Results 1 to 12 of 12

Math Help - A small problem about primes number

  1. #1
    Newbie
    Joined
    Jun 2009
    Posts
    9

    A small problem about primes number

    Conjecture.
    Does there exist a real number C such that for all i in N:
    tan(p_i)+cot(p_(i+1)) > 0 or tan(p_i)+cot(p_(i+1)) < C,
    where C = -0.14627836...

    If the conjecture was true, what is the exact value of C?

    Best Regards,
    Roberto Lepera
    Last edited by rrronny; June 26th 2009 at 02:44 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    What are tg and ctg?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jun 2009
    Posts
    9
    Quote Originally Posted by Bruno J. View Post
    What are tg and ctg?
    tg and ctg are tangent and cotangent functions
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    What makes you conjecture this?

    I doubt anyone is going to be able to tell you whether that is true or not.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by rrronny View Post
    Conjecture.
    Does there exist a real number C such that for all i in N:
    tan(p_i)+cot(p_(i+1)) > 0 or tan(p_i)+cot(p_(i+1)) < C,
    where C = -0.14627836...

    If the conjecture were true, what is the exact value of C?

    Best Regards,
    Roberto Lepera
    You've phrased it as a question, so I'm not sure what the conjecture is. Is this what you intend to state:
    There EXISTS a real number C (with conditions)

    It is not clear but for example:

     p_i is some prime such as 7
    &
     p_{i+1} is the next prime or 11

    Is that correct?

    --
    The  \tan &  \cot are reciprocals. When one of the two is greater than a unit the other will be a fraction. The sum will always exceed a unit.

    When  p_i AND  p_{i+1} are reduced to 0 < p <  2\pi
    THEN

    1st quadrant:
    the sum will always be greater than 1.

    2nd quadrant:
    the sum will always be less than -1.

    3rd quadrant:
    the sum will always be less than -1.

    4th quadrant:
    the sum will always be less than -1.

    When the tan--cotan pair occur in the 1st & 2nd quadrant or when they occur in the 4th and 1st quadrant will the sign of the two be different. But one will be exceedingly small and the other exceedingly large, since they are reciprocals.


    What am I missing here?

    From this quick look the value of 'c' is:

    c < -1
    or
    +1 < c

    Is there a C , such that -1 < c < +1
    Don't think so.

    Could you elaborate, and give a specific example so that your speculation can be fully understood?




    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jun 2009
    Posts
    9
    The conjecture is true for first 200.000.000 prime numbers.
    See also this code for Mathematica:
    Code:
    K = 10^8; c = -0.146278362977478; For[i=1, i≤K, i++,
    {q = Tan[Prime[i]] + Cot[Prime[i + 1]]; If[q>c && q<0, Print[{i, N[q, 15]}]]}]
    Last edited by rrronny; June 26th 2009 at 02:44 PM.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jun 2009
    Posts
    9
    The conjecture is false for i=886792302.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Super Member
    Joined
    Jan 2009
    Posts
    591
    Quote Originally Posted by rrronny View Post
    Code:
    K = 10^8; c = -0.146278362977478; For[i=1, i≤K, i++,
    {q = Tan[Prime[i]] + Cot[Prime[i + 1]]; If[q>c && q<0, Print[{i, N[q, 15]}]]}]
    After seeing the code, the meaning of the post became clear. A long while ago there was a problem about the gap between prime numbers. I have searched but have not been able to find the source of the information.

    Tom Nicely has a fantastic web site about the gap between prime numbers.
    I do not have it handy (you may use google to find it).

    Back to the problem.
    Roughly it was something like this:

    A is some odd number (not required to be prime).
    B is another odd number (or A + 2k, k being an natural number).


    tan(A) =  \frac {sin A}{cos A}
    tan(A) + cotan(A + 2k) = c


    that becomes:

     \frac { cos (2k) }{sin(A)cos(A)}

    A & 2k are integer values of radians. The sin/cos values will never be zero or one.

     0 < cos (2k) < 1 &  0 < \sin(A) \cos(A) < 1

    The question was how close to zero (or one or negative one) does this get?
    [Actually no integer values ever occur.]

    Your question then becomes, does being a prime number of radians create a special characteristic of the limits?

    This was answered in part by the indication that 2k (the gap between the prime numbers) needs to be
    710
    207986
    208696
    416682
    625378
    1667438
    and so forth.
    The gap between the prime numbers required increases somewhat rapidly.

    See Tom Nicely's web page for the size of primes with those gaps.

    The conjecture is false for i=886792302
    I do not know the values of
     Prime_{886792302} &  Prime_{886792303} but the gap is the critical factor.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    What I hardly undestand is the role of the constant c= - .146278362977478... Why this value and not any other value? ...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    The previous message has be sent two times... Very sorry!... I please a moderator to cancel this last post!... Thank you very much!...

    Kind regards

    \chi \sigma
    Follow Math Help Forum on Facebook and Google+

  11. #11
    Newbie
    Joined
    Jun 2009
    Posts
    9
    I know only that:
    C=2*tan(x*), where x* is the first root negative of the equation tan(x)*tan(x+8)=1.

    Best Regards,
    R.L.

    P.S.: If you speak the italian language, see also here: http://www.scienzematematiche.it/for...st=0&sk=t&sd=a
    Follow Math Help Forum on Facebook and Google+

  12. #12
    MHF Contributor chisigma's Avatar
    Joined
    Mar 2009
    From
    near Piacenza (Italy)
    Posts
    2,162
    Thanks
    5
    Quote Originally Posted by rrronny View Post
    Since I’m Italian I suppose to be able to speech Italian!… Regard to that some time ago somebody in mathelpforum asked me about my avatar. Now I have a good opportunity to answer to You and Him in only one post…


    The avatar is the reduction of this image...






    This stone plate is set at the main entrance of 'Collegio Ghislieri' in Pavia [Italy] , the college in which I followed my university studies a lot of years ago. For information about the college see...

    http://www.ghislieri.it/index.php

    The sentence on the plate is taken from the 'Divina Commedia' of Dante Alighieri, the most famous italian poet, and in english can be translated as...

    '... the more you know, the less you like wasting time...'

    The avatar is a [negative] detail of this photo...



    ... that my wife did an year ago at the annual college party...

    Regarding the ‘interesting’ problem You have proposed my opinion is that if in www.scienzematematiche.it there is a lot of people who love wasting time… the same is not here, so that please remember the sentence of Dante before propose to us an other problem…

    Kind regards

    \chi \sigma

    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Number theory(Primes)
    Posted in the Number Theory Forum
    Replies: 6
    Last Post: September 9th 2009, 04:54 PM
  2. Number theory (primes)
    Posted in the Number Theory Forum
    Replies: 2
    Last Post: September 6th 2009, 04:31 AM
  3. Replies: 2
    Last Post: April 11th 2008, 07:11 PM
  4. Replies: 3
    Last Post: November 1st 2007, 08:49 AM
  5. Find All Primes of a Number
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: February 18th 2007, 08:19 PM

Search Tags


/mathhelpforum @mathhelpforum