April 5th, 2018, 03:01 PM | #1 |
Newbie Joined: Apr 2018 From: Canada Posts: 10 Thanks: 0 | Is it possible?
Hi, I am new here and I see this is a pretty active place! I've been wondering about how difficult it is to find the 2 squares that are the "difference of 2 squares" for a given number. Example 100 = (26 squared) - (24 squared) Given any number, is there a formula to find what those 2 squared numbers are? thanks for anyone replying to this, have a great weekend!! |
April 5th, 2018, 03:47 PM | #2 |
Senior Member Joined: Sep 2016 From: USA Posts: 500 Thanks: 277 Math Focus: Dynamical systems, analytic function theory, numerics |
The difference of two squares always factors like \[n^2 - m^2 = (n + m)(n - m) \] so if the term on the right side is equal to your number, then $n+m$ and $n-m$ are a pair of factors. In your example, $n = 26, m = 24$ and indeed $26 - 24 = 2$ and $26 + 24 = 50$ and $50 \cdot 2 = 100$. |
April 5th, 2018, 04:11 PM | #3 |
Newbie Joined: Apr 2018 From: Canada Posts: 10 Thanks: 0 |
Hey thank you for your reply, I appreciate it! ... however I am not sure I understand how I determine what the squares are by just having 100. What would the answer be for let's say... 57,671 ? Where would I start attempting to find the 1st square? Last edited by skipjack; April 28th, 2018 at 12:33 AM. |
April 5th, 2018, 04:47 PM | #4 | |
Math Team Joined: Dec 2013 From: Colombia Posts: 7,502 Thanks: 2511 Math Focus: Mainly analysis and algebra | Quote:
For any coprime, odd $s$ and $t$, $$a=\frac{s^2-t^2}2, \; b=st, \; c=\frac{s^2+t^2}2$$ is a primitive Pythagorean Triple (meaning that $a$, $b$ and $c$ do not all share a common factor. Moreover, all primitive Pythagorean Triples are generated by the scheme. | |
April 5th, 2018, 04:56 PM | #5 |
Newbie Joined: Apr 2018 From: Canada Posts: 10 Thanks: 0 |
Ok you just blew my mind, I'm not that advanced at Math. I suppose a simple basic answer would be best for me. I assume it's not a simple answer and determining either of the squares requires some sort of brute force technique ? |
April 5th, 2018, 05:17 PM | #6 | ||
Senior Member Joined: Sep 2016 From: USA Posts: 500 Thanks: 277 Math Focus: Dynamical systems, analytic function theory, numerics | Quote:
Quote:
Last edited by skipjack; April 28th, 2018 at 12:34 AM. | ||
April 5th, 2018, 05:17 PM | #7 | |
Senior Member Joined: Aug 2012 Posts: 2,076 Thanks: 593 | Quote:
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 question 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. For example we all know that $3^2 + 4^2 = 5^2$ because $9 + 16 = 25$. This is a very famous Pythagorean triple. We can rewrite it as $3^2 = 5^2 - 4^2$. So $9$ is a difference of squares. You can see that every Pythagorean triple gives rise to (a couple of) examples of numbers that are differences of squares. On the other hand, suppose $a^2 = b^2 - c^2$, a number that's a difference of squares.. Then you can see how it easily turns into a Pythagorean triple by moving the $c^2$ term to the left. Finding all the numbers that are the difference of squares is exactly the same problem as finding all the Pythagorean triples. This is a great example of how mathematicians think. You solve a problem by recognizing that it's actually some other problem that you already know the answer to. Hope this Cliff Notes to v8archie's post was helpful. [Do they still use Cliff Notes these days? Or does everyone just go with the 140-character summary? "Disabled guy hunts endangered whale and gets what's coming to him."] Last edited by skipjack; April 28th, 2018 at 12:37 AM. | |
April 5th, 2018, 05:27 PM | #8 | |
Newbie Joined: Apr 2018 From: Canada Posts: 10 Thanks: 0 | Quote:
I'm not a math student or even have much more than basic math skills, but I do very much look for anomalies in math by using patterns that I discovery often.. I found something interesting so I wanted to understand how easy it was to find the "difference of 2 squares" given the number. | |
April 5th, 2018, 05:32 PM | #9 |
Math Team Joined: May 2013 From: The Astral plane Posts: 1,907 Thanks: 771 Math Focus: Wibbly wobbly timey-wimey stuff. | |
April 5th, 2018, 05:35 PM | #10 |
Newbie Joined: Apr 2018 From: Canada Posts: 10 Thanks: 0 |
As for whether or not this number has an answer, yes it does but I have a very long winded way of doing it but it requires you to know the factors,, if you know the factors then getting the 2 squares is easy. I just want to know if it's possible to do it the other way. 57671 = 336(2) - 235(2) I suppose, if there was an easy way to get the squares without the factors then I could simply reverse the same pattern I found and get the factors. I don't talk "math" as I'm very basically educated in it, however, I do spend countless hours playing with patterns of math and I've found so many cool things about it.. it's simply discovery of things math, very satisfying. |