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General form of prime no.s

  1. Apr 13, 2009 #1
    is there any other general form of prime no.s known except 6n+/-5 and 6n+/-1. is there any general form of n such that 6n+/-5 or 6n+/-1 is a composite no.?
     
  2. jcsd
  3. Apr 13, 2009 #2
    4n+/-1
    6n+/-1 is composite iff there are nonzero integers a and b such that n = 6ab + a + b.
    for instance 6(4) + 1 is composite since 4 = 6(-1)(-1) + (-1) + (- 1)
     
  4. Apr 13, 2009 #3

    CRGreathouse

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    As many as you'd like. 2n + 1, for example.

    n = 141.

    n = 5k.

    n in {1, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19, 20, 21, 22, 23, 24, 25, ...}

    n in {k | k - 5, k - 1, k + 1, or k + 5 can be written as ab with 1 < a <= b}

    Assuming, of course, that we interpret your statement identically.
     
  5. Apr 14, 2009 #4
    but, then, if i am not wrong, this series doesn't have a pattern, does it?
    what i meant by my question was if there is a general form or, a set of general forms which can represent each and every prime no. exhaustively.
     
  6. Apr 14, 2009 #5

    CRGreathouse

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    Wow, impredicativity in real life!

    The sequence has a pattern, it's stated just below it.
     
  7. Apr 14, 2009 #6
    n in {k | k - 5, k - 1, k + 1, or k + 5 can be written as ab with 1 < a <= b}
    i should've asked this earlier, i have no idea what that line between n and k stands for. so i can't understand what this statement means.
     
  8. Apr 14, 2009 #7

    CRGreathouse

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    "such that"
     
  9. Apr 14, 2009 #8
    oh now i get it.but that just comes directly from the definition of prime no.s.
    let me rephrase my original question- is there any general form/set of forms an+/-b which expresses every prime no. exhaustively, barring the cases where n=ck+/-d where a,b,c,d are constants and n,k integers>=0. i repeat again, this set of forms should represent every prime no. exhaustively.
     
  10. Apr 14, 2009 #9

    CRGreathouse

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    Sure, r for r a real number. Also r^2 + pi/2 (but not r^2 + pi). Also a^2 + b^2 + c^2 + d^2 for a, b, c, and d integers.
     
  11. Apr 16, 2009 #10
    a^2 + b^2 + c^2 + d^2 for a, b, c, and d integers.
    but then if a=b=c=d=1, we get 4. how is this prime? and, is it r^(2+pi/2) or r^2+pi/2
     
  12. Apr 16, 2009 #11

    CRGreathouse

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    6n+1 for n = 4, how is this prime?

    You asked for forms that cover all the primes, not for forms that were only prime.

    I intended the second, but both work.
     
  13. Apr 16, 2009 #12

    matt grime

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    There is no choice of a and b so that an+b is prime for every n.

    There is no choice of a and b (except for the trivial a=1 b=0 type) so that every prime is of the form an+b for some n (above you'll note that you have a collection of choices that will give every prime, with exceptions such as 2 and 3 for the 6n+1 and 6n-1 case).

    The 'pattern' of the primes is entirely deterministic (sieve of what's-his-face) and simultaneously very hard to prove anything about (e.g. twin prime conjecture).

    At least that is what I think you're getting at.
     
    Last edited: Apr 16, 2009
  14. Apr 17, 2009 #13
    i'm not sure but i think i found a set of 8 forms an+b which express every prime except 2,3,5.
    and each of these forms have exceptions, ie values of n for which no. is composite, based on integer solutions of set of quadratic eqns in two variables. i will work them out probably in a day or two after the damned sessionals are over. is this a new approach or has someone already done this?
    ps- what is sieve of what's-his-face?
     
    Last edited: Apr 17, 2009
  15. Apr 17, 2009 #14

    CRGreathouse

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    By 8 you mean {1, 7, 11, 13, 17, 19, 23, 29} mod 30. Yes, all primes greater than 5 are of that form. What's more, almost all numbers of this form are composite -- only a tiny fraction are prime.

    You can go a step higher if you'd like. All primes greater than 7 are {1, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 169, 173, 179, 181, 187, 191, 193, 197, 199, 209} mod 210.

    Like (a + 1)(b + 1) for positive integers a, b?

    It's about two to three thousand years old. I'm fairly sure it wasn't known 4000 years ago.

    The sieve of Eratosthenes.
     
  16. Apr 17, 2009 #15
    Last edited by a moderator: Apr 24, 2017
  17. Apr 18, 2009 #16
    By 8 you mean {1, 7, 11, 13, 17, 19, 23, 29} mod 30. Yes, all primes greater than 5 are of that form. What's more, almost all numbers of this form are composite -- only a tiny fraction are prime.
    well, yes. it goes something like this-
    30n+7 is prime for all n except when n is of form
    30k1k2+7k1+k2
    30k1k2+17k1+k2+6
    30k1k2+23k1+29k2+22
    30k1k2+13k1+19k2+8
     
    Last edited: Apr 19, 2009
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