find all prime numbers p,q,r such that (p/q )- (4/(r+1))=1
Hi!
We first rewrite the equality equivalently as .
If , we get , which forces and , so we've found one triplet.
So suppose from now on that is an odd prime.
If we have . We can't have because it implies . Forced is and , which gives us another triplet.
So suppose from now on that is an odd prime.
From we see that is an odd prime.
We now know , so we can rewrite as .
We cannot have (else wouldn't be a prime) so we must have and .
In other words, and .
Let for some . Then and .
If , we have , and , so we've found another triplet.
If , then which implies . So we cannot have .
If , then which implies , which we excluded, so we cannot have .
We conclude that the only triplets are 7,3,2 and 3,2,7 and 5,3,5.