TITLE: Intersection of $\{2^a 3^b 5^c 7^d\}$ and its translates
QUESTION [5 upvotes]: Let $S$ be the set of positive integers of the form $2^a3^b 5^c  7^d$. I need information about the cardinality of the intersection of $S$ and its translates. In particular, is  $S \cap (S+t)$  infinite for every integer $t$? For some values of $t$?
Photo of the some of the solutions for $t=1$. Only integer solutions count, however.
This still lacks proof it has infinitely many solutions over the set of integers. The graph looks similar for the solutions to 10, 100, and pretty much any number, which leads me to believe there are infinite solutions. A proof for this, though, would be invaluable. 
Another way this can be worded is if the equation $2^A 3^B 5^C 7^D - 2^a 3^b 5^c 7^d = 1 \{A,B,C,D,a,b,c,d \in \mathbb{Z} \}$ has an upper bound. This would determine if it has infinite solutions, and may be able to be generalized for all of $n>0$ instead of 1. This function has way too many variables to be graphed so an algebraic way to determine if a function has bounds would be best. 

A very special similar problem would be to determine whether there are infinitely many pairs of numbers differing by $1$ whose only prime factors are $2$ and $3$. (The answer is "no" for pure powers of $2$ and $3$: 
distance between powers of 2 and powers of 3)

REPLY [6 votes]: Long before anyone spoke of S-unit equations, there was Stormer's Theorem of 1897: given any finite set of primes, there are only finitely many pairs of consecutive numbers divisible only by those primes. Moreover, Stormer gave a method for finding all the solutions, using Pell equations. Lehmer gave practical improvements in 1964. Many details, references, and links to OEIS and elsewhere, at https://en.wikipedia.org/wiki/Størmer%27s_theorem