Results 1 to 2 of 2

Thread: Find all prime numbers p,q,r such that (p/q)-(4/(r+1))=1

  1. #1
    Newbie
    Joined
    Sep 2009
    Posts
    16

    Find all prime numbers p,q,r such that (p/q)-(4/(r+1))=1

    Find all prime numbers p,q,r such that (p/q)-(4/(r+1))=1
    anybody can help me
    tq
    Last edited by CaptainBlack; Sep 23rd 2009 at 11:40 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Newbie
    Joined
    Sep 2009
    Posts
    12
    After multiplying your equation by q times (r+1) we obtain:
    $\displaystyle
    p(r+1)-4 q = q(r+1)
    $
    It's equivalent to:
    $\displaystyle
    (p-q)(r+1)=4 q
    $
    Because q has no divisors there are only following possibilities:
    $\displaystyle
    \left\{
    \begin{array}{l}
    p-q=1 \\
    r+1 = 4 q
    \end{array} \right.
    $
    $\displaystyle
    \left\{
    \begin{array}{l}
    p-q=2 \\
    r+1 = 2 q
    \end{array} \right.
    $
    $\displaystyle
    \left\{
    \begin{array}{l}
    p-q=4 \\
    r+1 = q
    \end{array} \right.
    $
    (r+1 can't equal 1 or 2 because r>1, p-q can't equal q so I've rejected 3 possibilities)

    i)If difference between two prime numbers is 1 one of them have to be 2 (it's only even prime).
    $\displaystyle
    \left\{
    \begin{array}{l}
    p-q=1 \\
    r+1 = 4 q
    \end{array} \right. \Longleftrightarrow
    \left\{
    \begin{array}{l}
    p=3 \\
    q=2 \\
    r = 7
    \end{array}\right.
    $

    ii)If 3 numbers q, q+2 and 2q-1 are prime the reminder from devison q by 3 cannot be 1 (in this case q+2 would be divisible by 3) or 2 (2q-1 would be divisible by 3). So q is divisible by 3 and q=3 (if q+2 =3 or 2q-1=3 one of numbers p, r is not prime).
    $\displaystyle
    \left\{
    \begin{array}{l}
    p-q=2 \\
    r+1 = 2 q
    \end{array} \right. \Longleftrightarrow
    \left\{
    \begin{array}{l}
    p = 5 \\
    q = 3 \\
    r = 5
    \end{array}\right.
    $

    iii)
    $\displaystyle
    \left\{
    \begin{array}{l}
    p-q=4 \\
    r+1 = q
    \end{array} \right. \Longleftrightarrow
    \left\{
    \begin{array}{l}
    p = 7 \\
    r = 2 \\
    q = 3
    \end{array}\right.
    $
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Oct 22nd 2011, 12:37 PM
  2. find prime numbers
    Posted in the Math Forum
    Replies: 2
    Last Post: Apr 17th 2011, 09:44 AM
  3. find the prime factor of two numbers
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Mar 5th 2010, 05:04 AM
  4. find all prime numbers
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: Sep 16th 2009, 02:38 PM
  5. Find all prime numbers
    Posted in the Number Theory Forum
    Replies: 7
    Last Post: Aug 22nd 2008, 04:48 AM

Search Tags


/mathhelpforum @mathhelpforum