topics: |  math | history | fermat | aczel-amir-1996_fermats-last-theorem

Pierre de Fermat and his theorems

Pierre de Fermat was a 17c French jurist, son of a leather merchant. Developed the main ideas of calculus, thirteen years before Newton's birth. Around 1637 AD, Fermat wrote in Latin in the margin of his copy of Diophantus' Arithmetica, next to a problem on breaking down a squared number into two squares:

On the other hand, it is impossible to separate a cube into two cubes, or a biquadrate into two biquadrates, or generally any power except a square into two powers with the same exponent. I have discovered a truly marvelous proof of this, which, however, the margin is not large enough to contain. - p.9

[Fermat had many other results relating to primes. Another simple yet elegant one is the (4n+1) theorem, which states that all odd primes expressible as 4n+1 (mod 4=1) is the sum of two squares, e.g. 13=2²+3² (see Simon Singh's Fermat's enigma p.63, though Singh incorrectly calls this the prime number theorem).

Another theorem due to Fermat is the Fermat's little theorem, which says that if p is prime, then a^p mod p = a. If this does not hold, p is certainly composite. The main statistical approach to primality checking is to generate a lot of a's and then test this; if it is false for any a, then p is composite, otherwise there is a good chance it may be a prime. ]

Ancient origins of indeterminate integer equations (diophantine)

Also traces earlier history of mathematics.
Euclid of Alexandria around 300 BC - first two vols of elements are
       believed to be the work of Pythagoras
Eudoxus of Cnidus 408-355 BC
Archimedes 287-212 BC

From the Babylonian site of Nippur, over 50,000 tablets were recovered
and are now in the collections of the museums at Yale, Columbia and
U. Penn, most of them lying in basements, unread and undeciphered.  One
tablet that was deciphered, called Plimpton 322, has 15 triples of
numbers.  Each triple has the property that all three nums are squares
and that the first number is the sum of the other two. - p.14

Pythagoras

Pythagoras was born on the Greek island Samos around 580 BC.  He
travelled extensively and visited Babylon, Egypt and possibly even
India.  Settled in Crotona, on the heel of the Italian "boot", at that
time part of the Greek world, "Magna Graecia."  Pythagoras founded a
secret society to study numbers.  Motto: "Number is everything".  The
concept of a perfect number - sum of factors = num itself, e.g. 6 =
1+2+3, also 28.
[IDEA: What of nums where sum of factors is > num, e.g. 24 or 12?
Divides nums into three groups hypo-perfect, perfect, hyper-perfect].

Discovered the irrationality of the number sqrt2 - diag of rt triangle
[1,1].

Not like 1/7 which is an infinite decimal fraction, but
recurring.  Rational and Irr nums are dense - any nbrhood of one
contains inf many of the other.  Cantor (1845-1918) : Irrationals are
infintely more numerous - orders of infinity - made fun of by Kronecker
(1823-1891), denied Cantor his professorship at Berlin, and Cantor ended
up in a mental institution. - p.23
Shocked the Pythagorean's sense of number, and swore never to tell anyone
outside the society; legend has it that Pythagoras himself killed by
drowning themember who divulged to the world the secret existence of
such strange numbers.

Pentagon - Pythagorean star

Special symbol of the Pythagoean order - five pointed star.  [Can be
drawn without lifting pencil, unlike 6-point star.]  The diagonals
intersect s.t. the whole to the larger = the larger to the smaller = the
Golden Ratio.  The inside is a smaller, inverted pentagon, which also
can have diagonals holding another one etc ad infinitum.

This was the star adapted by Islam - Pythagoras of course, was known in the
lands of Islam and there may be some cultural continuity involved.  As an
aside, the six-pointed star or hexagram, appears as a motif in Judaism (the
star of David or mogen david) and in Hinduism (the ShaRkon-yantra, the
balance between nara, man and Narayana, God).  When the asterisk was
first introduced into typography it was six-armed, but the one more common
today in printing including this "*" you see on your screen, is likely to be
five-sided.  Is this the result of Islamic opposition to the Star of David,
as I had once read somewhere?  Please let me know if you know more on this!!

Pythagoras died around 500 BC.  His center was destroyed by a rival
political group, the Sybaritics, who killed most of them, and dispersed
the others, who as refugees, influenced other centers such as Tarentum.

Diophantus and his equations

[The Egyptian] Diophantus of Alexandria (around AD 250) wrote the
Arithmetica, 15 vols, of which only six are available [rest were burnt
in the fire in the library of Alexandria].  After his death, the
Palatine Anthology contains this description of his life:

    Here you see the tomb containing the remains of Diophantus, it is
    remarkable: artfully it tells the measures of his life.  The sixth
    part of his life God granted him for his youth.  After a twelfth
    more his cheeks were bearded.  After an additional seventh he
    kindled the light of marriage, and in the fifth year he accepted a
    son.  Alas, a dear but unfortunate child, half his
    father's life he had lived
    when chill Fate took him.  He consoled his grief in the remaining
    four years of his life.  By this devise of numbers, tell us the
    extent of his life.  - p. 32 [solution: 84]

[Son was half of his father's eventual age; misunderstading in book:

    Alas, a dear but unfortunate child, half of his father he was
    when chill Fate took him.  He consoled his grief in the reemaining

If he was half his father's age when he died it leads to
to the solution 196/3 =~ 65 ]

[Integer equations were also intense objects of study in India; the
kuttaka method was used to solve diophantine equations of order 1,
i.e. integer solutions to ax+by=c.  The algorithm was presented by
Aryabhata in 4th c. AD, and elaborated by many others over the centuries.
see S Balachandra Rao's  Indian Mathematics and Astronomy: Some Landmarks
or http://abhidg.mine.nu/writings/kuttaka.html.   ]

It was Diophantus' Problem 8 in Volume II, asking for a way of dividing
a given square into the sum of two squares -- that inspired Fermat to
write his famous Last theorem on the margin.

Golden section

Golden Section = (sqrt5-1)/2 = 0.618... ; 1/g.s. = 1+g.s. = 1.618...
   	1/golden-section = 1.618 - 1 = 0.618

Take a rectangle in G.S. Take out a square.  The remaining rect is also
in G.S.  And so on.  Now draw a circle in each square ==> spiral.



x/1 = (1-x)/x  ==> x² + x - 1 = 0
g.s. : x = -1 += sqrt (1 + 4) / 2

CALCULATOR TRICK:
1+1 =	   2
1/x	   0.5		1/2
 +1 =	   1.5
1/x	   0.67		2/3
... ==> converges to G.S./inverse, alternately 0.618, 1.618
The nums in these series are fibonacci nums - 1/2, 2/3, 3/5, 5/8, 8/13,
    and their inverses ... in the limit is the G.S.  - p.25

Fibonacci : Leonardo of Pisa 1180-1250

Fibonacci = son of Bonaccio
travelled widely in N Africa, Constantinople etc

Liber Abaci:
   [Fibonacci introduces the "modus Indorum" (method of the Indians), which
   somehow came to be known as "Arabic Numerals" - place-value arithm, with 0-9]

How many pairs of rabbits will be produced in a year, beginning
with a single pair, if in every month each pair bears a new pair, which
becomes productive in the second month and so on?

Leaves on a branch grow at distances from one another that correspond to
the Fibonacci sequence.  Fibonacci numbers appear in flowers.
In most flowers, the number of petals are 3(lilies), 5(buttercups),
8(common for delphiniums), 13 (marigold), 21 (aster), 34, 55 or 89
(daisies usually have one of these three nums).

Seeds of the sunflower:Phyllotaxis

In sunflower and other composite flowers, seeds are produced centrally and
move radially outward. In their movement outward, they appear to form
overlapping spirals.  In the sunflower, the number of spirals may be
34 clockwise, and 55 counter-clockwise.  Yes! These are fibonacci numbers.
Other successive fibonacci number pairs - (55,89), or even (89, 144), are
possible.  [Ian Stewart, Nature's Numbers] - p.37

[Successive seeds develop at angles of 137.5 degrees, (the compelement from
360 degrees is 222.5, and 360/222.5 = 222.5/360 = 1.618 = golden ratio.
When new seeds are formed at regular intervals, one may assume that earlier
ones travel equal distances radially.  The resulting seeds can be shown to
lie along two spirals, whose ratios are in the proportion of two fibonacci
numers.  Here is an image with 21/34 spirals.

 [click to enlarge]
(img from http://www.fccps.k12.va.us/gm/faculty/knoke/MinhProject/Beauty_of_Nature.htm)

The solution to this problem is also an optimal solution to the problem of
packing circles on a cylinder; see The Algorithmic Beauty of Plants,
by Przemyslaw Prusinkiewicz and Aristid Lindenmayer (1990) (out of print;
online version at http://algorithmicbotany.org/papers/#abop).

    You can simulate sunflower growth in this free software developed by
    [www.geocities.com/kyletimmerman@sbcglobal.net/|Kyle Timmerman]

Euler

Leonhard Euler was at Catherine the Great's Academy in St Petersburg
when philosopher Denis Diderot was visiting her.  She arranged a debate
with Euler.  Diderot was told that Euler had a proof of God's existence
(Euler was a theology student and a pastor's son, and was religious).
Euler approached Diderot and said gravely: "Sir, a + b/n = x, hence God
exists; reply!"  Diderot, who knew nothing abt math, gave up and
immediately returned to France.

Euler loved the formula: e^i.pi + 1 = 0
Has 0,1 - the invariants; the two natural nums e and pi, and the
imaginary num i. - p.48

Sophie Germain aka Mr Leblanc and Gauss

Gauss corresponded a lot with a Monsieur Leblanc.  In 1807 he asked Mr
Leblanc to intercede with the French Consul on a tax matter.  When
Mr. Leblanc obliged, it came out that he was not a man at all, but the
lady Sophie Germain, who had hidden her gender so that she would be
taken seriously.  On finding out her identity after this, Gauss was
delighted.  However, they never met.

Sophie Germain showed that in Fermat's Last Theorem (a^n+b^n=c^n, n>2),
cannot hold for all primes less than 100, ulnless n is divisible by 5;
and if so, then a,b,c themselves are also divisible by 5. - p. 57

Fermat's proof: (p.43)

n=3 - Fermat himself abt 1635AD
n=4 - Fermat - using "method of infinite descent"
    - if holds for n, then also multiples of n.  Hence concentrate on primes
n=3,4 - Euler - independently
n=5 Peter G.L. Dirichlet 1828
n=7 Lame / Lebesque 1840

--

Fields:  Complex plane: Smallest number field that contains the
solutions of all quadratic equations.

--
Ernst Eduard Kummer (1810-1893) - worked on approaches using fields of
numbers (as did Cauchy).  These fields were inadequate, and Kummer
invented the notion of "ideal" numbers - and was able to prove FLT for
all "regular" primes.  Irregular primes less than 100 are 37, 59 and 67,
and for these as well, Kummer was able to prove FLT.

In 1816, the French Academy of Sciences offered a prize for FLT.  In
1850, it again offered a gold medal and 3000 Francs.  But in 1856 it
withdrew the award, since it did not seem a solution was imminent, and
decided to give the award to Kummer instead.

Henri Poincare

Henri Poincare' 1854-1912 born to a prominent family - cousin was
   president of France during WW1 - was utterly absentminded - would
   skip meals because he forget whether or not he had eaten.  At 17 in
   his exams, almost failed in math - but he was already famous as a
   mathematician -- The chief examiner said: "Any student other than
   Poincare would have been given a failing grade."
Symmetries in the complex plane: automorphic forms: f(z) ==>
   f(az+b/cz+d); elements a,b,c,d form an algebraic group. - p.82

   For fifteen days I strove to prove that there could not be any functions
   like those I have since called Fuchsian functions. I was then very
   ignorant; every day I seated myself at my work table, stayed an hour or
   two, tried a great number of combinations and reached no results. One
   evening, contrary to my custom, I drank black coffee and could not
   sleep. Ideas rose in crowds; I felt them collide until pairs interlocked,
   so to speak, making a stable combination. By the next morning I had
   established the existence of a class of Fuchsian functions, those which
   come from the hypergeometric series; I had only to write out the results,
   which took but a few hours.

[However, Amir fails to explain this function in sufficient depth; on p. 84 he
gives a figure of "tiling of the complex half-plane using these
symmetries" which I found completely inaccessible given the text so far.
Perhaps it would be better not to give such tantalizing details, or go the
whole hog and explain it properly, perhaps in an appendix... ]

Louis J. Mordell: Relation between algebra and topology

The number of solutions to an algebraic equation was seen to be the same as
the number of holes (the genus) of the 3D surface formed in the complex space
of solutions to the equations.

This space can be mapped in complex space - and if it has two or more holes
(genus > 2) then the equation has only finitely many solutions.  This
conjecture was proved in 1983 by Gerd Faltings at U. Wupperthal.

FLT: Genus is > 2 for n > 3; so finitely many solutions. 85

Granville and Heath-Brown ==> used Faltings' result to show that
num of solutions decrease as n increases; the percentage of n's for
which FLT holds approaches 100% as n increases.

Elliptic Curves : Cubic polynomials in two variables, e.g.
	 y³ = ax³ + bx² + cx, [a,b,c integer, or rational]

the rational points on the elliptic curve form a group - e.g. sum of any two
is also a solution.  ==> became one of the foremost research tools in number
theory.

[Zeta functions of the elliptic curve were part of the results that Srinivasa
Ramanujan, in his ignorance, had sent Hardy in that momentous letter.]

Shimura-Taniyama conjecture

Now, the scene shifts to post-war Tokyo, where Goro Shimura and Yutaka
Taniyama were students at U. Tokyo
all elliptical functions of a certain
class are modular.  If a solution to Fermat's exists, it results in an
elliptical function that is not modular ==> proved by Ken Ribet - if S-T
is true, then FLT is true.

In 1955, they org a conference, in which 36 problems were presented.
Problems 11 - 14 were written by Taniyama, and related zeta functions of
elliptic curves with Poincares automorphous functions.

The conjectures as stated in the proceedings of that conference were
ambiguous, and after Taniyama committed suicide in 1958, by which time
Shimura was at Princetons Inst for Advanced Study.

A number of controversies arose as to who had said what after this.  French
mathematician Andre Weil, then at Princeton, is sometimes credited with
Taniyama on this conjecture, but the evidence seems to suggest that he
actually did not believe it.

The linkages from the S-T conjecture to Fermat also took quite a few
brilliant minds - Gerhard Frey, Ken Ribet, and Barry Mazur are mentioned -
Frey's conjecture that proving S-T would lead to Fermat became known as the
epsilon conjecture, and it was proved by Ken Ribet in 1985.  This proof was
mentioned to Wiles, and he became interested in proving the S-T conjecture,
which many people expected might take decades before a proof emerged.

But Andrew Wiles locked himself up in his acttic, and a theorem in 1993,
followed by a correction in 1995, proved the S-T
conjecture, and thereby the Fermat's theorem.   p.119-127

Other asides

Triangular Numbers

A triangular number is figurate number obtained by adding all positive
integers less than or equal to a given positive integer n, i.e.,
Tn = SUM(1..n) = n(n+1)/2 = (n+1)C2

Pentagonal Number

A polygonal number of the form. The first few are 1, 5, 12, 22, 35, 51, 70,
... (Sloane's A000326). The generating function for the pentagonal numbers is

	  x(2x+1)/(1-x)³ = x + 5x² + 12x^ + 22x⁴ + . . .
Every pentagonal number is 1/3 of a triangular number.

Number Theory > Special Numbers > Figurate Numbers > Miscellaneous Figurate
Numbers v

Figurate Number

A figurate number, also (but mostly in texts from the 1500 and 1600s)
known as a figural number (Simpson and Weiner 1992, p. 587), is a
number that can be represented by a regular geometrical arrangement of
equally spaced points. If the arrangement forms a regular polygon, the
number is called a polygonal number. The polygonal numbers illustrated
above are called triangular, square, pentagonal, and hexagonal
numbers, respectively. Figurate numbers can also form other shapes
such as centered polygons, L-shapes, three-dimensional solids, etc.

The nth regular r-polytopic number is given by

		     Pr(n) = (n+r-1)Cr = n^(r)/r!

where n^(r) is the rising factorial n(n+1)...(n+r-1). So

triangular: P2(n) = n(n+1)/2
tetrahedral: P3(n) = 1/6. n(n+1)(n+2)
pentatope: P4(n) = 1/24.n.n+1.n+2.n+3 etc.

scinews/mathtut/figurate-numbers.gif

Generally: Tn = triangular number (n). Then,
			   n + T(n-1) = Tn
n+2T(n-1) = n² = Square(n)
n+3T(n-1) = n(3n-1)/2 = Pentagonal(n) [Pn] etc.

Other reviews

Kirkus:

After laying the groundwork for an understanding of the basic concept, Aczel
jumps back in time to the Babylonian era, when the foundations of mathematics
were just being discovered. We follow the history of mathematics through
various steps, growing ever closer to the time of Fermat. Aczel makes a
special point of showing how mathematics continually builds upon the
discoveries of earlier scholars, and he gives a lively sense of the
personalities of the great mathematicians of the past. He does not overload
the reader with equations and other mathematical expressions but gives enough
to indicate the complexity of the concepts at issue.

The modern assault on the problem began with an obscure Japanese conference
on algebraic number theory in 1955. Two of the participants, Y. Taniyama and
G. Shimura, offered a conjecture that an American theorist [three decades
later], Ken Ribet, [proved to be] equivalent to Fermat's theorem; if the one
could be proven, the other would follow. It fell to Andrew Wiles, of
Princeton, to connect the two after seven years of secret research. His
dramatic announcement of the solution in 1993 was followed by the discovery
of a flaw, which he retired to his study to repair, eventually publishing a
perfected proof of the theorem. An excellent short history of mathematics,
viewed through the lens of one of its great problems--and achievements.

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