Prime divisors of \(a^n-b^n\) (Q6546705)

Consider the sequence \(u_n=a^n-b^n\) with given integers \(a>b\geq 1\), and let \(P(u_n)\) denote the largest prime factor of this \(n\)-th term.\N\NBy the uniqueness of prime factorization, we express \(u_n=UV^{k+1}\) such that each prime factor of \(U\) has at most \(k\) multiplicity. The author states the following hypothesis: Assume that there exists an absolute constant \(\lambda\) such that \(V<(aU)^{\lambda}\) whenever \(k\geq \lambda\). Then it follows that \(P(u_n)\gg (n/\tau(n))^2\) with an absolute implied constant. Here, \(\tau(n)\) is the number of distinct positive divisors of \(n\).\N\NIn multiple ways, this claim is a clear improvement of earlier results in studying the lower-bounds of \(P(u_n)\). And the stated hypothesis above is itself a weaker condition related to an \(abc\)-conjecture form that has played a role in existing literature.