The first of five articles concentrating on whole number dynamics, ideas of general dynamical systems are introduced and seen in concrete cases.
This article extends the discussions in "Whole number dynamics I". Continuing the proof that, for all starting points, the Happy Number sequence goes into a loop or homes in on a fixed point.
In this third of five articles we prove that whatever whole number we start with for the Happy Number sequence we will always end up with some set of numbers being repeated over and over again.
``The don whom they were awaiting had already forseen, in
his aeronautical researches at the National Physical laboratory and
Imperial College, the potential of electronic computers, having
used Hollerith punched-card machines in a pseudo-programmable sense
rather than as pure tabulators on numerical values of formulae. He
thought then as he thinks now: that in the context of computers the
dreams of the present are the reality of a decade or so
``Naturally, he tried to pass on such thoughts to his
pupils, and so in that July he wondered how to exemplify the ideas
of iteration and binary trees. The tools then available were, of
course, only pencil and paper (and log tables); so restriction to
positive integers seemed sensible. He thus devised the following
iterative algorithm: if a number of an iterative sequence is even,
then it is halved, otherwise multiplied by a first odd constant to
which product is added a second odd constant.
``As it happened, this proved to be more than enough to
keep the seventeen interestedly occupied for the week; for it was
soon discovered that one particular pair of constants, 3 and 1
respectively, seemed to posses a unique property. Thus was The
Thwaites' Conjecture postulated - it is believed for the first
time, since it was only in the 1960s that it began to crop up
apparently independently elsewhere and to attract such as Collatz,
Syracuse, Kakutani and others.''
Those were the opening four paragraphs of the paper by Professor
Sir Bryan Thwaites of Milnthorpe, Winchester, Hampshire SO22 4NF,
England (Tel/fax (0) 1962-852394; E-mail: firstname.lastname@example.org)
printed in the Bulletin of The Institute of Mathematics and its
Applications, UK, vol 21, nos 3/4, March/April 1985 which readers
are encouraged to read.
The conjecture is as follows:
Take a positive integer N.
If it is even, divide it by 2; if it is odd, multiply it by 3 and
Then for all N, the sequence so produced includes the
EXAMPLE: 15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20,
10, 5, 16, 8, 4, 2, 1
In the UK, Bryan Thwaites has taken many opportunities to
broadcast this conjecture and it was in the 1970's that, through
The Times newspaper, he offered a prize of 1000 for a
rigorous proof (or disproof) of the conjecture. As a result he was
inundated with frustrated efforts. One of the most amusing was from
a compartment-ful of eight regular train commuters between Brighton
and London: they complained that they had wasted a whole month
investigating it but with no success! Similar frustrations have
appeared in world-wide mathematical journals.
One important point to grasp at the outset is that computing
will never be able to produce a proof - for the conjecture applies
to all numbers. With modern notation it is easy to
write down unimaginably huge numbers, for example 10 to the 10 to
the 10 to the 10 etc. Nevertheless, it is interesting to
use computers to the limit of their capabilities just to discover
some of the curious behaviour of the sequences which can be
By now there is a very considerable literature on the
conjecture, some of it with quite sophisticated mathematics.
Various theorems have been formulated, and the IMA paper quoted
above contains a few simple ones. The best source of this
literature is: ``3x+1 Problem and Annotated Bibliography'' by
J.C.Lagarias published as an internal paper of the AT&T Bell
Laboratories, Murray Hill, New Jersey 07974, USA dated February 6,
1997. It quotes no less than 93 papers.
I would be happy to receive and comment upon communications on
the conjecture - though they would have to be serious in content!
And my prize still stands.
10 March 1998