Proving or disproving that if $\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$ both $a$ and $b$ are integers.

Question

I found some formula about special function very complicated, so I am curious how you people solve this by hand.

$$\Gamma(a)+\Gamma(b)= 121\,645\,106\,635\,852\,800$$

but $a$ and $b$ are very small actually, condition: at least one of $a$ and $b$ are integers.

Can we prove or disprove both of them must be integer?

Reference: http://www.wolframalpha.com/input/?i=gamma%2814%29%2Bgamma%2820%29