Published December 1998,April 1998,December 2011,February 2011.

``At 4.00 P.M. on Monday 21 ^{st} July
1952, seventeen clever, but not particularly mathematical, boys
were probably wondering how their mathma don - as a mathematics
teacher was called in the school in question - was going to amuse
them during the last few periods with him. Those were happy days
when, at least in that place, teaching continued until the last
minute of the last hour of the last day of the last term of the
year. Examinations, such as they were, came and went without
disturbing the steady rhythm of education, and a boy could opt to
do some serious mathematics even if he had already obtained an
Oxbridge scholarship in some other subject.

``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 later.

``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: bthwaites@dial.pipex.com) 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
add 1.
Iterate.**

**Then for all N, the sequence so produced includes the
number 1.**

**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
generated.

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.

B.Thwaites

10 March 1998