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

  1. #1
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    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; September 23rd 2009 at 11:40 PM.
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  2. #2
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    After multiplying your equation by q times (r+1) we obtain:
    <br />
p(r+1)-4 q = q(r+1)<br />
    It's equivalent to:
    <br />
(p-q)(r+1)=4 q<br />
    Because q has no divisors there are only following possibilities:
    <br />
\left\{<br />
\begin{array}{l}<br />
p-q=1 \\<br />
r+1 = 4 q<br />
\end{array} \right.<br />
    <br />
\left\{<br />
\begin{array}{l}<br />
p-q=2 \\<br />
r+1 = 2 q<br />
\end{array} \right.<br />
    <br />
 \left\{<br />
 \begin{array}{l}<br />
 p-q=4 \\<br />
 r+1 =  q<br />
 \end{array} \right.<br />
    (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).
    <br />
\left\{<br />
\begin{array}{l}<br />
p-q=1 \\<br />
r+1 = 4 q<br />
\end{array} \right. \Longleftrightarrow<br />
\left\{<br />
 \begin{array}{l}<br />
 p=3 \\<br />
q=2 \\<br />
 r = 7<br />
 \end{array}\right.<br />

    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).
    <br />
 \left\{<br />
 \begin{array}{l}<br />
 p-q=2 \\<br />
 r+1 = 2 q<br />
 \end{array} \right. \Longleftrightarrow<br />
\left\{<br />
   \begin{array}{l}<br />
   p = 5 \\<br />
q = 3 \\<br />
r = 5 <br />
   \end{array}\right.<br />

    iii)
    <br />
  \left\{<br />
  \begin{array}{l}<br />
  p-q=4 \\<br />
  r+1 =  q<br />
  \end{array} \right. \Longleftrightarrow<br />
\left\{<br />
   \begin{array}{l}<br />
   p = 7 \\<br />
   r =  2 \\<br />
q = 3<br />
   \end{array}\right.<br />
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