Loading ...
Sorry, an error occurred while loading the content.

Never square

Expand Messages
  • Jon Perry
    Anyone know why 4ab+3(a+b)+2 is never a square? It comes from looking at the alternative 5-squares conjecture, find a set a,b,c,d such that ab-1=u^2 ac-1=v^2
    Message 1 of 2 , Oct 3, 2002
    • 0 Attachment
      Anyone know why 4ab+3(a+b)+2 is never a square?

      It comes from looking at the 'alternative' 5-squares conjecture, find a set
      a,b,c,d such that

      ab-1=u^2
      ac-1=v^2
      ad-1=w^2
      bc-1=x^2
      bd-1=y^2
      cd-1=z^2

      I think no such 4-tuple exists.

      Anyway...(4f+3)(4g+3)-1 = 16fg+12(f+g)+8 = 4[4fg+3(f+g)+2]

      (4f+3)(4g+1) is obviously never a square.

      Jon Perry
      perry@...
      http://www.users.globalnet.co.uk/~perry/maths
      BrainBench MVP for HTML and JavaScript
      http://www.brainbench.com
    • jbrennen
      ... Rewrite this as: N = (4*a+3) * b + (3*a+2) In order for N to be a square, we must have (3*a+2) be a quadratic residue modulo (4*a+3). Note that (3*a+2) is
      Message 2 of 2 , Oct 3, 2002
      • 0 Attachment
        --- In primenumbers@y..., "Jon Perry" <perry@g...> wrote:
        > Anyone know why 4ab+3(a+b)+2 is never a square?

        Rewrite this as:

        N = (4*a+3) * b + (3*a+2)

        In order for N to be a square, we must have (3*a+2) be a
        quadratic residue modulo (4*a+3).

        Note that (3*a+2) is equal to -1/4 (modulo 4*a+3).
        [It is the modular solution to 4x+1 == 0]

        -1/4 is a quadratic residue iff -1 is a quadratic residue.

        And finally, -1 is never a quadratic residue modulo 4*a+3;
        this is a basic result of quadratic reciprocity.

        Or for you PARI/GP types who don't care about all of the theory:

        ? for(a=0,99,print(kronecker(3*a+2,4*a+3)))

        ...prints out an endless stream of -1, which gives strong
        evidence that 3*a+2 can never be a quadratic residue
        modulo 4*a+3.
      Your message has been successfully submitted and would be delivered to recipients shortly.