I cannot find the solution to this problem. It is part of a larger homework question but I can't go on until I solve this question.
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2how may squares? – Asinomás Dec 13 '15 at 18:35
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There is no limit on how many squares can be used but all squares must be unique – user298365 Dec 13 '15 at 18:36
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When you say unique you mean distinct? – Asinomás Dec 13 '15 at 18:37
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Yes! That is what I mean. – user298365 Dec 13 '15 at 18:38
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a solution or all solutions. – fleablood Dec 13 '15 at 18:42
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Thank you! I realize was making a silly mistake...I was trying to multiply them and not add. I'm a little tired. Thanks again! – user298365 Dec 13 '15 at 18:50
4 Answers
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HINT: Find the largest square less than $100,000$, subtract it, and see what remains. (Note that in general you can’t expect this procedure necessarily to work, but it’s an obvious first thing to try, and in this case it’s helpful.)
Brian M. Scott
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Wow, I can't believe we thought of almost exactly the same thing. Although, I don't think this is what should be done in general, but it was worth a shot. – Asinomás Dec 13 '15 at 18:41
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1Oh my mind went elswhere. $100,000 = 10 * 10,000$ and $10=3^2+1^2$ so $100,000=300^2+100^2$. There's probably a dozen ways. – Jorik Dec 13 '15 at 18:43
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@Jorik There seem to be three ways to write 100000 as a sum of two squares, found by brute-force. – MonadBoy Dec 13 '15 at 18:44
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I was thinking $10,000 - m^2 = n^2$ means $(10,000 -m)(10,000 + m)= n^2$ but then I got lazy. – fleablood Dec 13 '15 at 18:46
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@A.Sh Oh that's less than I would have guessed. But I couldn't compute all that in my head ;). – Jorik Dec 13 '15 at 18:48
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@Jorik That's where computers are more than happy to help, I guess : ) – MonadBoy Dec 13 '15 at 18:54
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Let's start by placing the largest number we can as a summand, (because this will make the problem "smaller" for us, although it may not be the correct aproach, but it won't hurt to try).
Taking the square root we find it is $316^2=99,856$. So now we need to write $100,000-99,856=144$ as a square or sum of squares, which is easy as $12^2=144$.
So $100,000=12^2+316^2$
Asinomás
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A brute-force approach by computer yields $$100000=12^2+316^2=100^2+300^2=180^2+260^2$$
MonadBoy
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The complete answer, to express $10^5$ as sum of distinct squares in as many ways as possible, has to do with the factorization of $10$ in the Gaussian numbers, $10=(3+i)(3-i)=i(1-i)^2(2+i)(2-i)$. I’m too lazy and groggy to work it out completely, but since $(3+i)^2=8+6i$ and $(3+i)^4=28+96i$, you can replace the $10^4$ in the equation $10^5=100^2+300^2$ by $80^2+60^2$ as well as by $28^2+96^2$.
Lubin
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