Fermat numbers
♦
7 matching pages ♦
(0.001 seconds)
7 matching pages
1: 27.18 Methods of Computation: Primes
2: 27.16 Cryptography
…
►Thus, and .
…
3: 27.2 Functions
…
►This is the number of positive integers that are relatively prime to ; is Euler’s totient.
…
4: 24.17 Mathematical Applications
…
►
§24.17(iii) Number Theory
►Bernoulli and Euler numbers and polynomials occur in: number theory via (24.4.7), (24.4.8), and other identities involving sums of powers; the Riemann zeta function and -series (§25.15, Apostol (1976), and Ireland and Rosen (1990)); arithmetic of cyclotomic fields and the classical theory of Fermat’s last theorem (Ribenboim (1979) and Washington (1997)); -adic analysis (Koblitz (1984, Chapter 2)). …5: 27.19 Methods of Computation: Factorization
…
►Fermat’s algorithm is another; see Bressoud (1989, §5.1).
…
►As of January 2009 the largest prime factors found by these methods are a 19-digit prime for Brent–Pollard rho, a 58-digit prime for Pollard , and a 67-digit prime for ecm.
…
►These algorithms include the Continued Fraction Algorithm (cfrac), the Multiple Polynomial Quadratic Sieve (mpqs), the General
Number Field Sieve (gnfs), and the Special Number Field Sieve (snfs).
…The snfs can be applied only to numbers that are very close to a power of a very small base.
The largest composite numbers that have been factored by other Type II probabilistic algorithms are a 63-digit integer by cfrac, a 135-digit integer by mpqs, and a 182-digit integer by gnfs.
…
6: 27.8 Dirichlet Characters
§27.8 Dirichlet Characters
… ►An example is the principal character (mod ): … ►For any character , if and only if , in which case the Euler–Fermat theorem (27.2.8) implies . …If , then the characters satisfy the orthogonality relation … ►A divisor of is called an induced modulus for if …7: Bibliography C
…
►
Some congruences for the Bernoulli numbers.
Amer. J. Math. 75 (1), pp. 163–172.
►
-Bernoulli and Eulerian numbers.
Trans. Amer. Math. Soc. 76 (2), pp. 332–350.
►
A note on Euler numbers and polynomials.
Nagoya Math. J. 7, pp. 35–43.
►
Expansions of -Bernoulli numbers.
Duke Math. J. 25 (2), pp. 355–364.
…
►
Modular Forms and Fermat’s Last Theorem.
Springer-Verlag, New York.
…