# 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:
$\displaystyle p(r+1)-4 q = q(r+1)$
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)
$\displaystyle \left\{ \begin{array}{l} p-q=4 \\ r+1 = q \end{array} \right. \Longleftrightarrow \left\{ \begin{array}{l} p = 7 \\ r = 2 \\ q = 3 \end{array}\right.$