Department of Mathematics
Texas A & M - Commerce
Student/Faculty Colloquium
Thursday, November 9, 2006
Henderson 302, 2:00pm
Jeff Norris,
Paris Junior College,
will speak on:
The Collatz conjecture
Abstract:
The Collatz conjecture states that iteration of the function T,
given by T(n)=n/2 if n is even, T(n)=(3n+1)/2 if n is odd, always
eventually reaches 1, regardless of the initial choice of positive n.
More formally, the conjecture is that the iterates of T generate an
integer sequence which includes T(k)(n)=1
for all positive n, for some k (which depends on n). For instance, if
n=3 we have the sequence (5,8,4,2,1,2,1,2,1,2...); so for n=3, we could
choose k=5. Like Fermat's Last Theorem, the Collatz conjecture is a
problem that is easy to state
and understand, but hard to solve. The apparent intractability of the
problem caused Paul Erdos to state
that "mathematics is not yet ready for such problems." We will briefly
discuss the origins of the problem and efforts to resolve it. We
will also look at some equivalent conjectures and will show some
partial results for one of these equivalent statements. Our talk
will be number theoretic in nature, but no prior knowledge
is assumed.