Uniform explicit Stewart's theorem on prime factors of linear recurrences

Abstract

Stewart (2013) proved that the biggest prime divisor of the nth term of a Lucas sequence of integers grows quicker than n, answering famous questions of Erdős and Schinzel. In this note we obtain a fully explicit and, in a sense, uniform version of Stewart's result.

Big prime factors in orders of elliptic curves over finite fields

“…The following two theorems are, essentially, due to Stewart [13], though in the present form they can be found in [3], see Theorems 1.4 and 1.5 therein.…”

“…3) is easier, the proof follows the same lines as the proof of Theorem 1.2 in [3]. Case (3.4) is harder and requires more intricate arguments.…”

“…To estimate log x/ log * ω(n), we consider two cases. Assume first that ω(n) ≤ log n (log log n) 3 .…”

“…Erdős [7] conjectured that P (u n )/n tends to infinity. This was proved to be so by Stewart [13] who showed that P (u n ) > n exp(log n/(104 log log n)) holds for n > n 0 , where n 0 is a constant which Stewart did not compute and which depends on the discriminant of the field Q(α) and the number of distinct prime factors of s. Explicit values for n 0 were computed in [3] at the cost of replacing 1/104 by somewhat smaller constants (see Theorem s 2.1 and 2.2 below). It is also shown in [3] that n 0 depends only on the field Q(α), but is independent of the number of prime divisors of s.…”

Stewart's Theorem revisited: suppressing the norm $\pm 1$ hypothesis

“…In 2013, Stewart [10] gave a lower bound of the largest prime factor of u n , which is of the form n exp(log n/104 log log n). What Stewart actually proved is the following, see [1,Theorem 1.1].…”