2006 IMO Problems/Problem 4
Problem
Determine all pairs of integers such that
Solution
: LHS integer iff
, but then
.
is a solution.
for
no solution.
so assume
. LHS is odd, so writing
gives us
.
are coprime, and so are
. so
or vice versa, but both lead to a contradiction
NOTE: This solution does not seem to be correct. Take e.g. . Then
and
. The argument of the solution is promising, but the following is not true: Let
,
be co-prime and let
,
be also coprime and additionally let
. Then it must either be
or
.