On the Greatest Prime Factor of Some Divisibility Sequences

Abstract

ABSTRACT. Let P (m) denote the greatest prime factor of m. For integer a > 1, M. Ram Murty and S. Wong proved that, under the assumption of the ABC conjecture, P (a n − 1) ≫ ǫ,a n 2−ǫ for any ǫ > 0. We study analogues results for the corresponding divisibility sequence over the function field F q (t) and for some divisibility sequences associated to elliptic curves over the rational field Q. In honor of M. Ram Murty on his sixtieth birthday

Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant

“…The point P ′ of Notation 3.2 is P ′ = (t, √ r) ∈ E(K(t, √ r)), which is non-constant. By Lemma 2.2 this proves (1). Note that K[t] has at most three primes at which E has bad reduction (the roots of r) and for all fields K except F 2 and F 3 there are more than 3 rational points in A 1 (K), hence there is at least one rational place of good reduction.…”

“…In a broad context, Flatters and Ward [10] prove an analogue of Theorem 8.1 for divisibility sequences of Lucas type for polynomials and Akbar-Yazdani [1] study the greatest degree of the prime factors of certain Lucas polynomial divisibility sequences.…”

“…Let v be the restriction of v ′ to F . Then E has good or additive reduction at v by Lemma 4.1 (1). If E has good reduction, then n = m(P ′ , v ′ ) = m(P, v) by Lemma 4.2.…”

“…It remains to prove (1). After base-changing to F ′ v ′ , we get a Weierstrass equation E over K. As E is defined over K, it has good reduction at v ′ .…”

“…If E has additive reduction, then n = m(P ′ , v ′ ) | d v by Lemma 4.4(1). (1) for all integers n > 4 with p ∤ n, the term D n has a primitive valuation, and (2) for all integers n > 4, the term D pn has no primitive valuation.…”