Jonathan Fletcher
2004-10-08 11:05:38 |
Hi
Hi, I was bored at work. I always had a theory/thought about odd number squares and put it to test on a spreadsheet which blew my theory apart - not spectacularly and the answer was always 32 - I wanted to know why - that's why I searched and found this site. Unfortunately I'm not allowed to ask anyone anything. |
nikki
2004-10-08 11:25:08 |
Re: Hi
Hey jonathan,
What do you mean you aren't allowed to ask anyone anything?
You're right, we are asked not to posting problems in the forums. But posting a problem is when you know the answer and just want to post it without waiting for the regular submission queue. Or when you have a homework problem that you want someone else to answer for you.
But that's not the same as you asking some number theory questions. So ask away! |
Fletch
2004-10-08 11:41:11 |
Re: Hi
Hi Nikki - thanks
Since school days I was always fascinated with the fact that if you take an odd number, square it, divide by two and subtract and add a half to the result you achieve a pythagorean triple. (eg 1, 0, 1; 3, 4, 5; 5, 12, 13) I think I arrived at this theory myself and I always assumed it to be true. However I never had access to a spreadsheet before and I created a simple spreadsheet to test my theory. I wished I hadn't, when the spreadsheet gets to 28339 my theory falls apart.
Can you confirm that this is so or is it just the limitations of my spreadsheet package? It seems a bit silly to let "Fletch's Theory" get to 27661 before it falls apart. It also does so at 28339, 28589... Is there any reason why it should do so? Why is the answer always 32?
If you are too busy to answer the above, please reply that you are too busy and I will completely understand.
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nikki
2004-10-08 12:54:45 |
Re: Hi
Hey...
I'm not a mathematician, so even after you answer these questions I might not be able to help. But could you clarify the following?
1) So for your pythagorean triple, you have a, b, c such that a is odd, b=(a^2)/2 +1/2, and c = (a^2)/2 - 1/2. Is that correct? It was hard to follow.
2) What do you mean "why is the answer always 32?"? |
nikki
2004-10-08 13:12:14 |
Re: Hi
Looking again... I get as far as 44721 before it breaks down at 44723 for me. That makes me believe that it's just the memory space in the spreadsheet package.
I wrote it out as my first column was just an odd number (a). My next column is (a^2)/2. So that is going to be xxx.5, no matter what. But at 44723, my (a^2)/2 loses it's .5 decimal. Also, with a = 44721, (a^2)/2 = 999983920.5. But with a = 44723, (a^2)/2 "=" 1000073365
I don't think it's a coincidence that when the answer went from a 9 digit number (with decimal place) to a 10 digit number that the decimal disappeared. This further supports it's just a package limitation.
I would suggest you find a way to prove the theory, and then submit it into the problem queue.
Good luck! |
Rajal
2004-10-08 15:29:34 |
Re: Hi
Write out the equation, blow it out, then simplify. The starting equation is a^2 + b^2 = c^2. Set b = (a^2/2)-1/2 and c = (a^2/2)+1/2. You should get the identity (1=1) eventually. Since only odd a's will give integer values, it shows your intuition correct. |
Fletch
2004-10-09 21:18:06 |
Re: Hi
Thank you everybody. I'm not really a mathematician but I think I might have done my first proof (eventually) thanks to Rajal. Thanks too Nikki, the answer was 32 when I subtracted c^2 from a^2+b^2. When I look further the answer becomes 64 when my spreadsheet goes up another digit so your theory about the limitations of my spreadsheet package seem correct.
Thanks for making me welcome.
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Tristan
2004-10-10 18:44:26 |
Re: Hi
I remember noticing this property once to. This is the way I justified it.
Every perfect square k^2 is the sum of the first k positive odd numbers. Therefore, the difference of squares n^2-k^2 should be the sum of all the positive odd numbers between the kth and nth. In the simplest case, k+1=n, and the difference is just a single odd number. If this odd number were a perfect square of m, then k, n, and m would be a pythagorean triple. Your method of finding a pythagorean triple is just a way of choosing any m and then finding k and n.
I'd say it is definitely a limitation on excel. Excel will only calculate so many digits. |
Aspiring Novice
2004-10-10 19:12:28 |
Re: Hi
Re: Hi
Hey...
I'm not a mathematician, so even after you answer these questions I might not be able to help. But could you clarify the following?
1) So for your pythagorean triple, you have a, b, c such that a is odd, b=(a^2)/2 +1/2, and c = (a^2)/2 - 1/2. Is that correct? It was hard to follow.
2) What do you mean "why is the answer always 32?"?
|
nikki
2004-10-10 23:10:24 |
Re: Hi
That was weird. I wonder who that was, why the copied my earlier post, and why they didn't say anything else :)
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