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April 17th, 2015, 09:08 AM   #1
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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!
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April 17th, 2015, 11:00 AM   #2
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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.
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April 18th, 2015, 09:17 AM   #3
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Quote:
Originally Posted by Nienna426 View Post
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?
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April 27th, 2015, 12:30 AM   #4
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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.
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April 27th, 2015, 04:27 PM   #5
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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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