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Re: Lucas Sequences. Why is S0=4?????

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  • grostoon
    ... book I ... help!! ... Hi matt, Take a look at this paper : http://www.maths.tcd.ie/pub/ims/bull54/M5402.pdf You will understand that in fact, S0=4 is one
    Message 1 of 5 , Dec 7, 2005
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      --- In primenumbers@yahoogroups.com, "newjack56" <mrnsigepalum@a...>
      wrote:
      >
      > I have a quick question. How did Lucas determine s of 0 =4? The
      book I
      > am reading says he uses U of n and V of n for (P,Q). IF (P,Q)
      > determines that s of 0 =4, Which equation did he use or which (P,Q)
      > did he use? Or, is 4 used because it is the first composite number.
      > Because if you use 2 or 3, you get prime numbers. Thanks for any
      help!!
      >
      >
      > -matt
      >

      Hi matt,

      Take a look at this paper :
      http://www.maths.tcd.ie/pub/ims/bull54/M5402.pdf

      You will understand that in fact, S0=4 is one possibility but that
      there are in fact infinitely many others (look at theorem 4 on page
      71).

      For example, S0=10 or S0=52 or S0=724 work as well.

      Regards,

      Jean-Louis.
    • Jens Kruse Andersen
      ... I have not heard of searches for large splitable primes. One way to search them is to concatenate known primes. 27918*10^10000+1 was found in 1999 by Yves
      Message 2 of 5 , Dec 7, 2005
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        Ed Pegg Jr wrote:

        > 2^17 - 1 = 131071. Both 131 and 071 are prime, or 131~071,
        > or 3~(2^17-1), since it splits at the third position.
        >
        > What is the largest known splitable prime?

        I have not heard of searches for large splitable primes.
        One way to search them is to concatenate known primes.

        27918*10^10000+1 was found in 1999 by Yves Gallot.
        58789 is the smallest prime which can be put in front and give a prime:

        D:>pfgw -t -q"5878927918*10^10000+1"
        PFGW Version 1.2.0 for Windows [FFT v23.8]

        Primality testing 5878927918*10^10000+1 [N-1, Brillhart-Lehmer-Selfridge]
        Running N-1 test using base 3
        Calling Brillhart-Lehmer-Selfridge with factored part 69.83%
        5878927918*10^10000+1 is prime! (26.4531s+0.0023s)

        The form k*10^n+1 was chosen to make the concatenation easily provable.
        PrimeForm/GW prp tested and APTreeSieve sieved.

        --
        Jens Kruse Andersen
      • mrnsigepalum@aol.com
        I found two more splitable primes, 127,031 (127)(031) and 839809 (839)(809). -matt [Non-text portions of this message have been removed]
        Message 3 of 5 , Dec 7, 2005
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          I found two more splitable primes, 127,031 (127)(031) and 839809
          (839)(809).



          -matt


          [Non-text portions of this message have been removed]
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