Bell Numbers Modulo a Prime Number, Traces and Trinomials
نویسندگان
چکیده
منابع مشابه
Bell Numbers Modulo a Prime Number, Traces and Trinomials
Given a prime number p, we deduce from a formula of Barsky and Benzaghou and from a result of Coulter and Henderson on trinomials over finite fields, a simple necessary and sufficient condition β(n) = kβ(0) in Fpp in order to resolve the congruence B(n) ≡ k (mod p), where B(n) is the n-th Bell number, and k is any fixed integer. Several applications of the formula and of the condition are inclu...
متن کاملThe period of the Bell numbers modulo a prime
We discuss the numbers in the title, and in particular whether the minimum period of the Bell numbers modulo a prime p can be a proper divisor of Np = (pp − 1)/(p− 1). It is known that the period always divides Np. The period is shown to equal Np for most primes p below 180. The investigation leads to interesting new results about the possible prime factors of Np. For example, we show that if p...
متن کاملAbout the Period of Bell Numbers modulo a Prime
Let p be a prime number. It is known that the order o(r) of a root r of the irreducible polynomial xp−x−1 over Fp divides g(p) = p p−1 p−1 . Samuel Wagstaff recently conjectured that o(r) = g(p) for any prime p. The main object of the paper is to give some subsets S of {1, . . . , g(p)} that do not contain o(r).
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 کاملSet systems with I-intersections modulo a prime number
Let p be a prime and let L = {l1, l2, . . . , ls} and K = {k1, k2, . . . , kr} be two subsets of {0, 1, 2, . . . , p − 1} satisfying max lj < min ki. We will prove the following results: If F = {F1, F2, . . . , Fm} is a family of subsets of [n] = {1, 2, . . . , n} such that |Fi ∩ Fj | (mod p) ∈ L for every pair i 6= j and |Fi| (mod p) ∈ K for every 1 ≤ i ≤ m, then |F| ≤ ( n− 1 s ) + ( n− 1 s− 1...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: The Electronic Journal of Combinatorics
سال: 2014
ISSN: 1077-8926
DOI: 10.37236/3532