All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars    
perplexus dot info

Home > Just Math
Rational Square Sum (Posted on 2015-05-07) Difficulty: 3 of 5
Can the sum of squares of two rational numbers equal 14?
If so, give an example.
If not, prove it.

No Solution Yet Submitted by K Sengupta    
No Rating

Comments: ( Back to comment list | You must be logged in to post comments.)
No Subject | Comment 1 of 5
It's impossible for the sum of the squares of two rational numbers to equal 14.

Set x=a/b and y=c/d and multiply out to get A^2 + B^2 = 14C^2, where A=(ad)^2, B=(bc)^2, C=(bd)^2.

(A,B) will be both even or both odd.  In the first case, C will also be even, and after factoring 4 from each term an equation of the same form is the result.

So take (A,B) both odd.  Then necessarily 7C^2 = 1 mod4, 7C^2 = 21 mod4, and C^2 = 3 mod4.  But all squares are either =1 or =0 mod 4 so the conclusion follows.  

  Posted by xdog on 2015-05-07 13:53:09
Please log in:
Remember me:
Sign up! | Forgot password

Search body:
Forums (1)
Newest Problems
Random Problem
FAQ | About This Site
Site Statistics
New Comments (6)
Unsolved Problems
Top Rated Problems
This month's top
Most Commented On

Copyright © 2002 - 2019 by Animus Pactum Consulting. All rights reserved. Privacy Information