Editing Arithmetic (section)

==== Number theory ====
{{main|Number theory}}

Number theory studies the structure and properties of integers as well as the relations and laws between them.<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Grigorieva|2018|pp=[https://books.google.com/books?id=mEpjDwAAQBAJ&pg=PR8 viii–ix]}} | {{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 15]}} }}</ref> Some of the main branches of modern number theory include [[elementary number theory]], [[analytic number theory]], [[algebraic number theory]], and [[geometric number theory]].<ref>{{multiref | {{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34]}} | {{harvnb|Yan|2002|p=12}} }}</ref> Elementary number theory studies aspects of integers that can be investigated using elementary methods. Its topics include [[divisibility]], [[factorization]], and [[primality]].<ref>{{multiref | {{harvnb|Page|2003|pp=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 18–19, 34]}} | {{harvnb|Bukhshtab|Nechaev|2014}} }}</ref> Analytic number theory, by contrast, relies on techniques from analysis and calculus. It examines problems like [[prime number theorem|how prime numbers are distributed]] and the claim that [[Goldbach's conjecture|every even number is a sum of two prime numbers]].<ref>{{multiref | {{harvnb|Page|2003|p=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34]}} | {{harvnb|Karatsuba|2014}} }}</ref> Algebraic number theory employs algebraic structures to analyze the properties of and relations between numbers. Examples are the use of [[Field (mathematics)|fields]] and [[Ring (mathematics)|rings]], as in [[algebraic number field]]s like the [[ring of integers]]. Geometric number theory uses concepts from geometry to study numbers. For instance, it investigates how lattice points with integer coordinates behave in a plane.<ref>{{multiref | {{harvnb|Page|2003|pp=[https://www.sciencedirect.com/science/article/abs/pii/B0122274105005032 34–35]}} | {{harvnb|Vinogradov|2019}} }}</ref> Further branches of number theory are [[probabilistic number theory]], which employs methods from [[probability theory]],<ref>{{harvnb|Kubilyus|2018}}</ref> [[combinatorial number theory]], which relies on the field of [[combinatorics]],<ref>{{harvnb|Pomerance|Sárközy|1995|p=[https://books.google.com/books?id=5ktBP5vUl5gC&pg=PA969 969]}}</ref> [[computational number theory]], which approaches number-theoretic problems with computational methods,<ref>{{harvnb|Pomerance|2010}}</ref> and applied number theory, which examines the application of number theory to fields like [[physics]], [[biology]], and [[cryptography]].<ref>{{multiref | {{harvnb|Yan|2002|pp=12, 303–305}} | {{harvnb|Yan|2013a|p=[https://books.google.com/books?id=74oBi4ys0UUC&pg=PA15 15]}} }}</ref>

Influential theorems in number theory include the [[fundamental theorem of arithmetic]], [[Euclid's theorem]], and [[Fermat's Last Theorem]].<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Křížek|Somer|Šolcová|2021|pp=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA23 23, 25, 37]}} }}</ref> According to the fundamental theorem of arithmetic, every integer greater than 1 is either a prime number or can be represented as a unique product of prime numbers. For example, the [[number 18]] is not a prime number and can be represented as <math>2 \times 3 \times 3</math>, all of which are prime numbers. The [[number 19]], by contrast, is a prime number that has no other prime factorization.<ref>{{multiref | {{harvnb|Křížek|Somer|Šolcová|2021|p=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA23 23]}} | {{harvnb|Riesel|2012|p=[https://books.google.com/books?id=ITvaBwAAQBAJ&pg=PA2 2]}} }}</ref> Euclid's theorem states that there are infinitely many prime numbers.<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Křížek|Somer|Šolcová|2021|p=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA25 25]}} }}</ref> Fermat's Last Theorem is the statement that no positive integer values exist for <math>a</math>, <math>b</math>, and <math>c</math> that solve the equation <math>a^n + b^n = c^n</math> if <math>n</math> is greater than <math>2</math>.<ref>{{multiref | {{harvnb|Bukhshtab|Nechaev|2016}} | {{harvnb|Křížek|Somer|Šolcová|2021|p=[https://books.google.com/books?id=tklEEAAAQBAJ&pg=PA37 37]}} }}</ref>