Editing Number theory (section)

=== Early modern number theory ===
==== Fermat ====
[[File:Pierre de Fermat.png|thumb|upright=0.8|[[Pierre de Fermat]]]]

[[Pierre de Fermat]] (1607–1665) never published his writings but communicated through correspondence instead. Accordingly, his work on number theory is contained almost entirely in letters to mathematicians and in private marginal notes.{{sfn|Weil|1984|pp=45–46}} Although he drew inspiration from classical sources, in his notes and letters Fermat scarcely wrote any proofs—he had no models in the area.<ref>{{harvnb|Weil|1984|p=118}}. This was more so in number theory than in other areas ({{harvnb|Mahoney|1994|p=|pp=283-289}}). Bachet's own proofs were "ludicrously clumsy" {{harv|Weil|1984|p=33}}.</ref>

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]];<ref group="note">Perfect and especially amicable numbers are of little or no interest nowadays. The same was not true in earlier times—whether in the West or the Arab-speaking world—due in part to the importance given to them by the Neopythagorean [[Nicomachus]] (c.&nbsp;100&nbsp;AD), who wrote a very elementary but influential book entitled ''[[Introduction to Arithmetic]]''. See {{harvnb|van der Waerden|1961|loc=Ch. IV}}.</ref> 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.<ref>{{harvnb|Mahoney|1994|pp=48, 53–54}}. The initial subjects of Fermat's correspondence included divisors ("aliquot parts") and many subjects outside number theory; see the list in the letter from Fermat to Roberval, 22.IX.1636, {{harvnb|Tannery|Henry|1891|loc=Vol. II, pp. 72, 74}}, cited in {{harvnb|Mahoney|1994|p=54}}.</ref>
* In 1638, Fermat claimed, without proof, that all whole numbers can be expressed as the sum of four squares or fewer.<ref>{{Cite encyclopedia |url=https://books.google.com/books?id=5tFFDwAAQBAJ |title=Numbers and Measurements |last1=Faulkner |first1=Nicholas |last2=Hosch |first2=William L. |date=2017 |encyclopedia=Encyclopaedia Britannica |isbn=978-1-5383-0042-8 |access-date=2019-08-06}}</ref>
* [[Fermat's little theorem]] (1640):<ref>{{harvnb|Tannery|Henry|1891|loc=Vol. II, p. 209}}, Letter XLVI from Fermat to Frenicle, 1640,
cited in {{harvnb|Weil|1984|p=56}}</ref> if ''a'' is not divisible by a prime ''p'', then <math>a^{p-1} \equiv 1 \bmod p.</math><ref group="note">Here, as usual, given two integers ''a'' and ''b'' and a non-zero integer ''m'', we write <math>a \equiv b \bmod m</math> (read "''a'' is congruent to ''b'' modulo ''m''") to mean that ''m'' divides ''a''&nbsp;−&nbsp;''b'', or, what is the same, ''a'' and ''b'' leave the same residue when divided by ''m''. This notation is actually much later than Fermat's; it first appears in section 1 of [[Gauss]]'s ''{{lang|la|[[Disquisitiones Arithmeticae]]}}''. Fermat's little theorem is a consequence of the [[Lagrange's theorem (group theory)|fact]] that the [[Order (group theory)|order]] of an element of a group divides the [[Order (group theory)|order]] of the group. The modern proof would have been within Fermat's means (and was indeed given later by Euler), even though the modern concept of a group came long after Fermat or Euler. (It helps to know that inverses exist modulo ''p'', that is, given ''a'' not divisible by a prime ''p'', there is an integer ''x'' such that <math> x a \equiv 1 \bmod p</math>); this fact (which, in modern language, makes the residues mod ''p'' into a group, and which was already known to Āryabhaṭa; see [[#Indian school: Āryabhaṭa, Brahmagupta, Bhāskara|above]]) was familiar to Fermat thanks to its rediscovery by [[Bachet]] {{harv|Weil|1984|p=7}}. Weil goes on to say that Fermat would have recognised that Bachet's argument is essentially Euclid's algorithm.</ref>
* If ''a'' and ''b'' are [[coprime]], then <math>a^2 + b^2</math> is not divisible by any prime congruent to −1 modulo 4;<ref>{{harvnb|Tannery|Henry|1891|loc=Vol. II, p. 204}}, cited in {{harvnb|Weil|1984|p=63}}. All of the following citations from Fermat's ''Varia Opera'' are taken from {{harvnb|Weil|1984|loc=Chap. II}}. The standard Tannery & Henry work includes a revision of Fermat's posthumous ''Varia Opera Mathematica'' originally prepared by his son {{harv|Fermat|1679}}.</ref> and every prime congruent to 1 modulo 4 can be written in the form <math>a^2 + b^2</math>.{{sfn|Tannery|Henry|1891|loc=Vol. II, p. 213}} These two statements also date from 1640; in 1659, Fermat stated to Huygens that he had proven the latter statement by the [[method of infinite descent]].{{sfn|Tannery|Henry|1891|loc=Vol. II, p. 423}}
* In 1657, Fermat posed the problem of solving <math>x^2 - N y^2 = 1</math> as a challenge to English mathematicians. The problem was solved in a few months by Wallis and Brouncker.{{sfn|Weil|1984|p=92}} Fermat considered their solution valid, but pointed out they had provided an algorithm without a proof (as had Jayadeva and Bhaskara, though Fermat was not aware of this). He stated that a proof could be found by infinite descent.
* Fermat stated and proved (by infinite descent) in the appendix to ''Observations on Diophantus'' (Obs. XLV){{sfn |Tannery|Henry|1891|loc=Vol. I, pp. 340–341}} that <math>x^{4} + y^{4} = z^{4}</math> has no non-trivial solutions in the integers. Fermat also mentioned to his correspondents that <math>x^3 + y^3 = z^3</math> has no non-trivial solutions, and that this could also be proven by infinite descent.{{sfn|Weil|1984|p=115}} The first known proof is due to Euler (1753; indeed by infinite descent).{{sfn|Weil|1984|pp=115–116}}
* Fermat claimed ([[Fermat's Last Theorem]]) to have shown there are no solutions to <math>x^n + y^n = z^n</math> for all <math>n\geq 3</math>; this claim appears in his annotations in the margins of his copy of Diophantus.

==== Euler ====
[[File:Leonhard Euler.jpg|thumb|upright=0.8|Leonhard Euler]]

The interest of [[Leonhard Euler]] (1707–1783) in number theory was first spurred in 1729, when a friend of his, the amateur<ref group="note">Up to the second half of the seventeenth century, academic positions were very rare, and most mathematicians and scientists earned their living in some other way {{harv|Weil|1984|pp=159, 161}}. (There were already some recognisable features of professional ''practice'', viz., seeking correspondents, visiting foreign colleagues, building private libraries {{harv|Weil|1984|pp=160–161}}. Matters started to shift in the late seventeenth century {{harv|Weil|1984|p=161}}; scientific academies were founded in England (the [[Royal Society]], 1662) and France (the [[Académie des sciences]], 1666) and [[Russian Academy of Sciences|Russia]] (1724). Euler was offered a position at this last one in 1726; he accepted, arriving in St. Petersburg in 1727 ({{harvnb|Weil|1984|p=163}} and
{{harvnb|Varadarajan|2006|p=7}}).
In this context, the term ''amateur'' usually applied to Goldbach is well-defined and makes some sense: he has been described as a man of letters who earned a living as a spy {{harv|Truesdell|1984|p=xv}}; cited in {{harvnb|Varadarajan|2006|p=9}}). Notice, however, that Goldbach published some works on mathematics and sometimes held academic positions.</ref> [[Christian Goldbach|Goldbach]], pointed him towards some of Fermat's work on the subject.{{sfn|Weil|1984|pp=2, 172}}{{sfn|Varadarajan|2006|p=9}} This has been called the "rebirth" of modern number theory,{{sfn|Weil|1984|pp=1–2}} after Fermat's relative lack of success in getting his contemporaries' attention for the subject.<ref>{{harvnb|Weil|1984|p=2}} and {{harvnb|Varadarajan|2006|p=37}}</ref> Euler's work on number theory includes the following:<ref>{{harvnb|Varadarajan|2006|p=39}} and {{harvnb|Weil|1984|pp=176–189}}</ref>
* ''Proofs for Fermat's statements.'' This includes [[Fermat's little theorem]] (generalised by Euler to non-prime moduli); the fact that <math>p = x^2 + y^2</math> if and only if <math>p\equiv 1 \bmod 4</math>; initial work towards a proof that every integer is the sum of four squares (the first complete proof is by [[Joseph-Louis Lagrange]] (1770), soon improved by Euler himself{{sfn|Weil|1984|pp=178–179}}); the lack of non-zero integer solutions to <math>x^4 + y^4 = z^2</math> (implying the case ''n=4'' of Fermat's last theorem, the case ''n=3'' of which Euler also proved by a related method).
* ''[[Pell's equation]]'', first misnamed by Euler.<ref name="Eulpell">{{harvnb|Weil|1984|p=174}}. Euler was generous in giving credit to others {{harv|Varadarajan|2006|p=14}}, not always correctly.</ref> He wrote on the link between [[simple continued fraction|continued fractions]] and Pell's equation.{{sfn|Weil|1984|p=183}}
* ''First steps towards analytic number theory.'' In his work of sums of four squares, [[Partition function (number theory)|partitions]], [[pentagonal numbers]], and the [[Distribution (number theory)|distribution]] of prime numbers, Euler pioneered the use of what can be seen as analysis (in particular, infinite series) in number theory. Since he lived before the development of [[complex analysis]], most of his work is restricted to the formal manipulation of [[power series]]. He did, however, do some very notable (though not fully rigorous) early work on what would later be called the [[Riemann zeta function]].<ref>{{harvnb|Varadarajan|2006|pp=45–55}}; see also chapter III.</ref>
* ''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]].{{sfn|Varadarajan|2006|pp=44–47}}{{sfn|Weil|1984|pp=177–179}}{{sfn|Edwards|1983|pp=285–291}}
* ''Diophantine equations''. Euler worked on some Diophantine equations of genus 0 and 1.{{sfn|Varadarajan|2006|pp=55–56}}{{sfn|Weil|1984|pp=179–181}} 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.{{sfn|Weil|1984|p=181}} He did notice there was a connection between Diophantine problems and [[elliptic integral]]s,{{sfn|Weil|1984|p=181}} whose study he had himself initiated.

==== Lagrange, Legendre, and Gauss ====
[[File:Carl Friedrich Gauss 1840 by Jensen.jpg|upright=0.8|thumb|Carl Friedrich Gauss]]

[[Joseph-Louis Lagrange]] (1736–1813) was the first to give full proofs of some of Fermat's and Euler's work and observations; for instance, the [[four-square theorem]] and the basic theory of the misnamed "Pell's equation" (for which an algorithmic solution was found by Fermat and his contemporaries, and also by Jayadeva and Bhaskara II before them.) He also studied [[quadratic form]]s in full generality (as opposed to <math>m X^2 + n Y^2</math>), including defining their equivalence relation, showing how to put them in reduced form, etc.

[[Adrien-Marie Legendre]] (1752–1833) was the first to state the law of quadratic reciprocity. He also conjectured what amounts to the [[prime number theorem]] and [[Dirichlet's theorem on arithmetic progressions]]. He gave a full treatment of the equation <math>a x^2 + b y^2 + c z^2 = 0</math>{{sfn|Weil|1984|pp=327–328}} and worked on quadratic forms along the lines later developed fully by Gauss.{{sfn|Weil|1984|pp=332–334}} In his old age, he was the first to prove Fermat's Last Theorem for <math>n=5</math> (completing work by [[Peter Gustav Lejeune Dirichlet]], and crediting both him and [[Sophie Germain]]).{{sfn|Weil|1984|pp=337–338}}

[[Carl Friedrich Gauss]] (1777–1855) worked in a wide variety of fields in both mathematics and physics including number theory, analysis, differential geometry, geodesy, magnetism, astronomy and optics. The ''[[Disquisitiones Arithmeticae]]'' (1801), which he wrote three years earlier when he was 21, had an immense influence in the area of number theory and set its agenda for much of the 19th century. Gauss proved in this work the law of [[quadratic reciprocity]] and developed the theory of quadratic forms (in particular, defining their composition). He also introduced some basic notation ([[congruences]]) and devoted a section to computational matters, including primality tests.{{sfn|Goldstein|Schappacher|2007|p=14}} The last section of the ''Disquisitiones'' established a link between [[roots of unity]] and number theory:
<blockquote>The theory of the division of the circle...which is treated in sec. 7 does not belong by itself to arithmetic, but its principles can only be drawn from higher arithmetic.<ref>From the preface of ''Disquisitiones Arithmeticae''; the translation is taken from {{harvnb|Goldstein|Schappacher|2007|p=16}}</ref></blockquote>

In this way, Gauss arguably made forays towards [[Évariste Galois]]'s work and the area [[algebraic number theory]].