On Thursday 08 October 2009 21:10:02 Dan Asimov wrote:
Find all rational solutions of
x^2 + y^2 + z^2 + 3(x + y + z) + 5 = 0.
Since no one else has bitten yet, here's a solution after some spoiler space. THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK THIS SPACE INTENTIONALLY LEFT BLANK There are no rational solutions. Let x=a/n etc. with minimal n. Then we're solving x^2 + y^2 + z^2 + 3n(x+y+z) + 5n^2 = 0; not all of x,y,z,n are even. If 4|n then 4|x^2+y^2+z^2, whence x,y,z are all even, contradiction. If n is odd then x^2+3nx etc. are all even, so the LHS is odd, hence nonzero; contradiction. So n=2m, m odd. Complete the squares, writing u=x+3m etc.; we get u^2+v^2+w^2 = 7m^2 but the RHS is 7 (mod 8) which the LHS cannot be. (We could replace the last three paragraphs with "Now just try all possibilities for x,y,z,n mod 4 and observe that none of them works.") -- g