Let P (m) denote the greatest prime factor of an integer m > 1. It has been known since the 1900s that Pn:= P (an − bn) > n + 1 for integers a > b > 0 and n > 2. A conjecture of Stewart (1977) states that Pn ≫ φ(n)2 where the implied constant is absolute. He (2013) later proved that Pn ≫a,b n1+ 1 104 log log n. Earlier, Murty and Wong (2002) had shown that the usual abc-conjecture implies that Pn ≫a,b,ε n2−ε. Recently, Murty and Séguin (2019) formulated a conjecture concerning the p-adic valuation of af − 1 where p ∤ a, and f is the order of a in the multiplicative group (Z/pZ)×. Conditional on their conjecture, they confirmed the conjecture of Stewart in the case that b = 1 with the implied constant depending on a. We prove that a milder abc-conjecture implies that Pn ≫ (n/τ (n))2 where τ (n) is the number of distinct positive divisors of n, and crucially, the implied constant is independent of a and b. This is an improvement over the result of Murty and Wong. Furthermore, as a simple consequence, Stewart’s conjecture follows in the case that n is prime, thereby refining the result of Murty and Séguin. Additionally, we obtain a distribution result for the prime factors of gcd(n, Φn (a, b)), generalizing a similar result of Murty and Séguin. © 2023, Colgate University. All rights reserved.