2008 JBMO Problems/Problem 3
Problem
Find all prime numbers , such that
Solution
The given equation can be rearranged into the below form:
then we have
and and
then we have
and and
note that if , then which is a contradiction.
and
then we have
and and We have that exactly one of is a multiple of .
cannot be a multiple of since . Since is prime, then we have is a prime.
contradiction.
Also, cannot be a multiple of since, contradiction.
So,
and
Thus we have the following solutions: