On Periods modulo a Prime of Some Classes of Sequences of Integers
نویسنده
چکیده
Theorem 1: Let un, n > 0, be the general term of a given sequence of integers and define the transformation T^yk)(un) as T^xyJc){un) = xun+lc +yun for every n > 0, A: being a positive integer. Then, if x mdy are nonzero integers and there exists a positive prime number/? which divides T(x,y,k)(n) f° every n>0 and is relatively prime to x, the distribution of the residues of (un) modulo p is either constant or periodic with period k(p -1).
منابع مشابه
Nonuniform Distributions of Patterns of Sequences of Primes in Prime Moduli
For positive integers q, Dirichlet’s theorem states that there are infinitely many primes in each reduced residue class modulo q. Extending a proof of Dirichlet’s theorem shows that the primes are equidistributed among the φ(q) reduced residue classes modulo q. This project considers patterns of sequences of consecutive primes (pn, pn+1, . . . , pn+k) modulo q. Numerical evidence suggests a pre...
متن کاملProfinite automata
Many sequences of p-adic integers project modulo p to p-automatic sequences for every α ≥ 0. Examples include algebraic sequences of integers, which satisfy this property for every prime p, and some cocycle sequences, which we show satisfy this property for a fixed p. For such a sequence, we construct a profinite automaton that projects modulo p to the automaton generating the projected sequenc...
متن کاملThe Jacobsthal Sequences in Finite Groups
Abstract In this paper, we study the generalized order- Jacobsthal sequences modulo for and the generalized order-k Jacobsthal-Padovan sequence modulo for . Also, we define the generalized order-k Jacobsthal orbit of a k-generator group for and the generalized order-k Jacobsthal-Padovan orbit a k-generator group for . Furthermore, we obtain the lengths of the periods of the generalized order-3 ...
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 کامل# A 79 INTEGERS 13 ( 2013 ) ON THE RESIDUE CLASSES OF ⇡ ( n ) MODULO t Ping
The prime number theorem is one of the most fundamental theorems of analytic number theory, stating that the prime counting function, ⇡(x), is asymptotic to x/ log x. However, it says little about the parity of ⇡(n) as an arithmetic function. Using Selberg’s sieve, we give, for any fixed integers r and t, a positive lower bound on the proportion of positive integers n such that ⇡(n) is congruen...
متن کامل