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I'm interested in finding adjacent numbers, $A,A+1$ that are both products of a set of primes $P$. Specifically, given a finite set $P$, is there a maximum such $A$ that exists? Here's my work so far:

For the concrete example of $P = \{2,3\}$, I think the maximum is $A= 8, A+1 = 9$, but I have trouble when considering sets with more than 2 elements.

For a set $P = \{p_1,...,p_n\}$, then any such $A$ must be co-prime with $A+1$. Otherwise, if they share a factor $p_i$, then $1 = A+1 - A = p_i \times x - p_i \times y = p_i \times (x-y)$ for some integers $x,y$, which implies $p_i$ divides 1 which is clearly impossible.

So, given there are finitely many partitions of $P$, if I could show that each partition generates finitely many $A$ that would prove finitely many $A$ overall, and thus a maximum exists. But I'm not sure how to prove this.

Servaes
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mannerpots
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  • The case of two primes, $p,q$ amounts to $p^m=q^n\pm1.$ This is obviously impossible if both are odd, so you can assume $p=2.$ tThe case $2,3$ are is a known problem, and I vaguely recall here is a proof that $8,9$ is the only case, but I think the proof was not easy. – Thomas Andrews Jan 13 '25 at 01:11
  • I doubt there is a good way to approach the general problem for more than $2$ primes. Exponential diopantine equations are often very hard. – Thomas Andrews Jan 13 '25 at 01:15
  • Cf. this question and this question regarding powers of $2$ and $3$ differing by $1$ – J. W. Tanner Jan 13 '25 at 01:27
  • As mentioned in the linked questions, the case for 2 primes is a sub-case of Catalan's conjecture. https://en.wikipedia.org/wiki/Catalan%27s_conjecture So now I'm just wondering about the case with more than 2 primes. but it might be a hard problem. – mannerpots Jan 13 '25 at 01:55

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This is fully answered by Størmer's theorem:

Given a finite set of primes $P$, let $p\in P$ be the largest prime and let $Q$ be the set of $p$-smooth numbers. Then Størmer's theorem ensures that there are only finitely many pairs of consecutive numbers in $Q$, and it gives an effective method for finding them all. Your desired set $P$-smooth numbers is a subset of these.

Servaes
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    Much discussion of Størmer's Theorem, from a numerical point of view, in Lehmer, On a problem of Størmer, Illinois J. Math. 8(1): 57-79 (March 1964). DOI: 10.1215/ijm/1256067456 available at https://projecteuclid.org/journals/illinois-journal-of-mathematics/volume-8/issue-1/On-a-problem-of-St%C3%B8rmer/10.1215/ijm/1256067456.full – Gerry Myerson Jan 13 '25 at 17:35