GENERALIZED WALL-SUN-SUN PRIMES AND MONOGENIC POWER-COMPOSITIONAL TRINOMIALS.

For positive integers a and b, we let [Un] be the Lucas sequence of the first kind defined by U0 = 0, U1 = 1 and Un = aUn-1 + bUn−2 for n ≥ 2, and let π(m) ∶= π(a,b)(m) be the period length of [Un] modulo the integer m ≥ 2, where gcd(b, m) = 1. We define an (a, b)-Wall-Sun-Sun prime to be a prime p such that π(p² ) = π(p). When (a, b) = (1, 1), such a prime p is referred to simply as a Wall-Sun-Sun prime. We say that a monic polynomial f(x) ∈ Z[x] of degree N is monogenic if f(x) is irreducible over Q and {1, θ, θ², . . ., θN−1 } is a basis for the ring of integers of Q(θ), where f(θ) = 0. Let f(x) = x² − ax − b, and let s be a positive integer. Then, with certain restrictions on a, b and s, we prove that the monogenicity of f(xsn ) = x2sn − axsn − b is independent of the positive integer n and is determined solely by whether s has a prime divisor that is an (a, b)-Wall-Sun-Sun prime. This result improves and extends previous work of the author in the special case b = 1. [ABSTRACT FROM AUTHOR]