For my thesis I would like to find integers (lying in a certain moduloclass) in small intervals which have large prime divisors. And for some reason I decided that I want all bounds appearing in my thesis to be explicit, so I am looking for something like the following result:
Given a moduloclass $a \pmod{m}$ and an integer $x \ge c_0$ (where $c_0 = c_0(m)$ may depend on $m$, but is bounded above by some explicit function of $m$), there exists an integer $n \equiv a \pmod{m}$ in the interval $[x, x + x^{c_1}]$ having a prime divisor larger than $n^{c_2}$.
Now, I don't really care about the constants $c_1$ and $c_2$, as long as $c_2 > c_1$.
The theorem in this paper by Ramachandra is more or less what I need, except for the restriction on the moduloclass and the fact that it's not explicit. On the other hand, theorem 1 in this paper by Laishram and Shorey gives the above with $n^{c_2}$ replaced by $\frac{2}{m}x^{c_1}$ for $m \ge 3$, $x \ge 19$.
Does anyone have a reference (or proof) for me?
EDIT: I have now asked this question here as well.
I think both the function $f$ and the integer $x$ should have restrictions in order not have counterexamples such as the following.
Let $f$ be positive continuously decreasing towards $0$ and choose $m$ such that $f(m)\lt 1$ (there are infinitely many possible choices); then, according to your post, $x = 1$ is valid to choose and for all $c_1$ one has $[x,x+x^{c_1}]=[1,2]$. Hence the only possibilities for $n$ would be $1$ and $2$.
Am I wrong? I just want to help you.