Prime Numbers as the Difference of Two SquaresDate: 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/ |
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