Fermat–Catalan Conjecture
In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation has only finitely many solutions (''a'', ''b'', ''c'', ''m'', ''n'', ''k'') with distinct triplets of values (''a''''m'', ''b''''n'', ''c''''k'') where ''a'', ''b'', ''c'' are positive coprime integers and ''m'', ''n'', ''k'' are positive integers satisfying The inequality on ''m'', ''n'', and ''k'' is a necessary part of the conjecture. Without the inequality there would be infinitely many solutions, for instance with ''k'' = 1 (for any ''a'', ''b'', ''m'', and ''n'' and with ''c'' = ''a''''m'' + ''b''''n''), with ''m''=''n''=''k''=2 (for the infinitely many Pythagorean triple A Pythagorean triple consists of three positive integers , , and , such that . Such a triple is commonly written , a well-known example is . If is a Pythagorean triple, then so is for any positive integer . A triangle whose side lengths are ...
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Number Theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example, rational numbers), or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory can often be understood through the study of Complex analysis, analytical objects, such as the Riemann zeta function, that encode properties of the integers, primes or other number-theoretic objects in some fashion (analytic number theory). One may also study real numbers in relation to rational numbers, as for instance how irrational numbers can be approximated by fractions (Diophantine approximation). Number theory is one of the oldest branches of mathematics alongside geometry. One quirk of number theory is ...
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