If the 3x+1 problem is intractable, why should one bother to study it?
One answer is provided by the following aphorism: ``No problem is so
intractable that something interesting cannot be said about it.''
Study of the 3x+1 problem has uncovered a number of interesting phenomena;
I believe further study of it may be rewarded by the discovery of
other new phenomena.
It also serves as a benchmark to measure the progress of general
mathematical theories.
For example,
future developments in solving exponential Diophantine equations may lead to the
resolution of the Finite Cycles Conjecture.