>>12068824Wlog
Claim
Observe , so .
Hence if , then .
If then . Hence . The only solution to this is , which does not give a real solution.
Further or clearly give no solutions. We have proved the claim.
Now, dividing the original expresion,
.
As is at least 3, is even.
Thus exactly one of or must be odd.
As , the ratio is a multiple of .
So if the second is even, the first must be too. Hence is odd.
There are two such cases where this happens.
Case 1: , and odd.
This means .
As , we have .
Of course . So the lhs vanishes mod , but the rhs .
So this case cant happen.
Case 2: .
Then .
As , then .
Hence . As a product of consequtive numbers this only happens when .
Hence .
This has unique solution
Hence is the only solution.
