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henpen Group Title

Show that 168 cannot be expressed as the sum of the squares of two rational numbers.

  • one year ago
  • one year ago

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  1. henpen Group Title
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    \[13^2=\frac{a^2}{b^2}+\frac{m^2}{n^2}+1\]

    • one year ago
  2. henpen Group Title
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    \[13^2=\frac{(pm)^2}{(qn)^2}+\frac{m^2}{n^2}+1\]\[13^2=\frac{(pm)^2+(qm)^2}{(qn)^2}+1\]

    • one year ago
  3. henpen Group Title
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    \[a,b,m,n \in \mathbb{Z}\]

    • one year ago
  4. henpen Group Title
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    Curse you, Diophantus!

    • one year ago
  5. asnaseer Group Title
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    :)

    • one year ago
  6. asnaseer Group Title
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    Here are my thoughts - if:\[168=a^2+b^2\]then either a and b are both even or both odd. take the case of both odd so a=2m+1 and b=2n+1:\[168=4m^2+4m+4n^2+4n+2\]which leads to:\[166=4(m^2+n^2+m+n)\]but 166 is not evenly divisible by 4 so this case can be rejected. now take both a and b as even, so a=2m and b=2n leads to:\[168=4m^2+4n^2\]thus:\[42=m^2+n^2\]strange how the number 42 appears everywhere :) I can continue on this train of thought - but do you think it will lead anywhere good?

    • one year ago
  7. henpen Group Title
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    Are you assuming that a and b are integers?

    • one year ago
  8. asnaseer Group Title
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    ah! sorry - didn't read your question properly - let me think again...

    • one year ago
  9. henpen Group Title
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    Fair enough, I made the same mistake intially.

    • one year ago
  10. asnaseer Group Title
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    hmmm - this is beyond my current understanding. However, I did find this article that has a very similar problem - maybe you will be able to understand it better: https://docs.google.com/viewer?a=v&q=cache:zNNprPwolosJ:www.math.ucsd.edu/~okikiolu/104b/hws4.pdf+&hl=en&gl=uk&pid=bl&srcid=ADGEESjuK9ScKokS94ta6viGZM-sAr6CpRkDKkoZCaHDIYTKGyysQ3HV4V9WXpsfgaE3-1QNepW2Q-3KKfpDfhJpOfRb3e8q0wGZTkkcD9AfH-IYVEtu7DqwdSfxSG_443AnysJK3vDs&sig=AHIEtbQavqEWSt-SGk2LHnfyVAiU8D1IHA It is on the first page - problem 4: Show that 21 cannot be expressed as the sum of squares of two rational numbers.

    • one year ago
  11. henpen Group Title
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    This is from the very interesting http://www.komal.hu/verseny/feladat.cgi?a=honap&h=201211&t=mat&l=en if anyone's interested. @asnaseer , thanks for the link

    • one year ago
  12. asnaseer Group Title
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    thanks @henpen - gives me more things to learn! :)

    • one year ago
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