What is Birch and Swinnerton-Dyer conjecture?
{{no sources|date=October 2022}} [[File:Elliptic curve1.png|thumb|An example of an elliptic curve]] In [[mathematics]], the '''Birch and Swinnerton-Dyer conjecture''' (often called the '''Birch–Swinnerton-Dyer conjecture''') describes the set of [[Rational number|rational]] [[Root (math)|solutions to equations]] defining an [[elliptic curve]]. It is an open problem in the field of [[number theory]]. It is one of the most challenging mathematical problems in active research. == Background == An [[elliptic curve]] is a function of the form <math>y^2=x^3+ax+b</math>, where <math>a</math> and <math>b</math> are [[Rational number|rational numbers]]. The <math>x, y</math> values that satisfy the equation are called solutions of the elliptic curve. These solutions may be both rational, both [[Irrational number|irrational]], or one rational and the other irrational. The set of rational solutions is an [[abelian group]]. This means that given rational points <math>P</math> and <math>Q</math>, there is a way to produce the sum <math>P+Q</math>, and this is another rational point. In 1922, Louis Mordell proved that the set of rational solutions is a finitely generated abelian group. This means that any rational point <math>P</math> can be written as a finite combination of certain generating points. For example, the points <math>P_1=(-2,3)</math> and <math>P_2=(-1,4)</math> are generators of the rational points of <math>y^2=x^3+17</math>. This means that all rational points on it, even very complicated points, can be written in terms of these two points. For example, we have: <math>rac{170914680007433755273838144758012}{111693884047097647373766735954481}, -rac{5355483650472998648137959899973372475907238039375}{1180440469197546185983425190055773135812433974871}= -5 P_1 + 5 P_2</math> == References == {{L-functions-footer}} [[Category:Conjectures]] [[Category:Number theory]]