On Prime Factors of Terms of Linear Recurrence Sequences
نویسنده
چکیده
whenever (un) ∞ n=0 is a non-degenerate linear recurrence sequence. Mahler’s proof is not effective in the following sense. Given a positive integer m the proof does not yield a number C(m) which is effectively computable in terms of m, such that |un| > m whenever n > C(m). However, Schmidt [31, 32], Allen [1], and Amoroso and Viada [2] have given estimates in terms of t only for the number of times |un| assumes a given value when the recurrence sequence is non-degenerate. For any integer n let P (n) denote the greatest prime factor of n with the convention that P (0) = P (±1) = 1. Suppose that in (1) t > 1, f1, . . . , ft are ∗Research supported in part by the Canada Research Chairs Program and by Grant A3528 from the Natural Sciences and Engineering Research Council of Canada.
منابع مشابه
Primitive Prime Divisors in Polynomial Arithmetic Dynamics
The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a polynomial. For two classes of polynomials f(x) ∈ Z[x] and initial values a1 ∈ Z, we show that th...
متن کاملPrimitive Prime Divisors of First-Order Polynomial Recurrence Sequences
The question of which terms of a recurrence sequence fail to have primitive prime divisors has been significantly studied for several classes of linear recurrence sequences and for elliptic divisibility sequences. In this paper, we consider the question for sequences generated by the iteration of a polynomial. For two classes of polynomials f(x) ∈ Z[x] and initial values a1 ∈ Z, we show that th...
متن کاملPrimitive Prime Factors in Second-order Linear Recurrence Sequences
For a class of Lucas sequences {xn}, we show that if n is a positive integer then xn has a primitive prime factor which divides xn to an odd power, except perhaps when n = 1, 2, 3 or 6. This has several desirable consequences.
متن کاملPseudoprimes for Higher-order Linear Recurrence Sequences
With the advent of high-speed computing, there is a rekindled interest in the problem of determining when a given whole number N > 1 is prime or composite. While complex algorithms have been developed to settle this for 200-digit numbers in a matter of minutes with a supercomputer, there is a need for simpler, more practical algorithms for dealing with numbers of a more modest size. Such practi...
متن کاملON COMPOSITION FACTORS OF A GROUP WITH THE SAME PRIME GRAPH AS Ln(5)
The prime graph of a finite group $G$ is denoted by$ga(G)$. A nonabelian simple group $G$ is called quasirecognizable by primegraph, if for every finite group $H$, where $ga(H)=ga(G)$, thereexists a nonabelian composition factor of $H$ which is isomorphic to$G$. Until now, it is proved that some finite linear simple groups arequasirecognizable by prime graph, for instance, the linear groups $L_...
متن کامل