Editing Number theory (section)

==== 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]].