</math> to the former. It is also the
same as asking for all points with rational coordinates on the curve described by <math>x^2 + y^2 = 1</math> (a circle of radius 1 centered on the origin).

[[File:ECClines-3.svg|thumb|Two examples of [[elliptic curve]]s, that is, curves of genus 1 having at least one rational point.]]

The rephrasing of questions on equations in terms of points on curves is felicitous. The finiteness or not of the number of rational or integer points on an algebraic curve (that is, rational or integer solutions to an equation <math>f(x,y)=0</math>, where <math>f</math> is a polynomial in two variables) depends crucially on the [[genus (mathematics)|genus]] of the curve.<ref group="note">The ''genus'' can be defined as follows: allow the variables in <math>f(x,y)=0</math> to be complex numbers; then <math>f(x,y)=0</math> defines a 2-dimensional surface in (projective) 4-dimensional space (since two complex variables can be decomposed into four real variables; that is, four dimensions). The number of doughnut-like holes in the surface is called the ''genus'' of the curve of equation <math>f(x,y)=0</math>.</ref> A major achievement of this approach is [[Wiles's proof of Fermat's Last Theorem]], for which other geometrical notions are just as crucial.

There is also the closely linked area of [[Diophantine approximations]]: given a number <math>x</math>, determine how well it can be approximated by rational numbers. One seeks approximations that are good relative to the amount of space required to write the rational number: call <math>a/q</math> (with <math>\gcd(a,q)=1</math>) a good approximation to <math>x</math> if <math>|x-a/q|<\frac{1}{q^c}</math>, where <math>c</math> is large. This question is of special interest if <math>x</math> is an algebraic number. If <math>x</math> cannot be approximated well, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of [[Glossary of arithmetic and diophantine geometry#H|height]]) are critical both in Diophantine geometry and in the study of Diophantine approximations. This question is also of special interest in [[transcendental number theory]]: if a number can be approximated better than any algebraic number, then it is a [[transcendental number]]. It is by this argument that [[Pi|{{pi}}]] and [[e (mathematical constant)|e]] have been shown to be transcendental.

Diophantine geometry should not be confused with the [[geometry of numbers]], which is a collection of graphical methods for answering certain questions in algebraic number theory. [[Arithmetic geometry]] is a contemporary term for the same domain covered by Diophantine geometry, particularly when one wishes to emphasize the connections to modern algebraic geometry (for example, in [[Faltings's theorem]]) rather than to techniques in Diophantine approximations.

=== Other subfields ===
{{Main|Probabilistic number theory}}

Probabilistic number theory starts with questions such as the following: Take an integer {{mvar|n}} at random between one and a million. How likely is it to be prime? (this is just another way of asking how many primes there are between one and a million). How many prime divisors will {{mvar|n}} have on average? What is the probability that it will have many more or many fewer divisors or prime divisors than the average?{{Main|Arithmetic combinatorics|Additive number theory}}

Combinatorics in number theory starts with questions like the following: Does a fairly "thick" [[infinite set]] <math>A</math> contain many elements in arithmetic progression: <math>a</math>,
<math>a+b, a+2 b, a+3 b, \ldots, a+10b</math>? Should it be possible to write large integers as sums of elements of <math>A</math>?{{Main|Computational number theory}}[[File:Computer History Museum (4145886786).jpg|thumb|A [[Lehmer sieve]], a primitive [[digital computer]] used to find [[primes]] and solve simple [[Diophantine equations]]]]There are two main questions: "Can this be computed?" and "Can it be computed rapidly?" Anyone can test whether a number is prime or, if it is not, split it into prime factors; doing so rapidly is another matter. Fast algorithms for [[primality test|testing primality]] are now known, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.