RMO 2001

[zefuri]

Find all primes [unparseable or potentially dangerous latex formula] such that [unparseable or potentially dangerous latex formula]

[makashi]

For primes greater than three, [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]. But in [unparseable or potentially dangerous latex formula],
[unparseable or potentially dangerous latex formula]

The remaining cases consist of at least one prime being either two or three. But note that there is only one case in which [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula] in which the both sides of the equation are positive ([unparseable or potentially dangerous latex formula] which does not have equality). Hence, the only possibilities remain with [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula].

Suppose [unparseable or potentially dangerous latex formula]. We have already dealt with the cases in which [unparseable or potentially dangerous latex formula] or [unparseable or potentially dangerous latex formula]. For [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], so there are no solutions with [unparseable or potentially dangerous latex formula].

Suppose [unparseable or potentially dangerous latex formula]. There is one solution with [unparseable or potentially dangerous latex formula]. For [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula], so there are no solutions with [unparseable or potentially dangerous latex formula].

Therefore, the only solution is [unparseable or potentially dangerous latex formula].