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Re: [PrimeNumbers] 3+ question

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  • Jens Kruse Andersen
    ... Googling Law of Small Numbers shows something in good agreement with Decio s use of the term, e.g.:
    Message 1 of 6 , Mar 5, 2005
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      Milton Brown wrote:

      > There is a Law of Small Numbers in statistics,
      > but you seem to be using the term incorrectly;
      > just dropping it in where it makes no sense.

      Googling "Law of Small Numbers" shows something in good agreement with Decio's
      use of the term, e.g.:
      http://primes.utm.edu/glossary/page.php?sort=LawOfSmall

      Milton's posts on the other hand, seem to often make no sense.

      Harsh Aggarwal wrote:

      > I think the below is true, but I can't prove or disprove the
      > statement. Please help.
      >
      > If 3^(3^n)+3^(((3^n)+1)/2)+1 is prime then is 3^(3^n)-3^(((3^n)+1)/2)+1
      > also prime, and vice-versa? I can't find any counter examples.
      >
      > true for n=0,1,2 up to 10(Till where I checked).

      Only true for n=0 if 1 is considered a prime. Then n = 0, 1, 2 gives 6 small
      primes: (7, 1), (37, 19), (19927, 19441).
      There are no other primes for n<=10. The number of digits approximately
      triples for each n. This growth is so fast that there are probably no more
      primes at all. That means Harsh's guess is actually likely to be true, but in
      an uninteresting way.

      PS: Do not use the default primeform Fermat 3-prp test on these base 3
      expressions.

      --
      Jens Kruse Andersen
    • jim_fougeron
      ... Cosigned, since the Law of Small Numbers makes perfect since in this case. Looking at the first 3 tiny pairs of numbers could lead someone to believe
      Message 2 of 6 , Mar 5, 2005
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        --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
        <jens.k.a@g...>wrote:
        >Milton Brown wrote:
        >
        >>There is a Law of Small Numbers in statistics,
        >>but you seem to be using the term incorrectly;
        >>just dropping it in where it makes no sense.
        >
        >Googling "Law of Small Numbers" shows something in good agreement
        >with Decio's use of the term, e.g.:
        >http://primes.utm.edu/glossary/page.php?sort=LawOfSmall
        >
        >Milton's posts on the other hand, seem to often make no sense.

        Cosigned, since the "Law of Small Numbers" makes perfect since in
        this case. Looking at the first 3 tiny pairs of numbers could lead
        someone to believe that there is some law "causing" this to happen.
        Well, there "might" be such a law, but most likely it is due to
        blind luck, and such tiny numbers which are statistically rich in
        primes.

        One could also look at the "plus" form, and make a statement
        (although wrong), that since n=0 and n=1 and n=2 are all prime,
        then all will be prime (note a Great one, Fermat made such a wild
        claim in the past).

        Both of the above claims are cases where the "Law of Small Numbers"
        causes someone to make a false (or likely false) claim. However,
        for the claim of if "plus" prime, then "minus" prime, (if the
        stipulation of n>1 is added to the original question posed),
        "might" be true, due to the fact that it is likely there are no
        additional primes of either form. There is no "law" (that I am
        aware of), but simple blind luck happened to form these pairs.

        >Harsh Aggarwal wrote:
        >
        >>I think the below is true, but I can't prove or disprove the
        >>statement. Please help.
        >>
        >>If 3^(3^n)+3^(((3^n)+1)/2)+1 is prime then is 3^(3^n)-3^(((3^n)
        +1)/2)+1
        >>also prime, and vice-versa? I can't find any counter examples.
        >>
        >>true for n=0,1,2 up to 10(Till where I checked).
        >
        >Only true for n=0 if 1 is considered a prime. Then n = 0, 1, 2
        gives 6
        >small primes: (7, 1), (37, 19), (19927, 19441).
        >There are no other primes for n<=10. The number of digits
        approximately
        >triples for each n. This growth is so fast that there are probably
        no
        >more primes at all. That means Harsh's guess is actually likely to
        be
        >true, but in an uninteresting way.
        >
        >PS: Do not use the default primeform Fermat 3-prp test on these
        base 3
        >expressions.

        I don't know about that PS statement. It might be good to have some
        of the "newer" prime hunters use default Fermat-3 tests. Frequently
        a wrong answer (especially one which might smack someone right in the
        forehead) will help teach people willing to learn, better than a
        correct answer.

        >Jens Kruse Andersen
      • Milton Brown
        Responding people should surely do their research before making making such might be false statements: Well, there might be such a law, but most likely it
        Message 3 of 6 , Mar 5, 2005
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          Responding people should surely do their research before making
          making such "might" be false statements:

          "Well, there 'might' be such a law, but most likely it is due to
          blind luck ...

          Milton L. Brown
          miltbrown at earthlink.net

          > [Original Message]
          > From: jim_fougeron <jfoug@...>
          > To: <primenumbers@yahoogroups.com>
          > Date: 3/5/2005 5:54:29 AM
          > Subject: [PrimeNumbers] Re: 3+ question
          >
          >
          >
          > --- In primenumbers@yahoogroups.com, "Jens Kruse Andersen"
          > <jens.k.a@g...>wrote:
          > >Milton Brown wrote:
          > >
          > >>There is a Law of Small Numbers in statistics,
          > >>but you seem to be using the term incorrectly;
          > >>just dropping it in where it makes no sense.
          > >
          > >Googling "Law of Small Numbers" shows something in good agreement
          > >with Decio's use of the term, e.g.:
          > >http://primes.utm.edu/glossary/page.php?sort=LawOfSmall
          > >
          > >Milton's posts on the other hand, seem to often make no sense.
          >
          > Cosigned, since the "Law of Small Numbers" makes perfect since in
          > this case. Looking at the first 3 tiny pairs of numbers could lead
          > someone to believe that there is some law "causing" this to happen.
          > Well, there "might" be such a law, but most likely it is due to
          > blind luck, and such tiny numbers which are statistically rich in
          > primes.
          >
          > One could also look at the "plus" form, and make a statement
          > (although wrong), that since n=0 and n=1 and n=2 are all prime,
          > then all will be prime (note a Great one, Fermat made such a wild
          > claim in the past).
          >
          > Both of the above claims are cases where the "Law of Small Numbers"
          > causes someone to make a false (or likely false) claim. However,
          > for the claim of if "plus" prime, then "minus" prime, (if the
          > stipulation of n>1 is added to the original question posed),
          > "might" be true, due to the fact that it is likely there are no
          > additional primes of either form. There is no "law" (that I am
          > aware of), but simple blind luck happened to form these pairs.
          >
          > >Harsh Aggarwal wrote:
          > >
          > >>I think the below is true, but I can't prove or disprove the
          > >>statement. Please help.
          > >>
          > >>If 3^(3^n)+3^(((3^n)+1)/2)+1 is prime then is 3^(3^n)-3^(((3^n)
          > +1)/2)+1
          > >>also prime, and vice-versa? I can't find any counter examples.
          > >>
          > >>true for n=0,1,2 up to 10(Till where I checked).
          > >
          > >Only true for n=0 if 1 is considered a prime. Then n = 0, 1, 2
          > gives 6
          > >small primes: (7, 1), (37, 19), (19927, 19441).
          > >There are no other primes for n<=10. The number of digits
          > approximately
          > >triples for each n. This growth is so fast that there are probably
          > no
          > >more primes at all. That means Harsh's guess is actually likely to
          > be
          > >true, but in an uninteresting way.
          > >
          > >PS: Do not use the default primeform Fermat 3-prp test on these
          > base 3
          > >expressions.
          >
          > I don't know about that PS statement. It might be good to have some
          > of the "newer" prime hunters use default Fermat-3 tests. Frequently
          > a wrong answer (especially one which might smack someone right in the
          > forehead) will help teach people willing to learn, better than a
          > correct answer.
          >
          > >Jens Kruse Andersen
          >
          >
          >
          >
          >
          >
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          > The Prime Pages : http://www.primepages.org/
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