Edit to clarify and rephrase:
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 $2^A 3^B 5^C 7^D - 2^a 3^b 5^c 7^d = 1 \{A,B,C,D,a,b,c,d \in {Z} \}$ 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 $3$:
https://mathoverflow.net/questions/116840/distance-between-powers-of-2-and-powers-of-3)
The original question. I am leaving this here since it has an upvoted answer.
I was browsing some set theory theorems on infinite sets and I couldn't find one that could help me with my specific case. This is a question I'm not positive has an answer, but for an infinite set with random integers $$\{ n_1, n_2, n_3, ...\} \cap \{n_1 + a, n_2 + a, n_3+a...\}$$ where a is an integer, what circumstances would yield a finite set as supposed to an infinite set. Alternatively, how could I prove that this set would be infinite instead of saying that it is 'probably' infinite. The distribution is all numbers greater than 7 who don't have any prime factors strictly greater than 7, and aren't prime. Some of the terms are $$\{80, 81, 84, 96, 100, 105, 108 ... \}$$ To clarify, these are values down the line. The first term is 8, the second 9, the third 10, etc. An answer for all of a is what I'm looking for. I do know that each progressive n is larger than the last. I'm sorry if this question is too vague, I don't have much mathematical experience. Thank you for anything you can provide.