There is this well-known elementary theorem:

*Every pair of integers $ a $ and $ b $ has a common divisor $ d $ the form $ d = ax + by $, In addition, every common divisor of $ a $ and $ b $ share this $ d $,*

This is how the largest common divisor can be represented *linear* as a function of $ a $ and $ b $,

But it also seems to me to be an interesting question when the greatest common divisor can be represented as follows $ ax + by + alpha xy $, Where $ alpha in mathbb Z $ is a constant.

It seems obvious to some $ alpha $ the greatest common divisor $ d $ will not be expressible in the form $ ax + by + alpha xy $ but for some others $ alpha $ it will be expressible.

At least, $ d $ can be expressed as $ ax + by + alpha xy $ if $ alpha = 0 $ but sure for some $ a, b in mathbb Z $ the vote $ alpha = 0 $ is not the only one.

So let's assume that for $ (a, b) in mathbb Z times mathbb Z $ the sentence $ beta ((a, b)) $ is the crowd of all $ alpha $ so that $ d = ax + by + alpha xy $, for example if for some $ (a, b) $ we have that $ d = ax + by + alpha_r xy $ to the $ r = 1, …, $ then $ alpha_r in beta ((a, b)) $,

If $ text {noe} ( beta ((a, b))) $ designated *the number of elements of the set $ beta ((a, b)) $* then I would like to know at least some facts about the function $ (a, b) to text {noe} ( beta ((a, b))) $ and what is generally known about this function, for example some properties of it or some rules that govern its behavior?