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](https://i.stack.imgur.com/JxFaF.png) 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](https://mathoverflow.net/questions/116840/distance-between-powers-of-2-and-powers-of-3))



Intersection of $\{2^a 3^b 5^c 7^d\}$ and its translates
This is an example of an $S$-unit equation. For ones of a shape similar to this, the solutions can be found rather easily using bounds for linear forms in logarithms and lattice basis reduction. By way of example, Theorem 5.5 of de Weger's thesis (from 1989) explicitly determines the $605$ relatively prime solutions to the inequality
$$
0 < x-y < \sqrt{y}
$$
with $x$ and $y$ with prime factors entirely in $\{ 2, 3, 5, 7, 11, 13 \}$. From this result, at a glance it seems that the last example for the problem you're interested in corresponds to $4375-4374=1$.


If you replace the $1$ by an arbitrary constant $k$, you still can solve the problem effectively (though the number of solutions may grow somewhat). 


