TITLE: Undecidable easy arithmetical statement
QUESTION [11 upvotes]: Is there a basic arithmetic statement which is known to be undecidable ?
By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture .  By the way is there some reasons to believe that the Collatz conjecture is undecidable ?

REPLY [15 votes]: The "mortal matrix" problem: Given a set of $n\times n$-matrices with integer entries, decide whether they can be multiplied, in any order and possibly with repetition, to give the $0$-matrix. If I remember correctly, it is already undecidable for $n\geq 3$.
I decided to put this here, since matrix multiplication is a sequence of simple arithmetical operations.