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Re: primes modulo 4

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  • Robert
    ... Hmm, must have been drunk when I wrote this. For modulo read modulo order, or multiplicative order. No base that is square produces a p-1 multiplicative
    Message 1 of 2 , Aug 17, 2007
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      --- In primenumbers@yahoogroups.com, "Robert" <robert_smith44@...> wrote:
      >
      > No prime p is (p-1)modulo 4, as primes that are(p-1)modulo 2 are
      > [(p-1)/2]modulo4, and if the order of pmodulo2 is less than p-1, then
      > the modulo4 of p is either the same as the modulo 2 value or is 1/2 of
      > the value.
      >
      > It should be possible therefore to determine values k in the power
      > series k*4^n+/1, which generate, for all values of n, no factors
      > smaller than any given prime value through using a modular sieve
      > process which is much more efficient that choosing the equivalent
      > primorial or even payam.
      >
      > As a result, the potential for long Cunningham Chains base 4 becomes
      > apparent.
      >
      > So far, the longest I have found (and found after only 3 minutes of
      > checking!!) is length 13, namely k= 6703351518. A further 5 hours of
      > checking produced a fair selection of length 10 chains.
      >
      > Whilst maybe it will be difficult to find one of length 17, it seems
      > that the trade off between the modular sieve and working in base 4
      > instead of base 2 might work in favour of the sieve, although I have
      > no way of proving this.
      >

      Hmm, must have been drunk when I wrote this. For modulo read modulo
      order, or multiplicative order.

      No base that is square produces a p-1 multiplicative order, 4 is the
      smallest square base, it works just as well with bases 9,16,25...
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