Given,
$$aX + bY = c$$
where,
$$c > b > a > 0;\quad X, Y > 0;\quad b\nmid c, a\nmid c$$
I want to find out if a solution exists as efficiently as possible (I'm not interested in the solutions). Are there any calculations I can make before (or without the need for) finding $\operatorname{gcd}(a, b)$ that can possibly save some time (even if for only few special cases)? $c, b, a$ can be very large numbers.
"Probably not" still counts as answer for me. You don't have to be 100% certain. I just want to make sure I'm not missing something that's very obvious.
P.S., English is not my first language.
If gcd(a,b) divides c, then there are infinitely many solutions for the equation
aX + bY = c
Divide the equation by gcd(a,b) to get the reduced equation
sX + tY = u
where gcd(s,t)=1
With the extended algorithm for the gcd-calculation, you can calculate a solution (m,n).
All solutions are given by
X = kt + m
Y = -ks + n
for some integer k.
If there is an integer k, such that X and Y satisfy all conditions, a solution is found. In order to check this, you have to find bounds for the number k by solving the desired inequalities and check, if the intervals found have a non-empty intersection.
So, this is the way to find solutions.
Now, I come to your additional restrictions. Since 3x + 4y = 5 is an equation not having integer solutions with x>0 and y>0, it is not enough to check, whether gcd(a,b) divides c, even if the additional restrictions hold.
But the way I described is very efficient even for very large numbers, since it only requires modulo-calculations.
I doubt that there are faster ways to decide if there are solutions of the desired form.