There is this well-known elementary theorem:

*Each pair of integers $ to $ Y $ b $ has a common divisor $ d $ Shape $ d = ax + for $. In addition, each common divisor of $ to $ Y $ b $ divide this $ d $.*

Then, the greatest common divisor can be represented *linearly* as a function of $ to $ Y $ b $.

But, it seems to me that also an interesting question is when the greatest common divisor can be represented as follows $ ax + for + alpha xy $, where $ alpha in mathbb Z $ It is something constant.

It seems obvious that for some $ alpha $ the greatest common divisor $ d $ will not be expressible in the form $ ax + for + alpha xy $ but for some other $ alpha $ It will be expressible.

At least, $ d $ is expressible as $ ax + for + alpha xy $ Yes $ alpha = 0 $ but surely for some $ a, b in mathbb Z $ the chose $ alpha = 0 $ It's not the only one.

Assume that by $ (a, b) in mathbb Z times mathbb Z $ set $ ((a, b)) $ it is the set of all $ alpha $ such that $ d = ax + for + alpha xy $so, for example, if for some $ (a, b) $ we have that $ d = ax + for + alpha_r xy $ for $ r = 1, …, $ then $ alpha_r in beta ((a, b)) $.

Yes $ text {noe} ( beta ((a, b))) $ of note *the number of elements in the set $ ((a, b)) $* so I would like to know at least some information about the function $ (a, b) to text {noe} ( beta ((a, b))) $ And what is generally known about that function, for example, some properties of it or some rules that govern its behavior?