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Trouble with deducing the contradiction

  1. Jul 6, 2012 #1
    Let x,y,z be integers satisfying a specific condition, which boils down to
    [itex]5|(x+y-z)[/itex] and [itex]2*5^{4}k=(x+y)(z-y)(z-x)((x+y)^2+(z-y)^2+(z-x)^2)[/itex]
    or equivalently [itex]5^{4}k=(x+y)(z-y)(z-x)((x+y-z)^2-xy+xz+yz)[/itex]
    I want to show that GCD(x,y,z)≠1, starting with the assumption 5 dividing (x+y), (z-y), or (z-x) results in x,y or z being divisible by 5. then it's easy to show that 5 divides another term, implying 5 divides all three.
    I run into trouble assuming 5 divides the latter part, [itex]2((x+y)^2+(z-y)^2+(z-x)^2)=((x+y-z)^2-xy+xz+yz)[/itex] and showing the contradiction from that point.
    Any hints?
     
  2. jcsd
  3. Jul 8, 2012 #2

    haruspex

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    Suppose 5 | a2+b2+c2. Each term individually must be 0, 1 or 4 mod 5. The only ways for three such to add to 0 mod 5 involve at least one of them being 0 mod 5, no?
     
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