Got Homework?
Connect with other students for help. It's a free community.
Here's the question you clicked on:
 0 viewing
henpen
Group Title
Show that 168 cannot be expressed as the sum of the squares of two rational numbers.
 2 years ago
 2 years ago
henpen Group Title
Show that 168 cannot be expressed as the sum of the squares of two rational numbers.
 2 years ago
 2 years ago

This Question is Closed

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

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

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

henpen Group TitleBest ResponseYou've already chosen the best response.1
Curse you, Diophantus!
 2 years 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?
 2 years ago

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

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

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

asnaseer Group TitleBest ResponseYou've already chosen the best response.2
thanks @henpen  gives me more things to learn! :)
 2 years ago
See more questions >>>
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.