VIKAS GODARA, DIVYUM SHARMA
5 min
Abstract
Let $C_n=n2^n+1$ denote the $n$th Cullen number. There has been recent interest in finding all Cullen numbers having a given Diophantine property. We prove that, for a fixed integer $k$ and bounded integers $a_1,\ldots,a_k$, the greatest prime divisor of $C_n-a_1m_1!-\cdots-a_km_k!$ tends to infinity, in an effective way. We prove this for some more general families of ternary recurrence sequences as well. We also solve the Diophantine equation $$C_n = m_1! + m_2! + s,$$ where $s$ is a positive integer composed of primes $2,3,5,7$.
Alex: So it's like measuring how much of a particular prime is baked into a number?
Sam: Exactly. Now, factorials are extremely "heavy" by this measure, because as factorials grow, they accumulate more and more copies of every prime. The key insight is this: if a Cullen number is supposed to equal a factorial plus an S-unit, then when you subtract the factorial from the Cullen number, what's left — the remainder — must have a tightly constrained weight. If that remainder turns out to be too heavy, the equation simply cannot balance. That possibility is eliminated.
Alex: So it acts as a filter. Most candidates fail the weight test immediately, and you can rule them out without checking further.
Sam: Right. And they combine this with a separate tool called linear forms in logarithms. Without getting into the machinery, this is a well-established method in number theory for showing that certain quantities can't be too close to zero — which translates into hard limits on how large the relevant indices can be. Together, these two tools squeeze the problem from both sides. The paper shows that the largest prime factor dividing the difference between these sequences must keep growing without bound. That growth forces the factorial indices to stay small. And once the indices are bounded, the infinite search collapses into a finite one.
Alex: That's a meaningful shift in how you approach the problem. You're not searching for solutions — you're proving that the search has a definite stopping point.
Sam: That is the core contribution. Once you have a finite search space, you can either find all solutions or confirm there are none. The paper demonstrates this concretely — for instance, by determining exactly when a Cullen number can equal the sum of two factorials and an S-unit. Questions like that had previously resisted a clean answer.
Alex: What makes this approach worth paying attention to beyond this one result?
Sam: The framework itself is transferable. The combination of p-adic valuations and linear forms in logarithms isn't tailored only to Cullen numbers. Other sequences with similar growth properties could potentially be analyzed the same way. So the paper opens a path for tackling a whole family of related open questions in number theory, not just this one.
Alex: So the real value isn't just the answer — it's the method that produced it. A way of turning an open-ended problem into something with clear boundaries. Thanks for walking through that, Sam. And thanks to everyone listening to ResearchPod.