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Proving Perfect Squares


Date: 07/05/98 at 16:53:34
From: Gary Brisebois
Subject: Algebra

Suppose a, b, and c are positive integers, with no factor in common, 
where 1/a + 1/b = 1/c. Prove that a + b, a - c, and b - c are all 
perfect squares.

I have spent the last three days of my spare time trying to solve 
this one. Maybe it has been too long since my university days...

The most promising route I devised was to consider each integer as 
the product of primes, where no prime in one could be found in both 
of the others:

Since a + b = ab/c it follows that
a + b = P1^(x1+y1-z1) * ...Pn^(xn+yn+zn)
where P1 thru Pn is the set of all primes found as factors of a, b, 
or c, and where xm, ym, and zm are the exponents of each of those 
primes found in each of the integers respectively. It remains to show 
that for each m, xm+ym-zm is even.

This question was published in a local paper as an example of 
questions asked of Canada's best high school students. 
I am stumped; please help!


Date: 07/07/98 at 12:24:27
From: Doctor Wilkinson
Subject: Re: Algebra

This is a pretty tricky problem. You have the right general idea: you 
want to show, for example, that in the prime factorization of a + b, 
every prime appears to be an even power. Let's write our equation in 
the form

   (a+b)c = ab

Note that if a prime p divides a+b, then it divides the left side of 
the equation so it must divide the right side also, so it must divide 
ab and therefore must divide either a or b. But if it divides a and 
a+b then it divides b, and similarly if it divides b and a+b then it 
divides a, so it must divide a and b in any case.  

Now let a = p^k a' and b = b^l b', where p does not divide either a' 
or b'.  Now I'm going to let you take over. Substitute back into the 
original equation and try to show that k = l and therefore that p^2k 
is the highest power of p dividing a + b. The other two parts of the 
problem are exactly similar. 

Let me know if you get stuck.  

- Doctor Wilkinson, The Math Forum
  http://mathforum.org/dr.math/   
    
Associated Topics:
High School Number Theory

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