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New Mathematics for an Old Problem By Mr Ottewill
The Thwaites Conjecture
A few weeks back, Professor Sir Bryan Thwaites (OA) gave a lunchtime talk at the College which covered, among other things, a conjecture which is sometimes called the Thwaites conjecture, due to the fact that for many years now Sir Byan has offered a £1000 prize for a proof or disproof of the conjecture. (The conjecture is also called the Collatz conjecture, or simply the 3 n + 1 conjecture). Stated as simply as possible, the conjecture says that if you start with a positive integer and then repeatedly follow the formula ‘halve the number if it is even, multiply by three and add one if it is odd’, reapplying the formula to each new number formed, then you eventually end up with 1. The conjecture has been tested for billions of numbers and always seems to be true, but to date no-one (including Sir Bryan) has proved whether or not it is true for all positive integers. On the day of the talk, Mr Kulatunge showed Sir Bryan a paper written by Craig Alan Feinstein which claims to prove that the conjecture is unprovable. The methods used in Feinstein’s paper are rather unusual. I hope here to indicate a rough outline of the methods which Feinstein uses, along with some brief comments on whether they are successful. For example, starting with 7 gives: 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
‘Standard’ work on the conjecture
It will be useful first of all to look briefly at some of the more standard approaches to the conjecture.
The first thing to do is to note that, as multiplying by 3 and adding 1 always produces an even number, we can combine this step with the step which always follows, namely halving, giving rise to the rule that next number after an odd
2 1 3 + n , not just
number will be
1 3 + n .
The next observation is often a rough, though fairly plausible, reason why the conjecture is likely to be true. The argument goes that a randomly chosen integer is equally likely to be odd or even. We can also argue that for large
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