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Infinite set of coprimes

  1. Feb 2, 2007 #1
    [SOLVED] infinite set of coprimes

    does any one know an infinite set of coprimes except for the elements of sylvester's sequence. S(n)=S(n-1)*(S(n-1)-1)+1, with s(0)=2
     
  2. jcsd
  3. Feb 2, 2007 #2
    The primes, maybe? :P
     
  4. Feb 2, 2007 #3
    yesss

    yes. thanks but... well... nevermind:rolleyes:
     
  5. Feb 2, 2007 #4

    CRGreathouse

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    The primes = 1 mod 4? The noncomposites? {2*3, 5*7, 11*13, 17*19, ...}?
     
  6. Feb 2, 2007 #5
    then i should change my question

    does any one know such a set AND its defining formula which gives us the nth term.
     
  7. Feb 3, 2007 #6

    CRGreathouse

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    The primes have many defining formulas, a fair number of which use only basic operations (say, addition, multiplication, factorials, and sine). To what end do you want this?
     
  8. Feb 5, 2007 #7

    matt grime

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    I don't have a reference to hand, but you can consider things like 2^n - 1. Indeed that is one proof that there are infintitely many primes.

    Note 2^n - 1 = 2*(2^{n-1} -1) + 1, proving that they are coprime.
     
  9. Feb 5, 2007 #8
    2^2-1=3 and 2^4-1=15 and 15/3=5 so they are not all coprimes am i wrong?
     
  10. Feb 5, 2007 #9

    matt grime

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    Sorry, my mistake - consecutive terms are coprime, not every term. Duh. But there is something to do with things like this that demonstrates infintely many primes by producing coprimes. I don't have the reference to hand (I read it in 'Proofs from the Book' by Aigner and Ziegler).
     
  11. Feb 5, 2007 #10
    a(n) = 5*2^(2n) +5*2^n + 1 is my guess. Prove or disprove
    One thing that is certain, they can only be divisible by a prime ending in 1 or 9
     
  12. Feb 6, 2007 #11
    Forget this too, If n = 8 2^n equals 9 and 2^2n = 4 mod 11.
    5*(4+9) + 1 = 66. also there are other powers of 2 with the same residue mod 11 so those values for n give a(n) which are not coprime.
    Maybe take n to be prime for the n in a(n) or something similar.
     
  13. Feb 6, 2007 #12
    ramsey2879 you're the man

    ramsey2879 your reply was the kind that i was looking for. thanks. but taking n to be prime makes this formula useless for me. because my purpose is to find a set of coprimes generated by a simple function . and you should see "a different approach to primes" thread for the reason of my asking for such a set.
     
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