Professor Stewart's Hoard of Mathematical Treasures (45 page)

. . . and if some cards are identical, some shuffles just swap identical cards - these are permutational symmetries of the pack.

Symmetries have come to dominate huge areas of mathematics. They are very general - it’s not only shapes that have symmetries. So do number systems, equations, and processes of all kinds. The symmetries of a mathematical ‘thing’ tell us a lot about it. For instance, Galois proved that you can’t solve the general equation of the fifth degree by an algebraic formula, and the main point of his proof is that the general equation of the fifth degree has the wrong kind of symmetries.

Symmetries are vital in physics, too. They classify the atomic lattices of crystals - there are 230 different symmetry types, or 219 if you consider mirror images to be the same. The ‘laws of nature’ turn out to be highly symmetric, mainly because the same laws operate at all points of space and all instants of time. The symmetries of the laws tell us a lot about the solutions. Quantum physics and relativity are both based on symmetry principles.

Front-back symmetry of a pacing giraffe. The front and back legs on each side hit the ground together.

Symmetries are even turning up in biology. Many important biological molecules are symmetric, and the symmetries affect how they work. But you can find symmetries in the shapes of animals, in their markings, and even in how they move. For example, when a giraffe paces, it moves both left legs together, then both right legs together. So the front legs do the same as the back legs, like two people walking one behind the other, in step with each other. The symmetry here is a permutation: swap front and back.

Perform this only in the abstract, please, or the giraffe will get upset.

Innumeratus wrote the nine non-zero digits down in order, with gaps, like this:

‘I want you to . . . ’ he began.

‘ . . . make 100 by inserting standard arithmetical symbols,’ said Mathophila. ‘That’s easy, it was in Professor Stewart’s Cabinet of Mathematical Curiosities, which you gave me for Christmas, but it goes back a lot further than that.’ And she wrote:

‘No, that’s cheating,’ said Innumeratus. ‘I left gaps! You can’t consider 1 2 3 to be one hundred and twenty-three, and ... ’

‘Oh. No concatenation of symbols allowed, then.’

‘Yeah. No caterwaulification . . . whatever.’

She thought for a moment, and wrote down

‘Sorry, no brackets,’ said Innumeratus.

Mathophila shrugged, and wrote

‘You don’t mind me using the rule that multiplication precedes addition, so I don’t need to put brackets round individual multiplications, do you?’

‘No, that’s OK. But ... uh . . . look, sorry, but no subtraction symbols either.’

There was a silence. ‘I’m not sure that’s possible,’ said Mathophila.

‘Wanna bet?’ asked Innumeratus smugly.

What should Mathophila do?

 

Euclid proved that there is no largest prime. Here’s a quick way to see this: if p is prime then
p
! + 1 is not divisible by any of the numbers 2, 3, . . . , p, since any such division leaves remainder 1. So all its prime factors are bigger than p. Here,
p
! =
p
× (
p
- 1)× (p - 2)×. . .×3×2×1.

Euclid’s proof was slightly different. He stated it geometrically, and in modern terms he used a typical example to show that if you have any finite list of primes, then you can get a bigger one by multiplying them all together, adding 1, and then taking any prime factor of the result.

This suggests an interesting sequence of primes, all guaranteed to be different:

For example,
(because 1807 = 13×139), and so on.

The first few terms are
and the sequence is highly irregular. Occasionally the product
p
₁
×
p
₂
× ‧‧‧ ×
p
_n
+ 1 is prime, and the size goes up enormously, but when it’s not prime, the smallest factor is often very small indeed. This behaviour is pretty much what you might expect, wild though it may be.

Despite (or perhaps because of) this tendency to swing madly between huge numbers and tiny ones, the first 13 terms include the first seven primes: 2, 3, 5, 7, 11, 13, 17. Which raises an interesting - and probably difficult - question: does every prime occur somewhere in this sequence?

I have no idea how to answer that, though if I had to guess I’d say it’s true.

The famous English puzzlist Henry Ernest Dudeney remarked that the fraction
is equal to 100, and uses every digit 1-9 exactly once. He found ten other ways to achieve this, one of which has only one digit before the fractional part. What was this solution?

 

But, by definition,

So,

which implies that

Therefore:

Readers of Douglas Adams’s The Hitch Hiker’s Guide to the Galaxy will recall the prominent role of the number 42 - the answer to the Great Question of Life, the Universe and Everything. The question turned out to be ‘what is six times nine?’, which was vaguely disappointing. Anyway, Adams chose 42 because a quick poll of his friends suggested that this was the most boring number they could think of.

It’s true that interesting properties of 42 don’t exactly trip off the tongue, but we know (Cabinet, page 105) that all numbers are interesting. However, the proof is non-constructive. So I was pleased to find out about a natural occurrence of 42 as an interesting number. It arises in a sequence of numbers introduced by F. Göbel. Suppose we define

There is no obvious reason why the
x
_n
should be whole numbers, but the first few terms of the sequence are
so you do begin to wonder whether, by some miracle, all the terms are integers.

The truth is, if anything, even more miraculous. Hendrik Lenstra put the equation on a computer, and discovered that the first term that is not an integer is
x
₄₃
. So 42 is the largest integer for which all terms of the sequence, up to and including that one, are integers.

Other sequences of this kind also seem to behave that way - a lot of integers to begin with, but at some point the pattern fails. With the same rule but using sums of cubes, the first term that is not an integer is
x
₈₉
. With fourth powers the first non-integer is
x
₉₇
, with fifth powers it is
x
₂₁₄
, with sixth powers it is the relatively feeble
x
₁₉
, but with seventh powers we get the astonishing
x
₂₃₉
. So here is a sequence with a nice pattern, such that the first 238 terms
⁴⁴
are integers, but the 239th is not.

As far as I know, no one really understands why these sequences behave like they do.

James Joseph Sylvester was a 19th-century mathematician who specialised in algebra and geometry. He worked a lot of the time with Arthur Cayley, whose day job was in the law. Cayley had a superb memory and knew almost everything that was going on in mathematics. Sylvester was the exact opposite.

On one occasion the American mathematician William Pitt Durfee sent some of his work to Sylvester, only to be informed that the first theorem in it was false, and had never even been stated, let alone proved. Durfee produced a paper whose main objective was to prove the theorem concerned, which it did successfully.

The paper had been written by Sylvester.

James Joseph Sylvester.

By definition,

Therefore,
where (*) is justified by the associative law
with a = (1 + 1), b = 1, c = 1.