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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
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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henpen Group TitleBest ResponseYou've already chosen the best response.1
\[13^2=\frac{a^2}{b^2}+\frac{m^2}{n^2}+1\]
 one year ago

henpen Group TitleBest ResponseYou've already chosen the best response.1
\[13^2=\frac{(pm)^2}{(qn)^2}+\frac{m^2}{n^2}+1\]\[13^2=\frac{(pm)^2+(qm)^2}{(qn)^2}+1\]
 one year ago

henpen Group TitleBest ResponseYou've already chosen the best response.1
\[a,b,m,n \in \mathbb{Z}\]
 one year ago

henpen Group TitleBest ResponseYou've already chosen the best response.1
Curse you, Diophantus!
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
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

henpen Group TitleBest ResponseYou've already chosen the best response.1
Are you assuming that a and b are integers?
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
ah! sorry  didn't read your question properly  let me think again...
 one year ago

henpen Group TitleBest ResponseYou've already chosen the best response.1
Fair enough, I made the same mistake intially.
 one year ago

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
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=ADGEESjuK9ScKokS94ta6viGZMsAr6CpRkDKkoZCaHDIYTKGyysQ3HV4V9WXpsfgaE31QNepW2Q3KKfpDfhJpOfRb3e8q0wGZTkkcD9AfHIYVEtu7DqwdSfxSG_443AnysJK3vDs&sig=AHIEtbQavqEWStSGk2LHnfyVAiU8D1IHA 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

henpen Group TitleBest ResponseYou've already chosen the best response.1
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

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
thanks @henpen  gives me more things to learn! :)
 one year ago
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