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April 5th, 2018, 05:44 PM   #11
SDK
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Quote:
Originally Posted by Maschke View Post
I will try to provide an explanatory walkthrough of @v8archie's post.



I have a question for you. How do you know this number can be written as a difference of squares at all?

Now that, as it turns out, is a very sophisticated question. It's the kind of questions mathematicians ask themselves.

What v8archie pointed out is that the numbers that can be written as the difference of squares are exactly those numbers that are members of Pythagorean triples.
But this isn't correct. For example, 57671 is not a perfect square so it isn't a member of any pythagorean triple but it is a difference of squares. Based on the factorization $N = (n+m)(n-m)$ it appears to be both necessary and sufficient that $N$ has a pair of divisors which are equal mod 2.

As an example, every odd prime satisfies this criterion and no prime is ever a member of any Pythagorean triple. An explicit formula is given by letting $p$ be an odd prime that $p = p\cdot 1$ so we have to solve $n+m = p, n-m=1$ which is easily seen to give $n = \frac{p-1}{2} + 1$ and $m = \frac{p-1}{2}$ and we have
\[ (\frac{p-1}{2} + 1)^2 - (\frac{p-1}{2})^2 = p \]
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April 5th, 2018, 05:48 PM   #12
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Not sure why it isn't correct?

336(2) = 112896
235(2) = 55225

112896 - 55225 = 57671

Am I missing something?
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April 5th, 2018, 05:49 PM   #13
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ohhh, nm.. you were referring to
"57671 is not a perfect square so it isn't a member of any pythagorean triple but it is a difference of squares"
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April 5th, 2018, 05:49 PM   #14
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But this isn't correct.
Yes I got carried away and got all this totally wrong. I'll quit while I'm behind now.
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April 5th, 2018, 05:51 PM   #15
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k, so it sounds like there is no magic bullet to determine these squares.. interesting

if you knew the factors of the number would that help in any "known" way now?
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April 5th, 2018, 06:13 PM   #16
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Originally Posted by Clonus View Post
336(2) = 112896
235(2) = 55225
For the record, it would be better to write this as
336^2 = 112896
235^2 = 55225

-Dan
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April 5th, 2018, 06:15 PM   #17
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Quote:
Originally Posted by topsquark View Post
For the record, it would be better to write this as
336^2 = 112896
235^2 = 55225

-Dan
Thanks Dan, sorry.. I couldn't find that little 2 to insert there... I'll use ^ next time.

Last edited by skipjack; April 28th, 2018 at 12:40 AM.
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April 5th, 2018, 06:56 PM   #18
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So, is this thread dead? lol

There was so much initial educational enthusiasm, then all of a sudden it zeroed out.

Last edited by skipjack; April 28th, 2018 at 12:43 AM.
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April 5th, 2018, 07:47 PM   #19
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So, is this thread dead? lol

There was so much initial educational enthusiasm, then all of a sudden it zeroed out.
It doesn't work like a chat room. People may respond days from now. Sometimes a thread lies dormant for years and someone revives it. Check back from time to time. The question will be here forever.

Last edited by skipjack; April 28th, 2018 at 12:43 AM.
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April 5th, 2018, 07:54 PM   #20
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Originally Posted by Clonus View Post
So, is this thread dead? lol

There was so much initial educational enthusiasm, then all of a sudden it zeroed out.
It's not dead but it does get a little complicated.

Your equation $\displaystyle x^2 - y^2 = C$ is what's called a Pell equation. It's a little hair-raising to read, but all the information you would need to solve the problem is right in there.

-Dan

Last edited by skipjack; April 28th, 2018 at 12:44 AM.
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