# diophantine equations

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• Sep 26th 2006, 04:57 PM
suedenation
diophantine equations
show that if n = ab - a - b, then there are no nonnegative solutions of ax + by = n.

Thanks
• Sep 27th 2006, 07:45 AM
ThePerfectHacker
Quote:

Originally Posted by suedenation
show that if n = ab - a - b, then there are no nonnegative solutions of ax + by = n.

Thanks

I did not solve the problem yet, but I was able to simplify what you are saying. :)
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You have,
aX+bY=ab-a-b
Equivaleny,
a(X+1)+b(Y+1)=ab
Call x=X+1 and y=Y+1
Thus,
ax+by=ab
Where x and y are positive integers
Because,
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,
ax+by=ab
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.
• Oct 6th 2006, 01:00 PM
ThePerfectHacker
Okay, I finally solved it.

Again, using my previous post it is equivalent to saying,
ax+by=ab
Has no positive solutions x,y.

But that is not true!
10x+8y=80
Has a solution,
x=4 y=5

Thus, what you need to add is that gcd(a,b)=1; a,b>0

Then we have,
ax+by=ab
A trivial solution is, (but not positive)
x=b y=0
Thus, all solutions are, for integer t
x=b-bt
y=at
We need that,
x,y>0
Thus,
b-bt>0
bt>0
Thus,
b>bt
bt<0
Thus,
0<t<1
Which is an impossible because t is an integer.