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Re: [PrimeNumbers] Puzzle: changing composite to prime

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  • Jack Brennen
    For one digit, 212159?
    Message 1 of 5 , Jun 18, 2012
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      For one digit, 212159?

      On 6/18/2012 2:29 PM, Mark wrote:
      > Puzzle: What is the smallest composite number ending in 1,3,7 or 9, which cannot be made prime by changing one of its digits? By changing two of its digits?
      >
      > Mark
      >
      > PS I don't know the answer(s), the question just occurred to me
      >
      >
      >
      >
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    • Maximilian Hasler
      a.k.a. A143641 (1) ;-) Maximilian ... [Non-text portions of this message have been removed]
      Message 2 of 5 , Jun 18, 2012
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        a.k.a. A143641 <https://oeis.org/A143641> (1)
        ;-)

        Maximilian



        On Mon, Jun 18, 2012 at 6:01 PM, Jack Brennen <jfb@...> wrote:

        > **
        >
        >
        > For one digit, 212159?
        >
        >
        > On 6/18/2012 2:29 PM, Mark wrote:
        > > Puzzle: What is the smallest composite number ending in 1,3,7 or 9,
        > which cannot be made prime by changing one of its digits? By changing two
        > of its digits?
        > >
        > > Mark
        > >
        > > PS I don't know the answer(s), the question just occurred to me
        > >
        > >
        > >
        > >
        > > ------------------------------------
        >
        > >
        > > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
        > > The Prime Pages : http://primes.utm.edu/
        > >
        > > Yahoo! Groups Links
        > >
        > >
        > >
        > >
        > >
        >
        >
        >


        [Non-text portions of this message have been removed]
      • Jack Brennen
        Note that heuristically, any number ending in 1,3,7,or 9 can be made prime by changing two digits. Note that I m assuming that changing the first digit or the
        Message 3 of 5 , Jun 18, 2012
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          Note that heuristically, any number ending in 1,3,7,or 9
          can be made prime by changing two digits. Note that I'm
          assuming that changing the first digit or the first two
          digits to zero is allowed, but I don't think that detail
          affects the heuristic much.




          On 6/18/2012 3:01 PM, Jack Brennen wrote:
          > For one digit, 212159?
          >
          > On 6/18/2012 2:29 PM, Mark wrote:
          >> Puzzle: What is the smallest composite number ending in 1,3,7 or 9, which cannot be made prime by changing one of its digits? By changing two of its digits?
          >>
          >> Mark
          >>
          >> PS I don't know the answer(s), the question just occurred to me
          >>
          >>
          >>
          >>
          >> ------------------------------------
          >>
          >> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
          >> The Prime Pages : http://primes.utm.edu/
          >>
          >> Yahoo! Groups Links
          >>
          >>
          >>
          >>
          >>
          >
          >
          >
          >
          > ------------------------------------
          >
          > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
          > The Prime Pages : http://primes.utm.edu/
          >
          > Yahoo! Groups Links
          >
          >
          >
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          >
        • Mark
          212159 appears to be it, thanks Jack! How uncanny that Maximilian would happen to be the one who would come up with this very same idea and put the sequence of
          Message 4 of 5 , Jun 19, 2012
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            212159 appears to be it, thanks Jack! How uncanny that Maximilian would happen to be the one who would come up with this very same idea and put the sequence of such "prime proof" numbers in OEIS (four years ago). Now I read that there is a variation of the theme, where one starts with a prime and sees if changing a digit will always result in a composite. These are called "weakly prime" numbers and none other than Terence Tao has proven them to be infinite in number. (wow!)

            Jack I just tried to come up with a heuristic myself, because frankly I thought that doubly prime proof numbers should exist. First, my calculations show that the chances of any given number ending in 1,3,7,9 being prime proof decreases only very slowly as the number of digits increases. For instance around the 10 digit mark the chances of a number being prime proof are about 1 in 6.4 x 10^5 . Around the 100 digit mark the chances of a number being prime proof are about 1 in 5.8 x 10^5.

            But the chances of a number around the 10 digit mark being doubly prime proof is about 1 in 10^350. Around the 100 digit mark the chances of a number being doubly prime proof falls to 1 in 10^3785. So it is essentially impossible to be double prime proof. In other words, as you say, one should always be able to change two digits in a number ending in 1,3,7,9 and obtain a prime.


            Mark


            --- In primenumbers@yahoogroups.com, Jack Brennen <jfb@...> wrote:
            >
            > Note that heuristically, any number ending in 1,3,7,or 9
            > can be made prime by changing two digits. Note that I'm
            > assuming that changing the first digit or the first two
            > digits to zero is allowed, but I don't think that detail
            > affects the heuristic much.
            >
            >
            >
            >
            > On 6/18/2012 3:01 PM, Jack Brennen wrote:
            > > For one digit, 212159?
            > >
            > > On 6/18/2012 2:29 PM, Mark wrote:
            > >> Puzzle: What is the smallest composite number ending in 1,3,7 or 9, which cannot be made prime by changing one of its digits? By changing two of its digits?
            > >>
            > >> Mark
            > >>
            > >> PS I don't know the answer(s), the question just occurred to me
            > >>
            > >>
            > >>
            > >>
            > >> ------------------------------------
            > >>
            > >> Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
            > >> The Prime Pages : http://primes.utm.edu/
            > >>
            > >> Yahoo! Groups Links
            > >>
            > >>
            > >>
            > >>
            > >>
            > >
            > >
            > >
            > >
            > > ------------------------------------
            > >
            > > Unsubscribe by an email to: primenumbers-unsubscribe@yahoogroups.com
            > > The Prime Pages : http://primes.utm.edu/
            > >
            > > Yahoo! Groups Links
            > >
            > >
            > >
            > >
            > >
            >
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