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

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  • newjack56
    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,
    Message 1 of 5 , Dec 7, 2005
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      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
    • ed pegg
      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. I tried to split other Mersenne primes up
      Message 2 of 5 , Dec 7, 2005
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        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.

        I tried to split other Mersenne primes up into two primes,
        but found no other examples up to 2^9689 - 1.

        What is the largest known splitable prime?

        -Ed Pegg Jr
      • 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 3 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 4 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 5 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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