Number Theory - Revision history

Khanh: /* History */

2022-01-06T00:23:20Z

History

Khanh at 00:20, 6 January 2022

2022-01-06T00:20:12Z

Khanh: /* Diophantine geometry */

2022-01-06T00:17:30Z

Diophantine geometry

Khanh: /* Early modern number theory */

2022-01-06T00:15:22Z

Early modern number theory

Khanh: /* Other subfields */

2022-01-06T00:13:04Z

Other subfields

Khanh: /* Diophantine geometry */

2022-01-06T00:09:01Z

Diophantine geometry

Khanh: /* Analytic number theory */

2022-01-06T00:05:29Z

Analytic number theory

Khanh: /* Elementary number theory */

2022-01-06T00:02:53Z

Elementary number theory

Khanh: /* Fermat */

2022-01-05T23:58:43Z

Fermat

Khanh: /* Western Europe in the Middle Ages */

2022-01-05T23:57:30Z

Western Europe in the Middle Ages

@@ Line 130: / Line 130: @@
 ===Maturity and division into subfields===
-[[Image:ErnstKummer.jpg|upright|thumb|[[Ernst Kummer]]]]
+[[Image:ErnstKummer.jpg|upright|thumb|Ernst Kummer]]
 [[Image:Peter Gustav Lejeune Dirichlet.jpg|upright|left|thumb|Peter Gustav Lejeune Dirichlet]]

@@ Line 75: / Line 75: @@
 ====Arithmetic in the Islamic golden age====
-[[File:Hevelius Selenographia frontispiece.png|upright|right|thumb|Al-Haytham as seen by the West: on the frontispiece of ''Selenographia'' Alhasen {{sic}} represents knowledge through reason and Galileo knowledge through the senses.]]
+[[File:Hevelius Selenographia frontispiece.png|upright|right|thumb|Al-Haytham as seen by the West: on the frontispiece of ''Selenographia'' Alhasen [sic] represents knowledge through reason and Galileo knowledge through the senses.]]
 In the early ninth century, the caliph Al-Ma'mun ordered translations of many Greek mathematical works and at least one Sanskrit work (the ''Sindhind'',
@@ Line 249: / Line 249: @@
 Some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven, as a solution to Hilbert's 10th problem, that there is no Turing machine which can solve all Diophantine equations. In particular, this means that, given a computably enumerable set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (We would necessarily be speaking of Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. We cannot prove that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.)

@@ Line 209: / Line 209: @@
 of genus 1 having at least one rational point. (Either graph can be seen as a slice of a torus in four-dimensional space.)]]
-The rephrasing of questions on equations in terms of points on curves turns out to be 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—turns out to depend crucially on the ''genus'' of the curve. The ''genus'' can be defined as follows:<ref group="note">If we want to study the curve <math>y^2 = x^3 + 7</math>. We allow ''x'' and ''y'' to be complex numbers: <math>(a + b i)^2 = (c + d i)^3 + 7</math>. This is, in effect, a set of two equations on four variables, since both the real
+The rephrasing of questions on equations in terms of points on curves turns out to be 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—turns out to depend crucially on the ''genus'' of the curve. 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). If we count the number of (doughnut) holes in the surface; we call this number the ''genus'' of <math>f(x,y)=0</math>. Other geometrical notions turn out to be just as crucial.
 There is also the closely linked area of Diophantine approximations: given a number <math>x</math>, then finding how well can it be approximated by rationals. (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: 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 well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) turn out to be 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 better approximated than any algebraic number, then it is a transcendental number. It is by this argument that <math< \pi </math> and e have been shown to be transcendental.

@@ Line 111: / Line 111: @@
 *''Quadratic forms''. Following Fermat's lead, Euler did further research on the question of which primes can be expressed in the form <math>x^2 + N y^2</math>, some of it prefiguring quadratic reciprocity.
 *''Diophantine equations''. Euler worked on some Diophantine equations of genus 0 and 1. In particular, he studied Diophantus's work; he tried to systematise it, but the time was not yet ripe for such an endeavour—algebraic geometry was still in its infancy. He did notice there was a connection between Diophantine problems and elliptic integrals, whose study he had himself initiated.
-[[File:Andrew wiles1-3.jpg|thumb|upright|"Here was a problem, that I, a 10 year old, could understand, and I knew from that moment that I would never let it go. I had to solve it."<ref name=pbs>{{cite web|url=https://www.pbs.org/wgbh/nova/physics/andrew-wiles-fermat.html|title=Andrew Wiles on Solving Fermat|publisher=[[WGBH-TV|WGBH]]|access-date=16 March 2016}}</ref> – Sir [[Andrew Wiles]] about [[Wiles's proof of Fermat's Last Theorem|his proof]] of [[Fermat's Last Theorem]].]]
+[[File:Andrew wiles1-3.jpg|thumb|upright|"Here was a problem, that I, a 10 year old, could understand, and I knew from that moment that I would never let it go. I had to solve it." – Sir Andrew Wiles about his proof of Fermat's Last Theorem.]]
 ====Lagrange, Legendre, and Gauss====

@@ Line 222: / Line 222: @@
 ==Other subfields==
-The areas below date from no earlier than the mid-twentieth century, even if they are based on older material. For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of [[computability]] dates only from the 1930s and 1940s, and [[computational complexity theory]] from the 1970s.
+The areas below date from no earlier than the mid-twentieth century, even if they are based on older material. For example, as is explained below, the matter of algorithms in number theory is very old, in some sense older than the concept of proof; at the same time, the modern study of computability dates only from the 1930s and 1940s, and computational complexity theory from the 1970s.
 ===Probabilistic number theory===
-Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually [[statistical independence|independent]]. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite.
+Much of probabilistic number theory can be seen as an important special case of the study of variables that are almost, but not quite, mutually independent. For example, the event that a random integer between one and a million be divisible by two and the event that it be divisible by three are almost independent, but not quite.
-It is sometimes said that [[probabilistic combinatorics]] uses the fact that whatever happens with probability greater than <math>0</math> must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.
+It is sometimes said that probabilistic combinatorics uses the fact that whatever happens with probability greater than <math>0</math> must happen sometimes; one may say with equal justice that many applications of probabilistic number theory hinge on the fact that whatever is unusual must be rare. If certain algebraic objects (say, rational or integer solutions to certain equations) can be shown to be in the tail of certain sensibly defined distributions, it follows that there must be few of them; this is a very concrete non-probabilistic statement following from a probabilistic one.
-At times, a non-rigorous, probabilistic approach leads to a number of [[heuristic]] algorithms and open problems, notably [[Cramér's conjecture]].
+At times, a non-rigorous, probabilistic approach leads to a number of heuristic algorithms and open problems, notably Cramér's conjecture.
 ===Arithmetic combinatorics===
 If we begin from a fairly "thick" infinite set <math>A</math>, does it contain many elements in arithmetic progression: <math>a</math>,
 <math>a+b, a+2 b, a+3 b, \ldots, a+10b</math>, say? Should it be possible to write large integers as sums of elements of <math>A</math>?
-These questions are characteristic of ''arithmetic combinatorics''. This is a presently coalescing field; it subsumes ''[[additive number theory]]'' (which concerns itself with certain very specific sets <math>A</math> of arithmetic significance, such as the primes or the squares) and, arguably, some of the ''[[geometry of numbers]]'',
+These questions are characteristic of ''arithmetic combinatorics''. This is a presently coalescing field; it subsumes ''additive number theory'' (which concerns itself with certain very specific sets <math>A</math> of arithmetic significance, such as the primes or the squares) and, arguably, some of the ''geometry of numbers'',
-together with some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with [[ergodic theory]], [[finite group theory]], [[model theory]], and other fields. The term ''additive combinatorics'' is also used; however, the sets <math>A</math> being studied need not be sets of integers, but rather subsets of non-commutative [[Group (mathematics)|groups]], for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of [[ring (mathematics)|rings]], in which case the growth of <math>A+A</math> and <math>A</math>·<math>A</math> may be
+together with some rapidly developing new material. Its focus on issues of growth and distribution accounts in part for its developing links with ergodic theory, finite group theory, model theory, and other fields. The term ''additive combinatorics'' is also used; however, the sets <math>A</math> being studied need not be sets of integers, but rather subsets of non-commutative groups, for which the multiplication symbol, not the addition symbol, is traditionally used; they can also be subsets of rings, in which case the growth of <math>A+A</math> and <math>A</math>·<math>A</math> may be
 compared.
 ===Computational number theory===
-{{main|Computational number theory}}[[Image:Lehmer sieve.jpg|thumb|A [[Lehmer sieve]], a primitive [[digital computer]] used to find [[Prime number|primes]] and solve simple [[Diophantine equations]].]]While the word ''algorithm'' goes back only to certain readers of [[al-Khwārizmī]], careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
+[[Image:Lehmer sieve.jpg|thumb|A Lehmer sieve, a primitive digital computer used to find primes and solve simple Diophantine equations.]]
-An early case is that of what we now call the [[Euclidean algorithm]]. In its basic form (namely, as an algorithm for computing the [[greatest common divisor]]) it appears as Proposition 2 of Book VII in ''[[Euclid's Elements|Elements]]'', together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation <math>a x + b y = c</math>,
+While the word ''algorithm'' goes back only to certain readers of al-Khwārizmī, careful descriptions of methods of solution are older than proofs: such methods (that is, algorithms) are as old as any recognisable mathematics—ancient Egyptian, Babylonian, Vedic, Chinese—whereas proofs appeared only with the Greeks of the classical period.
-or, what is the same, for finding the quantities whose existence is assured by the [[Chinese remainder theorem]]) it first appears in the works of [[Aryabhata#Indeterminate equations|Āryabhaṭa]] (5th–6th century CE) as an algorithm called
+An early case is that of what we now call the Euclidean algorithm. In its basic form (namely, as an algorithm for computing the greatest common divisor) it appears as Proposition 2 of Book VII in ''Elements'', together with a proof of correctness. However, in the form that is often used in number theory (namely, as an algorithm for finding integer solutions to an equation <math>a x + b y = c</math>,
 ''kuṭṭaka'' ("pulveriser"), without a proof of correctness.
-There are two main questions: "Can we compute this?" and "Can we compute it 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. We now know fast algorithms for [[primality test|testing primality]], but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.
+There are two main questions: "Can we compute this?" and "Can we compute it 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. We now know fast algorithms for testing primality, but, in spite of much work (both theoretical and practical), no truly fast algorithm for factoring.
-The difficulty of a computation can be useful: modern protocols for [[cryptography|encrypting messages]] (for example, [[RSA (algorithm)|RSA]]) depend on functions that are known to all, but whose inverses are known only to a chosen few, and would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorized. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems.
+The difficulty of a computation can be useful: modern protocols for encrypting messages (for example, RSA) depend on functions that are known to all, but whose inverses are known only to a chosen few, and would take one too long a time to figure out on one's own. For example, these functions can be such that their inverses can be computed only if certain large integers are factorized. While many difficult computational problems outside number theory are known, most working encryption protocols nowadays are based on the difficulty of a few number-theoretical problems.
-Some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven, as a solution to [[Hilbert's 10th problem]], that there is no [[Turing machine]] which can solve all Diophantine equations.<ref>{{cite book | editor=Felix E. Browder | editor-link=Felix Browder | title=Mathematical Developments Arising from Hilbert Problems | series=[[Proceedings of Symposia in Pure Mathematics]] | volume=XXVIII.2 | year=1976 | publisher=[[American Mathematical Society]] | isbn=978-0-8218-1428-4 | pages=323–78 | first1=Martin | last1=Davis | author-link1=Martin Davis (mathematician) | first2=Yuri | last2=Matiyasevich | author-link2=Yuri Matiyasevich | first3=Julia | last3=Robinson | author-link3=Julia Robinson | chapter=Hilbert's Tenth Problem: Diophantine Equations: Positive Aspects of a Negative Solution | zbl=0346.02026 }} Reprinted in ''The Collected Works of Julia Robinson'', [[Solomon Feferman]], editor, pp. 269–378, American Mathematical Society 1996.</ref> In particular, this means that, given a [[computably enumerable]] set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (We would necessarily be speaking of Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. We cannot prove that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.)
+Some things may not be computable at all; in fact, this can be proven in some instances. For instance, in 1970, it was proven, as a solution to Hilbert's 10th problem, that there is no Turing machine which can solve all Diophantine equations. In particular, this means that, given a computably enumerable set of axioms, there are Diophantine equations for which there is no proof, starting from the axioms, of whether the set of equations has or does not have integer solutions. (We would necessarily be speaking of Diophantine equations for which there are no integer solutions, since, given a Diophantine equation with at least one solution, the solution itself provides a proof of the fact that a solution exists. We cannot prove that a particular Diophantine equation is of this kind, since this would imply that it has no solutions.)

@@ Line 191: / Line 191: @@
 ===Diophantine geometry===
-The central problem of ''Diophantine geometry'' is to determine when a [[Diophantine equation]] has solutions, and if it does, how many. The approach taken is to think of the solutions of an equation as a geometric object.
+The central problem of ''Diophantine geometry'' is to determine when a Diophantine equation has 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 ''n''-dimensional space. In Diophantine geometry, one asks whether there are any ''[[rational points]]'' (points all of whose coordinates are rationals) or
+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 curve, a surface or some other such object in ''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 if there are finitely
 or infinitely many rational points on a given curve (or surface).
-In the [[Pythagorean theorem|Pythagorean equation]] <math>x^2+y^2 = 1,</math>
+In the Pythagorean equation <math>x^2+y^2 = 1,</math>
 we would like to study its rational solutions, that is, its solutions
 <math>(x,y)</math> such that
@@ Line 206: / Line 206: @@
 described by <math>x^2 + y^2 = 1</math>. (This curve happens to be a circle of radius 1 around the origin.)
-[[Image:ECClines-3.svg|right|thumb|300px|Two examples of an [[elliptic curve]], that is, a curve
+[[Image:ECClines-3.svg|right|thumb|300px|Two examples of an elliptic curve, that is, a curve
-of genus 1 having at least one rational point. (Either graph can be seen as a slice of a [[torus]] in four-dimensional space.)]]
+of genus 1 having at least one rational point. (Either graph can be seen as a slice of a torus in four-dimensional space.)]]
 The rephrasing of questions on equations in terms of points on curves turns out to be 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—turns out to depend crucially on the ''genus'' of the curve. The ''genus'' can be defined as follows:<ref group="note">If we want to study the curve <math>y^2 = x^3 + 7</math>. We allow ''x'' and ''y'' to be complex numbers: <math>(a + b i)^2 = (c + d i)^3 + 7</math>. This is, in effect, a set of two equations on four variables, since both the real
 and the imaginary part on each side must match. As a result, we get a surface (two-dimensional) in four-dimensional space. After we choose a convenient hyperplane on which to project the surface (meaning that, say, we choose to ignore the coordinate ''a''), we can
 plot the resulting projection, which is a surface in ordinary three-dimensional space. It
-then becomes clear that the result is a [[torus]], loosely speaking, the surface of a doughnut (somewhat
+then becomes clear that the result is a torus, loosely speaking, the surface of a doughnut (somewhat
 stretched). A doughnut has one hole; hence the genus is 1.</ref> 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). If we count the number of (doughnut) holes in the surface; we call this number the ''genus'' of <math>f(x,y)=0</math>. Other geometrical notions turn out to be just as crucial.
-There is also the closely linked area of [[Diophantine approximations]]: given a number <math>x</math>, then finding how well can it be approximated by rationals. (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: 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 well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of [[Glossary of arithmetic and diophantine geometry#H|height]]) turn out to be 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 better approximated 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.
+There is also the closely linked area of Diophantine approximations: given a number <math>x</math>, then finding how well can it be approximated by rationals. (We are looking for approximations that are good relative to the amount of space that it takes to write the rational: 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 well approximated, then some equations do not have integer or rational solutions. Moreover, several concepts (especially that of height) turn out to be 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 better approximated than any algebraic number, then it is a transcendental number. It is by this argument that <math< \pi </math> and 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'', however, is a contemporary term
+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'', however, is a contemporary term
 for much the same domain as that covered by the term ''Diophantine geometry''. The term ''arithmetic geometry'' is arguably used
-most often when one wishes to emphasise the connections to modern algebraic geometry (as in, for instance, [[Faltings's theorem]]) rather than to techniques in Diophantine approximations.
+most often when one wishes to emphasise the connections to modern algebraic geometry (as in, for instance, Faltings's theorem) rather than to techniques in Diophantine approximations.
 ==Other subfields==

@@ Line 152: / Line 152: @@
 ===Analytic number theory===
-[[Image:Complex zeta.jpg|right|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]].]]
+[[Image:Complex zeta.jpg|right|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.]]
-[[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]].]]
+[[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'' may be defined
@@ Line 160: / Line 160: @@
 * in terms of its concerns, as the study within number theory of estimates on size and density, as opposed to identities.
-Some subjects generally considered to be part of analytic number theory, for example, [[sieve theory]], are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory.
+Some subjects generally considered to be part of analytic number theory, for example, sieve theory, are better covered by the second rather than the first definition: some of sieve theory, for instance, uses little analysis, yet it does belong to analytic number theory.
-The following are examples of problems in analytic number theory: the [[prime number theorem]], the [[Goldbach conjecture]] (or the [[twin prime conjecture]], or 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.
+The following are examples of problems in analytic number theory: the prime number theorem, the Goldbach conjecture (or the twin prime conjecture, or the Hardy–Littlewood conjectures), the Waring problem and the Riemann hypothesis. Some of the most important tools of analytic number theory are the circle method, sieve methods and L-functions (or, rather, the study of their properties). The theory of modular forms (and, more generally, automorphic forms) also occupies an increasingly central place in the toolbox of analytic number theory.
-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 ideals]] (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. 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.
+One may ask analytic questions about algebraic numbers, and use analytic means to answer such questions; it is thus that algebraic and analytic number theory intersect. For example, one may define prime ideals (generalizations of prime numbers in the field of algebraic numbers) and ask how many prime ideals there are up to a certain size. This question can be answered by means of an examination of Dedekind zeta functions, which are generalizations of the Riemann zeta function, a key analytic object at the roots of the subject.[81] 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.
 ===Algebraic number theory===

@@ Line 146: / Line 146: @@
 ==Main subdivisions==
 ===Elementary number theory===
-The term ''[[elementary proof|elementary]]'' generally denotes a method that does not use [[complex analysis]]. 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]]. The term is somewhat ambiguous: for example, proofs based on complex [[Tauberian theorem]]s (for example, [[Wiener–Ikehara theorem|Wiener–Ikehara]]) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an ''elementary'' proof may be longer and more difficult for most readers than a non-elementary one.
+The term ''elementary'' generally denotes a method that does not use complex analysis. For example, the prime number theorem was first proven using complex analysis in 1896, but an elementary proof was found only in 1949 by Erdős and Selberg. The term is somewhat ambiguous: for example, proofs based on complex Tauberian theorems (for example, Wiener–Ikehara) are often seen as quite enlightening but not elementary, in spite of using Fourier analysis, rather than complex analysis as such. Here as elsewhere, an ''elementary'' proof may be longer and more difficult for most readers than a non-elementary one.
-[[File:Paul Erdos with Terence Tao.jpg|thumb|270px|Number theorists [[Paul Erdős]] and [[Terence Tao]] in 1985, when Erdős was 72 and Tao was 10.]]
+[[File:Paul Erdos with Terence Tao.jpg|thumb|270px|Number theorists Paul Erdős and Terence Tao in 1985, when Erdős was 72 and Tao was 10.]]
 Number theory has the reputation of being a field many of whose results can be stated to the layperson. At the same time, 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.

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 Over his lifetime, Fermat made the following contributions to the field:
-* One of Fermat's first interests was [[perfect number]]s (which appear in Euclid, ''Elements'' IX) and [[amicable numbers]]; these topics led him to work on integer [[divisor]]s, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.
+* One of Fermat's first interests was perfect numbers (which appear in Euclid, ''Elements'' IX) and amicable numbers; these topics led him to work on integer divisors, which were from the beginning among the subjects of the correspondence (1636 onwards) that put him in touch with the mathematical community of the day.
 * In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.
 * Fermat's little theorem (1640): if ''a'' is not divisible by a prime ''p'', then <math>a^{p-1} \equiv 1 \bmod p.</math>

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 ====Western Europe in the Middle Ages====
-Other than a treatise on squares in arithmetic progression by [[Fibonacci]]—who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late [[Renaissance]], thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' ''[[Arithmetica]]''.<ref>[[Claude Gaspard Bachet de Méziriac|Bachet]], 1621, following a first attempt by [[Guilielmus Xylander|Xylander]], 1575</ref>
+Other than a treatise on squares in arithmetic progression by Fibonacci—who traveled and studied in north Africa and Constantinople—no number theory to speak of was done in western Europe during the Middle Ages. Matters started to change in Europe in the late Renaissance, thanks to a renewed study of the works of Greek antiquity. A catalyst was the textual emendation and translation into Latin of Diophantus' ''Arithmetica''.
 ===Early modern number theory===