show that if n = ab - a - b, then there are no nonnegative solutions of ax + by = n.
Call x=X+1 and y=Y+1
Where x and y are positive integers
X,Y where non-negative implies X+1,Y+1 are positive.
Thus, you need to show for a,b not equal to zero,
The linear diophantine equation,
Has no positive solutions.
Begin by noting the trivial solutions:
(x,y)=(b,0) and (x,y)=(0,a)
Then use them to construct your basis of solutions.
But I did not get any further.
Okay, I finally solved it.
Again, using my previous post it is equivalent to saying,
Has no positive solutions x,y.
But that is not true!
Has a solution,
Thus, what you need to add is that gcd(a,b)=1; a,b>0
Then we have,
A trivial solution is, (but not positive)
Thus, all solutions are, for integer t
We need that,
Which is an impossible because t is an integer.