منابع مشابه
#A16 INTEGERS 12A (2012): John Selfridge Memorial Issue THE SEARCH FOR AURIFEUILLIAN-LIKE FACTORIZATIONS
We searched the Cunningham tables for new algebraic factorizations similar to those discovered by Aurifeuille. A naive search would have been too slow. We accelerated it enough to make it feasible. Many interesting results were found. –Dedicated to the memory of John Selfridge, who loved the integers.
متن کاملAurifeuillian factorizations and the period of the Bell numbers modulo a prime
We show that the minimum period modulo p of the Bell exponential integers is (pp−1)/(p−1) for all primes p < 102 and several larger p. Our proof of this result requires the prime factorization of these periods. For some primes p the factoring is aided by an algebraic formula called an Aurifeuillian factorization. We explain how the coefficients of the factors in these formulas may be computed.
متن کاملAurifeuillian factorization
The Cunningham project seeks to factor numbers of the form bn±1 with b = 2, 3, . . . small. One of the most useful techniques is Aurifeuillian Factorization whereby such a number is partially factored by replacing bn by a polynomial in such a way that polynomial factorization is possible. For example, by substituting y = 2k into the polynomial factorization (2y2)2+1 = (2y2−2y+1)(2y2+2y+1) we ca...
متن کاملComputing Aurifeuillian Factors
For odd square-free n > 1, the cyclotomic polynomial Φn(x) satisfies an identity Φn(x) = Cn(x) ± nxDn(x) of Aurifeuille, Le Lasseur and Lucas. Here Cn(x) and Dn(x) are monic polynomials with integer coefficients. These coefficients can be computed by simple algorithms which require O(n2) arithmetic operations over the integers. Also, there are explicit formulas and generating functions for Cn(x...
متن کاملConstructive Bounds on Ordered Factorizations
The number of ways to factor a natural number into an ordered product of integers, each factor greater than one, is called the ordered factorization of n and is denoted H(n). We show upper and lower bounds on H(n) with explicit constructions.
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ژورنال
عنوان ژورنال: Indagationes Mathematicae (Proceedings)
سال: 1987
ISSN: 1385-7258
DOI: 10.1016/1385-7258(87)90009-6