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Prime Numbers as the Difference of Two Squares

Date: 08/07/2002 at 20:17:01
From: Rachael
Subject: Prime numbers as the difference of two squares

I can't figure out how to express the prime number 7, the prime number 
15, and the prime number 261 as the difference of two squares.

If you would help me I would be very appreciative. Thanks.


Date: 08/07/2002 at 23:51:47
From: Doctor Paul
Subject: Re: Prime numbers as the difference of two squares

   4^2 - 3^2 = 16 - 9 = 7

   4^2 - 1^1 = 16 - 1 = 15

   19^2 - 10^2 = 361 - 100 = 261

Now, I didn't just pull these answers out of thin air. They come from 
the factorizations of 7, 15, and 261:

    7 = 7*1 = (4+3)*(4-3) = 4^2 - 3^2

   15 = 3*5 = (4 - 1)*(4 + 1) = 4^2 - 1^2

  261 = 9*29 = (19-10)*(19+10) = 19^2 - 10^2

Notice also that

   261 = 3*87 = (45 - 42)*(45 + 42) = 45^2 - 42^2

So 261 can be written as a difference of two squares in two different 
ways.

I hope this helps. Please write back if you'd like to talk about this 
some more.

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

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