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There is this well-known elementary theorem:

Every pair of integers $a$ and $b$ has a common divisor $d$ of the form $d=ax+by$. Moreover, every common divisor of $a$ and $b$ divides this $d$.

So, the greatest common divisor can be represented linearly as a function of $a$ and $b$.

But, it seems to me that also an interesting question is when the greatest common divisor can be represented in the following way $ax+by +\alpha xy$, where $\alpha \in \mathbb Z$ is some constant.

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

At least, $d$ is expressible as $ax+by +\alpha xy$ if $\alpha=0$ but surely for some $a,b \in \mathbb Z$ the choice $\alpha=0$ is not the only one.

So suppose that for $(a,b) \in \mathbb Z \times \mathbb Z$ the set $\beta ((a,b))$ is the set of all $\alpha$ such that $d=ax+by +\alpha xy$, so, for example, if for some $(a,b)$ we have that there are $(x_r,y_r)$ such that $d=ax_r+by_r+\alpha_r x_r y_r$ for $r=1,...,$ then $\alpha_r \in \beta((a,b))$.

If $\text{noe}(\beta((a,b)))$ denotes 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 that function, for example, some properties of it or some rules governing its behaviour?

  • Why do you believe that such a representation is interesting? – Bill Dubuque Feb 14 '20 at 06:10
  • @BillDubuque One of the reasons is that it would be a generalization of the stated well-known elementary theorem but also there are some other reasons. –  Feb 14 '20 at 06:12
  • But why that particular form? One could ask infinitely many similar questions about more general nonlinear representations and almost all of them will likely not prove very fruitful. Why should this be more interesting than any other nonlinear rep, or whjy should any nonlinear rep be of interest? – Bill Dubuque Feb 14 '20 at 06:15
  • @BillDubuque This particular form I have chosen exactly because it is, it could be said, a first step towards what you typed in a comment, a first step towards a more general non-linear representations. –  Feb 14 '20 at 06:17
  • Linearity is essential in results in this area. Are you familiar with ideals? – Bill Dubuque Feb 14 '20 at 06:19
  • @BillDubuque I know some facts about ideals but I did not study them thoroughly and in detail. –  Feb 14 '20 at 06:20
  • I highly recommend that before embarking on exploring generalizations that you first master the basics, esp. topics such as Euclidean and Bezout domains and PIDs. Then the crucial role played by linearity should be much clearer. Even experts find it difficult to make fruitful generalizations so it will be hopeless without first mastering the basics. – Bill Dubuque Feb 14 '20 at 06:25
  • @BillDubuque I understand that much, I firmly believe that fruitful generalizations can be obtained but at this moment I do not know how, so basically I could first study elementary essentials in this area. –  Feb 14 '20 at 06:27
  • Re: linearity, ideals, and modules, I highly recommend that you read this, from the Handbook of Algebra. – Bill Dubuque Feb 14 '20 at 06:38
  • @BillDubuque We could move this discussion to chatroom, how much would you like that? –  Feb 14 '20 at 06:40

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