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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

AND

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.
Thanks.
```

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

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)

1-0=1
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
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
High School Basic Algebra

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