2005 IMO Problems/Problem 4
Determine all positive integers relatively prime to all the terms of the infinite sequence
Solution
For all primes greater than
, by Fermat's last theorem,
mod
if
and
are relatively prime. This means that
mod
. Plugging
back into the equation, we see that the value mod
is simply
. Thus, the expression is divisible by
. Because the expression is clearly never divisible by
or
, our answer is all numbers of the form
.