Editing Number theory (section)

==== 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.