Glimpses of History 2 (mostly Number Theory)

by Martin Huxley

Raise two to a prime p and then
Take off one --- is it prime, asks MERSENNE?
   What MERSENNE never knew,
   It looks nice in base two,
And an analogue's stated base ten.

Nor FERMAT nor MERSENNE really shone
With a listing of primes that's spot on,
   And students now snigger
   At an incorrect figure,
Dividing by six forty-one.

A reduction step acting on pairs
Gets a prime to be sum of two squares;
   FERMAT's own device,
   That he used once or twice,
As he listened to legal affairs.

"No sum of two cubes makes a third,
And higher pow'rs, also absurd,
   But the space in this marge
   Being not very large...";
No more from FERMAT has been heard.

"On the shoulders of giants," said haughty
Sir ISAAC. Few heard this as naughty.
   But he based calculation
   On whose observations?
Astronomer HOOKE was a shorty.    [NEWTON]

Sir ISAAC had many pursuits,
But at school we learned two of his fruits:
   The Laws about Motion,
   And (opposite notion)
The methods for putting down roots.    [NEWTON]

A fact that has long been suppressed
Is that NEWTON's old polygon's best.
   Its factor detection
   Proceeds by inspection,
Anticipates EISENSTEIN's test.

Each night an hour longer in bed
Was the plight of DE MOIVRE, it's said.
   "In two weeks, it's plain,
   I shall not wake again
By induction." And then he was dead.

His astronomy book must be shewn
To the King, and LAPLACE stands alone.
   "We regard it as odd,
   "there's no mention of GOD!"
"I neglect higher Pow'rs of Unknown!"

Buying horses and cows at the fair,
One sev'n seventy, nothing to spare.
   "How many bought now?"
   "Twenty-one for a cow,
Thirty-one roubles each for a mare!"    [EULER]

Is insight notationed or notioned?
The parts out of n, what's the quotient?
   In EULER's construction,
   No further reduction,
Phi (n) - but why call it the totient?

There are several proofs that GAUSS saw
For the Great Reciprocity Law,
   But none had the look
   Of the Pearly-Bound Book,
So he sat down to find even more.

A progression from zero, steps q,
Reduce mod p, stop halfway through.
   How many are minus
   Determines the sign as
Pow'r p minus one over two.    [GAUSS]

The last proof of GAUSS reciprocity
Depends on the sort of atrocity
   Each Little-Go student,
   However imprudent,
Was accustomed to solve with velocity.

You innocents, listen in time:
Not ev'rything's true that's sublime.
   If it's certain that GAUSS
   Did not keep a pet mouse,
We would still make it up for the rhyme.

No integral's safe from his probe. He
   Filled leather-bound Works
   With notations and quirks,
That still can't be read in Adobe.

Left Potsdam to travel the globe, he
   Crossed over the Oder
   To train under YODA
As Jedi, or be one can no be.

The fractions we now know as FAREY,
He looked at a table, and there he
   At properties hinted
   Which CAUCHY had printed;
Nomenclature's quite arbitrary.

Although Baron CAUCHY had founded
Analysis, rigorous, grounded,
   His papers were tending
   Towards the unending;
The Academy wanted them bounded.

In ABEL's `Quintessence' one stage
Was quoted from CAUCHY the sage,
   "Comptes, last year". You look:
   It's a very thick book,
And CAUCHY wrote every page.

Of NAPOLEON's soldiers, most fell;
Field of honour or snowfield, both Hell.
   That campaign, it gave birth
   To Operations Research,
And to MERTENS's mother as well.

LEGENDRE wrote `easily seen';
We know what such weasel words mean.
   To set matters straight
   Made DIRICHLET create
L(s, chi) and the modern machine.

Just the primes in A. P. are officially
Ascribed to the younger DIRICHLET.
   Both residue symbol
   And HOOLEY-fied nimble
Interchange were DIRICHLET initially.

DIRICHLET assumed the existence
If it comes from some physical systems,
   Of pulleys and pistons,
   Or water in cisterns
Or forces that act at a distance.

In the integral studied by AIRY,
The sine of x cubed appears scary.
   There's an interval small
   Which approximates all,
Where the angle, or phase, doesn't vary.

Conspiracy theorists thrill
How GALOIS wrote out, not a will,
   But his ultimate draft
   (Two rejected as daft)
On Groups, and then went to the kill.    (see BORGES, `The Legend of the Hero and the Assassin')

If GALOIS had made an apology,
Or shot first and gone back to College, he
   Would have done so much more,
   And, surer than SCHUR,
Beaten STEINROD to found cohomology.

Is RIEMANN's last joke still a laugh?
He carefully plots on a graph
   The sum of the series,
   Deducing nine zeros,
"And probably x is one half."

Under gravity, gas in a ball,
Do the free oscillations stay small?
   The stable condition:
   First order HERMIT'ian,
And RIEMANN discovered it all.

If RIEMANN had spoken his heart,
We would have POLYA's clue from the start:
   His manuscript revels
   In energy levels,
Then zeta is taken apart.    (thanks to KEATING for finding this)

"What are numbers and what should they be?"
Was DEDEKIND's philosophy.
   He cut to the real,
   Imposed the ideal,
And his sum is a rum cup of tea.

To accompany JACOBI's theta,
And its transform that RIEMANN called zeta,
   This DEDEKIND chap
   Filled a much-needed gap
In the alphabet: DEDEKIND's eta.

When LINDEL"OF lightly let slip
A conjecture on zeta, his tip
   Was carefully sounded:
   "I don't think it's bounded
All over the critical strip."

Success seemed to bless ARTHUR CAYLEY,
Who climbed ev'ry mountain-top gaily,
   He ate strawberry jam
   In his house on the Cam,
And published on Algebra daily.

Academicals honour his leg,
He gave us three square roots of neg,
   Was rude to his wife
   For most of his life,
In his papers they found a fried egg.    [HAMILTON]

HERMITE was picked up as a spy,
"No, it's Maths. I'll explain. Let me try..."
   "What I understand
   Can't be Maths" (gun in hand).
Good thing some top brass wandered by.

When Logic abandoned its lodging
At the hand of the artful CHARLES DODGSON,
   Was `Alice' a cover
   For trysts with her mother,
An urgent erogenous sojourn?

LEWIS CARROLL found permanent fame
When he played a determinant game,
   When studying minors,
   Attaching a sign, as
The Matrix herself was his aim.    [DODGSON]

To render the ancients their due,
It has long been regarded as true,
   (Implicit in GAUSS
   And some work of LANDAU's)
That unity doubled makes two.

Is the RIEMANN Hypothesis true?
How helpful is CHEBYSHEV's clue?
   "The numbers called `primes'
   You can't build up from `times'
Can be found from factorials too".

Was to fathom where no zeros are,
   A domain which essentially
   Goes back for a century,
But it doesn't go back very far.

The inversions that M"OBIUS knew:
The surfaces, one side or two,
   With az+b
   Over cz+d,
And mu (n), which does not require glue.

By a process he cannot discuss,
As POINCARE sat on the bus,
   He saw the deduction
   Of laws theta-FUCHS-ian.
What if he used the Metro like us?

Great ideas without names we can see
In the busy nineteenth century.
   GRASSMAN used tensors daily,
   There are quotients in CAYLEY,
And SCHUR found Co-Homology.

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