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  2. Fermat's factorization method - Wikipedia

    en.wikipedia.org/wiki/Fermat's_factorization_method

    Fermat's factorization method, named after Pierre de Fermat, is based on the representation of an odd integer as the difference of two squares: N = a 2 − b 2 . {\displaystyle N=a^{2}-b^{2}.} That difference is algebraically factorable as ( a + b ) ( a − b ) {\displaystyle (a+b)(a-b)} ; if neither factor equals one, it is a proper ...

  3. Integer factorization - Wikipedia

    en.wikipedia.org/wiki/Integer_factorization

    For example, if n = 171 × p × q where p < q are very large primes, trial division will quickly produce the factors 3 and 19 but will take p divisions to find the next factor. As a contrasting example, if n is the product of the primes 13729, 1372933, and 18848997161, where 13729 × 1372933 = 18848997157, Fermat's factorization method will ...

  4. Fermat number - Wikipedia

    en.wikipedia.org/wiki/Fermat_number

    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 ).

  5. Glossary of number theory - Wikipedia

    en.wikipedia.org/wiki/Glossary_of_number_theory

    factorization Factorization is the process of splitting a mathematical object, often integers or polynomials, into a product of factors. Fermat's last theorem Fermat's last theorem, one of the most famous and difficult to prove theorems in number theory, states that for any integer n > 2, the equation a n + b n = c n has no positive integer ...

  6. Fermat's little theorem - Wikipedia

    en.wikipedia.org/wiki/Fermat's_little_theorem

    For example, if a = 2 and p = 7, then 2 7 = 128, and 128 − 2 = 126 = 7 × 18 is an integer multiple of 7. If a is not divisible by p, that is, if a is coprime to p, then Fermat's little theorem is equivalent to the statement that a p − 11 is an integer multiple of p, or in symbols: [1] [2] ().

  7. Pierre de Fermat - Wikipedia

    en.wikipedia.org/wiki/Pierre_de_Fermat

    It was while researching perfect numbers that he discovered Fermat's little theorem. He invented a factorization methodFermat's factorization method—and popularized the proof by infinite descent, which he used to prove Fermat's right triangle theorem which includes as a corollary Fermat's Last Theorem for the case n = 4.

  8. Jerry Rice critical of 49ers WR Deebo Samuel: 'You cannot ...

    www.aol.com/jerry-rice-critical-49ers-wr...

    Samuel managed three catches for 16 receiving yards and added two rushes for three yards. The 49ers faithful made their frustration known after the dropped potential touchdown, showering down boos.

  9. Quadratic sieve - Wikipedia

    en.wikipedia.org/wiki/Quadratic_sieve

    To factorize the integer n, Fermat's method entails a search for a single number a, n 1/2 < a < n−1, such that the remainder of a 2 divided by n is a square. But these a are hard to find. The quadratic sieve consists of computing the remainder of a 2 / n for several a , then finding a subset of these whose product is a square.