What is the Name of This Problem? Given two integers $a$ and $b$, determine if $a$ and $b$ are equivalent modulo any $k \, (k: 1 \lt k \lt |a - b|)$.         


If $\exists$ an integer $ (k: 1 \lt k \lt |a - b|) \text{ and }a \equiv b \mod k$, then the algorithm would return true, otherwise it would return false.

I thought of this problem today.     

Before attempting to solve it, I want to know if there is existing literature on the problem. However, I would need to know the problem to do research on it.
You just need to check if $|a-b|$ is divisible by a some integer $k$. You can reduce it to the integer factorization problem if you are not interested in the trivial divisors 1 and $|a-b|$.