Linear Combinations of Factorial and $S$-unit in a Ternary recurrence sequence with a double root

TL;DR

The paper investigates Diophantine equations of the form $u_n \pm m! = s$ where $\{u_n\}$ is a ternary recurrence with a double root, and in particular extends Cullen and Woodall numbers by writing $u_n = p(n) α^n + b β^n$ with integers $α,β$ and gcd(α,β)=1. Using height theory, Matveev’s theorem, and a p-adic version due to Yu, the authors derive explicit, finite bounds for $n$ and the size of the S-unit $s$, distinguishing degenerate and nondegenerate cases. They obtain an explicit general bound $n < e^{12X}$ and show that, for the prime set ${2,3,5,7}$, all nondegenerate solutions require $n \le 8$ and $m \le 7$, with the largest-n instance giving $(8\cdot 2^8+1)-4! = 3^4\cdot 5^2$. Theorem 3 is then combined with a computational check to list all small solutions and to confirm that the maximal $n$ in this restricted setting is $8$, together with a complete solution set including degenerate cases.

Abstract

Here, we show that if $u_n=n2^n\pm 1$, then the largest prime factor of $u_n\pm m!$ for $n\ge 0,~m\ge 2$ tends to infinity with $\max\{m,n\}$. In particular, the largest $n$ participating in the equation $u_n\pm m!=2^a3^b5^c7^d$ with $n\ge 1,~m\ge 2$ is $n=8$ for which $(8\cdot 2^8+1)-4!=3^4\cdot 5^2$.

Theorems & Definitions (19)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 4
• Lemma 5: See Theorem 9.4 in Bu
• Lemma 6: Yu Y
• Lemma 7
• Lemma 8
• proof
• Lemma 9