Linear Diophantine Equation

Date: 02/05/2003 at 11:00:20
From: Jane
Subject: Linear Diophantine equation

Given positive integers a and b such that a|b^2, b^2|a^3, a^3|b^4, 
b^4|a^5,....., prove that a = b.

I do not know how to approach this question. I thought about 
contradiction or induction but I'm not too familiar with either 
method.

Date: 02/06/2003 at 09:56:16
From: Doctor Jacques
Subject: Re: Linear Diophantine equation

Hi Jane,

If both a and b are equal to 1, we certainly have a=b. We may 
therefore assume that at least one of {a,b} is not equal to 1.

Let us assume that a <> b.

If a<>b, their prime factorisations are different. It means that we 
can find a prime p such that p^j | a, p^i | b, i and j are the 
greatest such exponents, and i <> j. Let us assume, for example, that 
i < j (the other case is similar). Note that this includes the case 
i = 0.

The given relations imply (we use only every other relation):

  j  <= 2i
  3j <= 4i
  5j <= 6i
  ....
  (2n - 1) j <= 2n*i for all n.

If we divide the last equation by 2n, we find:

  j - (j/2n) <= i

As this relation must be true for all n, we may choose n > j.

The inequation then becomes:

  j - (1/2) < i

On the other hand, we assumed that i < j

Now, remember that i and j are integers. This should allow you to 
find a contradiction.

Please feel free to write back if you are still stuck.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/