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

Thanks

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- Sep 26th 2006, 05:57 PMsuedenationdiophantine equations
show that if n = ab - a - b, then there are no nonnegative solutions of ax + by = n.

Thanks - Sep 27th 2006, 08:45 AMThePerfectHacker
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, 02:00 PMThePerfectHacker
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.