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Fermat's theorem gives only a necessary condition for extreme function values, as some stationary points are inflection points (not a maximum or minimum). The function's second derivative , if it exists, can sometimes be used to determine whether a stationary point is a maximum or minimum.
Fermat's theorem on sums of two squares, about primes expressible as a sum of squares; Fermat's theorem (stationary points), about local maxima and minima of differentiable functions; Fermat's principle, about the path taken by a ray of light; Fermat polygonal number theorem, about expressing integers as a sum of polygonal numbers
Euler's theorem is a generalization of Fermat's little theorem: For any modulus n and any integer a coprime to n, one has (), where φ(n) denotes Euler's totient function (which counts the integers from 1 to n that are coprime to n).
By Fermat's theorem, all local maxima and minima of a continuous function occur at critical points. Therefore, to find the local maxima and minima of a differentiable function, it suffices, theoretically, to compute the zeros of the gradient and the eigenvalues of the Hessian matrix at these zeros.
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, b, and c satisfy the equation a n + b n = c n for any integer value of n greater than 2. The cases n = 1 and n = 2 have been known since antiquity to have infinitely many solutions. [1]
[1] [2] [3] Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions. As defined in set theory , the maximum and minimum of a set are the greatest and least elements in the set, respectively.
Fermat's theorem on sums of two squares is strongly related with the theory of Gaussian primes. A Gaussian integer is a complex number a + i b {\displaystyle a+ib} such that a and b are integers. The norm N ( a + i b ) = a 2 + b 2 {\displaystyle N(a+ib)=a^{2}+b^{2}} of a Gaussian integer is an integer equal to the square of the absolute value ...
Euler proved Newton's identities, Fermat's little theorem, Fermat's theorem on sums of two squares, and made distinct contributions to the Lagrange's four-square theorem. He also invented the totient function φ(n) which assigns to a positive integer n the number of positive integers less than n and coprime to n.