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In mathematics, the interior extremum theorem, also known as Fermat's theorem, is a theorem which states that at the local extrema of a differentiable function, its derivative is always zero. It belongs to the mathematical field of real analysis and is named after French mathematician Pierre de Fermat .
If 2 k + 1 is prime and k > 0, then k itself must be a power of 2, [1] so 2 k + 1 is a Fermat number; such primes are called Fermat primes. As of 2023 [update] , the only known Fermat primes are F 0 = 3 , F 1 = 5 , F 2 = 17 , F 3 = 257 , and F 4 = 65537 (sequence A019434 in the OEIS ).
every element x of GF(2) satisfies x + x = 0 and therefore −x = x; this means that the characteristic of GF(2) is 2; every element x of GF(2) satisfies x 2 = x (i.e. is idempotent with respect to multiplication); this is an instance of Fermat's little theorem. GF(2) is the only field with this property (Proof: if x 2 = x, then either x = 0 or ...
If two primes which end in 3 or 7 and surpass by 3 a multiple of 4 are multiplied, then their product will be composed of a square and the quintuple of another square. In other words, if p, q are of the form 20 k + 3 or 20 k + 7 , then pq = x 2 + 5 y 2 .
To prove the Fermat's Last Theorem for a strong irregular prime p is more difficult (since Kummer proved the first case of Fermat's Last Theorem for B-regular primes, Vandiver proved the first case of Fermat's Last Theorem for E-regular primes), the most difficult is that p is not only a strong irregular prime, but 2p + 1, 4p + 1, 8p + 1, 10p ...
The sum of one odd square and one even square is congruent to 1 mod 4, but there exist composite numbers such as 21 that are 1 mod 4 and yet cannot be represented as sums of two squares. Fermat's theorem on sums of two squares states that the prime numbers that can be represented as sums of two squares are exactly 2 and the odd primes congruent ...
(u 2 − v 2) 2 = w 4 − 16s 4. But as Fermat proved, there can be no integer solution to the equation x 4 − y 4 = z 2, of which this is a special case with z = u 2 − v 2, x = w and y = 2s. The first step of Fermat's proof is to factor the left-hand side [30] (x 2 + y 2)(x 2 − y 2) = z 2
Eisenstein discovered that the Fermat quotient with base 2 could be expressed in terms of the sum of ... could be reduced from 1/2 to 1/4, 1/5, or even 1/6: ...
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