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Perfect Square: Solving Two Equations

Date: 6/21/96 at 20:58:1
From: Sheldon Wilson
Subject: Find x so that Two Expressions are Perfect Squares

Find the value of x such that

        x^2 + 5 is a perfect square


        x^2 - 5 is also a perfect square.

The value of x must not contain surds; that is,
x must equal M / N where M = {1 or 2 or 3 or 4 or . . .} 
and N = {1 or 2 or 3 or . . .}

I have solved this problem correctly but I am not satisfied 
with my solution. I am looking for a more elegant solution. 

Date: 6/26/96 at 11:5:38
From: Doctor Brian
Subject: Re: Find x so that Two Expressions are Perfect 

Try looking at some of the differences between perfect 
squares. You're trying to find a difference of ten (between 
x^2 - 5 and x^2 + 5)

4-0=4   4-1=3
9-0=9   9-1=8   9-4=5
16-0=16 16-1=15 16-4=12 16-9=7
25-0=25 25-1=24 25-4=21 25-9=16 25-16=9

At this point, the difference between *consecutive* perfect 
squares is more than ten:
(x + 1)^2 - x^2 = x^2 + 2x + 1 - x^2 = 2x + 1, which for x>5 
gives an answer greater than or equal to 11.

Finally, since squaring a number is an increasing function, 
then any larger gap between the numbers must give a larger 
difference that between consecutive numbers.  For instance, 
the gap between the squares of 6 and 8 must be larger than the 
difference in the squares of 6 and 7.

So the only possible way to give a difference less than 10 
between the numbers would be to use numbers less than five.  
But the above table shows that there is no pair of numbers 
less than five whose squares give a difference of ten.

-Doctor Brian,  The Math Forum
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Associated Topics:
High School Basic Algebra

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