Books, papers & preprints
A Prime Rhyme
If you want to show the primes go on for ever,
There's a trick that Euclid taught us long ago.
Suppose not: then multiply these primes together,
And add one to get a number, call it rho.
Just take a careful look at this big number,
Rho has to have a factor which is prime,
And it can't be one of those you 've got already,
So there's your contradiction shown in rhyme.
A Week in the Life of a Mathematician
(with apologies to Michael Flanders and Donald Swann)
'Twas on a Monday morning I had a bright idea,
I was lying in the bath tub and the strategy seemed clear,
For a problem posed by Erdös back in nineteen forty nine,
On sequences dilated into subsets of the line
'Twas on a Tuesday morning I jotted down my thoughts,
I covered backs of envelopes with surds and aleph noughts.
After several cups of coffee I began to feel inspired,
And a lengthy calculation gave the answer I desired.
'Twas on a Wednesday morning I wrote the details out.
My lemmas and corollaries left little room for doubt.
I filled up many pages just to get the logic right,
And with epsilons and deltas I made it watertight.
'Twas on a Thursday morning I typed the paper up,
With "slash subset" and "slash mapsto" to say nothing
of "slash cup".
My LaTeXing was perfect, printed out it looked so good,
Should I send it to the Annals? I rather thought I would!
'Twas on a Friday morning I read the paper through,
I checked out every detail as good authors ought to do.
At the bottom of page twenty in an integral I found,
I'd divided through by zero and the proof crashed to the ground.
On Saturday and Sunday I was too depressed to care,
So 'twas on a Monday morning that I had my next idea.
(This poem appeared in The London Mathematical Society
A Nightmare Seminar
(with apologies to WS Gilbert and Arthur Sullivan)
When you go to a talk, speaker, blackboard and chalk,
at the end of a hard day of teaching,
It's no great surprise that you half close your eyes
and you can't concentrate on the preaching.
You very soon find that your wandering mind
with ideas for strange theorems is teeming,
But you're somewhat bemused and your thoughts get confused
and before very long you're just dreaming.
For you meet a long line of numbers, all prime,
that pass by with a nod now and then,
With big gaps in the queue, the primes become few,
a proportion one over log n.
The gaps vary in size and you want to know why,
so you write as an infinite sum
Riemann's function called zeta, and what could be neater?
A product o'er primes it becomes.
You set out on a train 'cross the vast complex plane
as you search for the zeros of Riemann,
There are plenty to find on the critical line
and you note them all down as you see them.
Then the train hits a pole and you tumble and roll
(near the pole it's as cold as an icicle),
But you pick yourself up and you find you're in luck
for nearby is a rusty old bicycle.
You pedal like mad, though the weather's turned bad,
keeping lookout for rogue Riemann zeros,
But before you get far you arrive at a bar
full of great mathematical heroes.
Next to Hilbert and Gauss, you see Erdös and Straus
who're absorbed in some very hard thinking,
Then there's Riemann and Rayleigh, and Cauchy and Cayley,
and Euler with Fermat is drinking.
You buy Riemann a beer, and he says: "To be clear
why my zeros all lie in a row,
You will need to consider with very great rigour
the way eigenvalues can grow."
Although you implore, he won't tell any more
and so you depart on your travels,
You keep scratching your head at what Riemann said
but the problem you still can't unravel.
By a river that's deep you encounter a heap
of some very large matrices random,
And their eigenvalues and the Riemann zeros
wander off to infinity in tandem.
You then get the point that if one's self-adjoint
eigenvalues all lie on a line,
And the same should be so for the Riemann zeros -
an idea that is somewhat sublime.
You allow a faint smile as you search through the pile
for such matrices, hoping you'll see them,
You spot one midst the trash - and you see in a flash
how to prove the Hypothesis Riemann.
You start work right away and by night and by day
you fill hundreds of pages with writing,
The lemmas are tricky, the details are sticky,
and getting it right's quite exciting.
You break out in a sweat for you mustn't forget
any parts of the proof you're recording,
But at last comes the end and you put down your pen ...
and wake up to your colleagues' applauding ...
You're a regular wreck with a crick in your neck,
Your hair's in a mess and your head's on the desk,
Your mouth's open wide and your tie's to one side,
The board's covered with chalk and you've missed all the talk,
You've forgotten your proof which must be a spoof,
Your face has turned red, you've an ache in your head,
With a throb that's intense and a general sense
That you'll take a long time to recover.
But the seminar's past, you can go home at last,
And the day has been long, ditto ditto my song,
And thank goodness they're both of them over!
(A version of this poem appears in The Mathematical
Intelligencer, 31(2009) No.3, p.9)
A Poem of Academic Prejudice
Our Pure Mathematicians are wise and profound:
They prove deep new theorems with arguments sound.
They solve old conjectures and work problems out,
And their elegant methods leave nothing in doubt.
But Applied Mathematicians are dubious types,
They study strange flows yet can't sort out our pipes.
They get hefty grants just to show the Sun's hot,
When up here in Scotland it's patently not.
The Statistics Division, well what do they do?
They count furry mammals and shoals of fish, too.
From vast piles of data find medians and mids,
And conclude everybody has two point three kids.
In Computing, where life is controlled by machines,
And eyesight is ruined by staring at screens,
They write complex programs that may never stop,
And argue if P equals NP or not.
Then over in Physics they do tricks with light,
Show near absolute zero things don't work quite right.
In accelerators costing us millions each week,
With quarks and Higgs bosons they play hide and seek.
And going into town, one can only despair
At Divinity, Russian and other schools there.
For History's happened and Latin is dead,
And Philosophers do all their thinking in bed.
The English department could scarcely be worse,
They don't even try to put rhymes in their verse.
Then Geographers blame us for greenhouse effects,
Whilst Psychologists put all we do down to sex.
As for other departments, and I've missed out a few,
To list their shortcomings I'll leave up to you.
But Pure Mathematicians are noble and good,
And wonderful people, though misunderstood!
First Year Mathematics Exam
You should answer all the questions on this paper,
You'll find two hours will be sufficient time.
You must give full derivations of your answers,
Your solutions should, of course, all be in rhyme.
Question one will test your skill with complex numbers:
Take the pair 4 + 3i and i + 1.
First add, then multiply, these two together,
Divide the second by the first once this is done.
Next, write down the argument and modulus
Of i + 1; thus find its thirteenth power.
And after that we'd like to know its cube roots,
This should not take more than quarter of an hour!
Here's a matrix A that has to be inverted:
The top row has the entries 1, 2, 3,
In the middle are the numbers 2, 5, 20,
At the bottom 3, 6, 10 is what you see.
To round this off, some linear equations:
Can you solve the system Ax equals c,
Where c's the column vector (2, 6, 7)?
Then what d'you think the vector x must be?
Next, please define the hyperbolic functions.
Find the difference of the squares of cosh and shine.
Use this fact to do a simple integration
Of 1 over root 4 x squared minus 9.
After this, you should evaluate the limit
Of x squared over cosh x minus 1,
As the variable x approaches zero;
L'Hopital might help you get this question done.
Now here's a homogeneous equation:
y double dot plus y dot equals nought.
You must calculate the general solution,
Which shouldn't take up too much time or thought.
Now change this differential equation,
By putting on the right hand side sin t.
Solve this, with the initial conditions
Of y dot equals 5, y equals 3.
S'pose a's the vector i + j +2 k,
And b's 2 i + j + 7 k.
Then what d'you think the product a dot b is?
Also the vector product, b cross a?
And next perhaps you'd work out the equation
Of the plane containing vectors a and b.
And finally, we'd like to know its distance
From the point that has coordinates (1, 2, 3).
Show the cubes of all the first n natural numbers
Sum to quarter of n (n + 1) all squared.
We suggest you use mathematical induction -
A technique on which you should be well-prepared.
Now what about the first n even numbers,
And after that the first n odd ones too?
What do the cubes of odds and evens sum to?
It shouldn't be too hard to think this through!
Our final question deals with infinite series:
Give reasons why they converge, if they do.
First, how about the series where the n-th term
Is 3 n over n cubed minus 2?
Next think about the case of sine of n squared
Divided by n squared - what happens then?
And finally, maybe a little harder,
The summation of 1 over n log n?
A Toast to Pi
The distance round a circle,
Whose diameter is d,
Is quickly found by multiplying
By not much more than three.
Well-known by the Egyptians,
Four thousand years ago;
Indeed, the Rhind Papyrus
Gives three point one six oh.
Along came Archimedes,
Who, not to be outdone,
With polygons and rigour,
Got three point one four one.
Then in the sixteen sixties,
James Gregory came to town,
And for the inverse tan of x,
A series he wrote down.
It rapidly converges,
If x is not too big.
Taking x as root one third
We get far more sig figs.
This number didn’t have a name
Till seventeen oh six,
When Jones, a Welshman, came along,
This oversight to fix.
He soon rejected nu and mu
And omicron and xi,
Then musing on his apple tart,
Cried “Ah - Let's call it pie!”.
But now we use computers,
To save us time and space.
Last year one took four months to get
The sixty-trillionth place.
We rightly toast this number,
So lift your glasses high,
As on this day we celebrate,
The ratio known as pi.
(Written for the SUMS Pi Day Ball, 14-03-2022)