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

1. ## 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

2. After multiplying your equation by q times (r+1) we obtain:
$
p(r+1)-4 q = q(r+1)
$

It's equivalent to:
$
(p-q)(r+1)=4 q
$

Because q has no divisors there are only following possibilities:
$
\left\{
\begin{array}{l}
p-q=1 \\
r+1 = 4 q
\end{array} \right.
$

$
\left\{
\begin{array}{l}
p-q=2 \\
r+1 = 2 q
\end{array} \right.
$

$
\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).
$
\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).
$
\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)
$
\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.
$