[UNSOVED] What is the largest number (in terms of a, b, c) that can not be expressed in the form $$ax+by+cz$$
where $$x,y,z≥0$$$$gcd(a,b,c)=1$$and all values are non-negative integers?

As an example, take $$6x+10y+15z$$Here, the largest integer that can't be expressed in this form given $$x, y, z \geq 0$$is 29.

If instead you're solving the same problem, only with $$ax+by$$instead of$$ax+by+cz$$ with the same conditions that every value is a non-negative integer, the largest integer not expressible in this form is given by the formula$$ab-a-b$$

Question

[UNSOVED] What is the largest number (in terms of a, b, c) that can not be expressed in the form $$ax+by+cz$$
where $$x,y,z≥0$$$$gcd(a,b,c)=1$$and all values are non-negative integers?

As an example, take $$6x+10y+15z$$Here, the largest integer that can't be expressed in this form given $$x, y, z \geq 0$$is 29.

If instead you're solving the same problem, only with $$ax+by$$instead of$$ax+by+cz$$ with the same conditions that every value is a non-negative integer, the largest integer not expressible in this form is given by the formula$$ab-a-b$$

anonymous · Answer

IMO problem?

anonymous · Answer

A similar problem appear in IMO 1983,

Let  $ a, b,$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc − ab − bc − ca$ is the largest integer that cannot be expressed in the form $xab+yca+zab$, where$ x, y,$ and $z$ are non-negative integers.

kinggeorge · Answer

no, the very last part of a homework problem from our textbook. I've figured out the two variable case without too much trouble, but for some reason, I just don't know how to generalize it.

kinggeorge · Answer

Well, that appears to be almost the generalization I'm looking for...

anonymous · Answer

Yes, I think we still need to do some work, btw what level of mathematics is this?

anonymous · Answer

Like high-school , undergrad, etc...

kinggeorge · Answer

undergrad - intro to number theory

anonymous · Answer

Cool, I thought so.

kinggeorge · Answer

I've worked through a few other cases while varying c. So far I've gotten 25 if a=6, b=10, c=11. And 23 if a=6, b=10, c=9.

kinggeorge · Answer

Here a few more cases 
11 if (a, b, c) = (4, 5, 6) 
also 11 if (a, b, c) = (4, 5, 7)
also 11 if (a, b, c) = (4, 6, 9)

kinggeorge · Answer

My best guess currently, for those of you looking at this, is that the solution is the smallest integer that satisfies these equations$$\begin{matrix} ab-a-b-d = cx \ ac-a-c-d=by \ bc-b-c-d=az \end{matrix}$$where d is the solution we're looking for, and x, y, z are still all positive.