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[2] [3] The statement that every prime p of the form + is the sum of two squares is sometimes called Girard's theorem. [4] For his part, Fermat wrote an elaborate version of the statement (in which he also gave the number of possible expressions of the powers of p as a sum of two squares) in a letter to Marin Mersenne dated December 25, 1640 ...
Two-square theorem — Denote the number of divisors of as (), and write () for the number of those divisors with . Let n = 2 f p 1 r 1 p 2 r 2 ⋯ q 1 s 1 q 2 s 2 ⋯ {\displaystyle n=2^{f}p_{1}^{r_{1}}p_{2}^{r_{2}}\cdots q_{1}^{s_{1}}q_{2}^{s_{2}}\cdots } where p i ≡ 1 mod 4 , q i ≡ 3 mod 4 {\displaystyle p_{i}\equiv 1{\bmod {4}},\ q_{i ...
Girard was the first to state, in 1625, that each prime of the form 1 mod 4 is the sum of two squares. [3] (See Fermat's theorem on sums of two squares.) It was said that he was quiet-natured and, unlike most mathematicians, did not keep a journal for his personal life. In the opinion of Charles Hutton, [4] Girard was
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.
The works of the 17th-century mathematician Pierre de Fermat engendered many theorems. Fermat's theorem may refer to one of the following theorems: Fermat's Last Theorem, about integer solutions to a n + b n = c n; Fermat's little theorem, a property of prime numbers; Fermat's theorem on sums of two squares, about primes expressible as a sum of ...
Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation + = if n is an integer greater than two (n > 2).. Over time, this simple assertion became one of the most famous unproved claims in mathematics.
Fermat's little theorem states that if p is prime and a is not divisible by p, then a p − 1 ≡ 1 ( mod p ) . {\displaystyle a^{p-1}\equiv 1{\pmod {p}}.} If one wants to test whether p is prime, then we can pick random integers a not divisible by p and see whether the congruence holds.
In number theory, Fermat's little theorem states that if p is a prime number, then for any integer a, the number a p − a is an integer multiple of p. In the notation of modular arithmetic , this is expressed as a p ≡ a ( mod p ) . {\displaystyle a^{p}\equiv a{\pmod {p}}.}