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henpen
 2 years ago
Show that 168 cannot be expressed as the sum of the squares of two rational numbers.
henpen
 2 years ago
Show that 168 cannot be expressed as the sum of the squares of two rational numbers.

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

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

henpen
 2 years ago
Best ResponseYou've already chosen the best response.1\[a,b,m,n \in \mathbb{Z}\]

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2Here 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?

henpen
 2 years ago
Best ResponseYou've already chosen the best response.1Are you assuming that a and b are integers?

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

henpen
 2 years ago
Best ResponseYou've already chosen the best response.1Fair enough, I made the same mistake intially.

asnaseer
 2 years ago
Best ResponseYou've already chosen the best response.2hmmm  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.

henpen
 2 years ago
Best ResponseYou've already chosen the best response.1This 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

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