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24523Re: 1+1+1+2 selfridge composite test and a question

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  • paulunderwooduk
    Oct 4, 2012
      The characteristic equation of [x+2,-2;2,-x+2]
      is L^2-4*L-x^2+8==0

      Let
      P=4
      Q=-(x^2-8)

      > Then
      > v=P^2/Q-2 == -2*x^2/(x^2-8)
      >
      > For this new test of odd n, find x such that:
      > gcd(x,n)==1
      > gcd(x^2-8,n)==1
      > kronecker(x^2-4)==-1
      >
      > and perform the sub-tests:
      > (x+2)^((n-1)/2)==kronecker(x+2,n) (mod n)
      > (x-2)^((n-1)/2)==kronecker(x-2,n) (mod n)
      > x^(n-1)==1 (mod n)
      > L^(n+1)==1 (mod n, L^2-v*L+1)
      >
      > I am testing all x against psp-2 n below 2^32.
      >

      The gremlins score another goal with their counterexample:
      n==741470549 and x==68216238.

      I refuse to take their bait of x^((n-1)/2)!=+-1 (mod n) and "recant"

      Paul
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