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  • Prime Question n||p

    From Carl G.@21:1/5 to All on Fri Jan 6 10:32:29 2023
    I had the following thought when I was trying to fall asleep the other day:

    The prime numbers 2 and 5 have the property that when a number is formed
    by concatenating an integer from 1 to infinity with the prime, the
    number is always composite (for 2: 12, 22, 32, 42, ... 102, 112, ...;
    and for 5: 15, 25, 35, 45, ...). Using "||" as a concatenation
    operator, then this can be expressed as: If p is the prime and n is in
    the set of integers from 1 to infinity, then n||p is composite. For
    most primes, some of the numbers in the set formed by concatenation
    would be composite and some would be prime. For example, for 11: 111 is composite, 211 is prime, 311 is prime, 411 is composite, etc. Are there
    primes other than 2 and 5 in which all the numbers would be composite?

    --
    Carl G.

    --
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  • From Gareth Taylor@21:1/5 to Carl G. on Fri Jan 6 20:25:01 2023
    In article <tp9pft$37ug2$1@dont-email.me>,
    Carl G. <carlgnewsDELETECAPS@microprizes.com> wrote:

    Are there primes other than 2 and 5 in which all the numbers would be composite?

    No, by Dirichlet's theorem on primes in arithmetic progression, which
    says that if a and d are coprime integers then there are infinitely many
    primes of the form a+nd.

    Your sequences have this form: e.g., 111, 211, 311, 411, ... is 11+100n.

    Any prime other than 2 or 5 is coprime to that 10^k term, and so the
    sequence will have many primes in it.

    https://en.wikipedia.org/wiki/Dirichlet%27s_theorem_on_arithmetic_progressions

    Gareth

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  • From Doc O'Leary ,@21:1/5 to Carl G. on Sat Jan 7 19:32:34 2023
    For your reference, records indicate that
    "Carl G." <carlgnews@microprizes.com> wrote:

    I had the following thought when I was trying to fall asleep the other day:

    The prime numbers 2 and 5 have the property that

    they are factors of 10. I think anything beyond that is numerology.

    It’s like realizing that the digits for multiples of 9 add up to multiples of 9.

    --
    "Also . . . I can kill you with my brain."
    River Tam, Trash, Firefly

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  • From Richard Tobin@21:1/5 to droleary.1022@2022.subsume.com on Sun Jan 8 12:32:01 2023
    In article <tpchci$3j024$1@dont-email.me>,
    Doc O'Leary , <droleary.1022@2022.subsume.com> wrote:

    I had the following thought when I was trying to fall asleep the other day: >>
    The prime numbers 2 and 5 have the property that

    they are factors of 10. I think anything beyond that is numerology.

    That explains why they have that property, but shows nothing about
    the case for other digits, which is a much more interesting problem.

    -- Richard

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  • From Doc O'Leary ,@21:1/5 to Richard Tobin on Mon Jan 9 04:15:41 2023
    For your reference, records indicate that
    richard@cogsci.ed.ac.uk (Richard Tobin) wrote:

    In article <tpchci$3j024$1@dont-email.me>,
    Doc O'Leary , <droleary.1022@2022.subsume.com> wrote:

    I had the following thought when I was trying to fall asleep the other day:

    The prime numbers 2 and 5 have the property that

    they are factors of 10. I think anything beyond that is numerology.

    That explains why they have that property, but shows nothing about
    the case for other digits, which is a much more interesting problem.

    I disagree. There’s nothing to “show” for other digits. It’s just math,
    and/or a quirk of our common base-10 representation. I mean, feel free
    to explore a 3 * 7 = base-21 system to see what “interesting” things may hold true. Nothing wrong with finding new ways to count sheep. :-)

    --
    "Also . . . I can kill you with my brain."
    River Tam, Trash, Firefly

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  • From Gareth Taylor@21:1/5 to droleary.1022@2022.subsume.com on Mon Jan 9 09:35:51 2023
    In article <tpg4dd$3ek8$1@dont-email.me>,
    Doc O'Leary , <droleary.1022@2022.subsume.com> wrote:

    I disagree. There’s nothing to “show” for other digits. It’s just math, and/or a quirk of our common base-10 representation. I mean,
    feel free to explore a 3 * 7 = base-21 system to see what
    “interesting” things may hold true. Nothing wrong with finding new
    ways to count sheep. :-)

    It may be just maths, but it's interesting and challenging maths! For
    all primes other than 2 or 5, the sequence described contains infinitely
    many primes and infinitely many composites.

    Yes, the "other than 2 or 5" bit is to do with our number base being 10.
    If you worked in base 21 then it would be "other than 3 or 7".

    As for something to show, it's answering what the original question
    asked.

    Gareth

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