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 April 17th, 2015, 09:08 AM #1 Newbie   Joined: Apr 2015 From: United States Posts: 8 Thanks: 0 Odd numbers that can be represented in only one/two ways as difference of squares a) Find (with proof) all odd numbers that can be represented in only one way as a difference of two squares. I know that the answer to this part lies in the primes, because there is only one way to multiply to get a prime number. Just not sure how to prove it. b) Find (with proof) all odd numbers that can be represented in only two ways as a difference of two squares. I believe that the proof to this part would have something to do with four factors, (let's say a,b,c,d) where n is an odd number and a(d)=n, and b(c)=n. But I'm not too sure where to go after that? Thank you in advance!
 April 17th, 2015, 11:00 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,932 Thanks: 1127 Math Focus: Elementary mathematics and beyond a) primes and perfect squares that are semiprimes. b) semiprimes that are not perfect squares. This excludes using 0 as a perfect square. Consider solving x - a = one factor, x + a = another factor. Last edited by greg1313; April 17th, 2015 at 12:22 PM.
April 18th, 2015, 09:17 AM   #3
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Quote:
 Originally Posted by Nienna426 a) Find (with proof) all odd numbers that can be represented in only one way as a difference of two squares. I know that the answer to this part lies in the primes, because there is only one way to multiply to get a prime number. Just not sure how to prove it. b) Find (with proof) all odd numbers that can be represented in only two ways as a difference of two squares. I believe that the proof to this part would have something to do with four factors, (let's say a,b,c,d) where n is an odd number and a(d)=n, and b(c)=n. But I'm not too sure where to go after that? Thank you in advance!
This might point you in a useful direction.

Prime Numbers: Is the difference of two squares always composite?

 April 27th, 2015, 12:30 AM #4 Senior Member   Joined: Nov 2013 From: Germany Posts: 179 Thanks: 1 Math Focus: Number Theory all odd numbers are difference of two perfect squares a*b = ((a+b)/2)²-((a-b)/2)² a*b = odd number q.e.d.
 April 27th, 2015, 04:27 PM #5 Senior Member     Joined: May 2013 From: España Posts: 151 Thanks: 4 Hello. If you may be interested. Machine of test of prime numbers There is a demonstration, compatible with your question. Regards.

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