The theorem's history, however, is more complex than this legend
would suggest. The use of the 3-4-5 triangle for constructing a
right angle, for instance, goes back to much earlier times in Egypt,
Babylon, and China. In his textbook The History of Mathematics,
Roger Cooke of the University of Vermont describes how the
Babylonians might have discovered the Pythagorean theorem more than
1,000 years before Pythagoras.
Basing his account on a passage in Plato's dialogue Meno, Cooke
suggests that the discovery arose when someone, either for a
practical purpose or perhaps just for fun, found it necessary to
construct a square twice as large as a given square. Simply doubling
a square's side actually quadruples the square's area. If you
contemplate the quadrupled square for a while, you might think to
join the midpoints of adjacent sides--in effect, drawing the
diagonals of the four copies of the original square.
"Since these diagonals cut the four
squares in half, they will enclose a square twice as big as the
original one," Cooke notes. Someone 'playing' with the figure might
then consider the effect of joining points on adjacent sides when
they are no longer the midpoints but at a given distance from the
corners of a square.
"Doing so creates a square in the center of the larger square
surrounded by four copies of a right triangle whose hypotenuse
equals the side of the center square; it also creates the two
squares on the legs of that right triangle and two rectangles that
together are equal in area to four copies of the triangle," Cooke
writes. This construction adds up to the
Pythagorean theorem.
"The
Pythagorean theorem was an
early example of an important fact rediscovered independently and
often," Veljan remarks. Moreover, more than 400 different proofs of
the theorem are known today
PYTHAGOREAN TRIPLES
The famous Babylonian clay tablet known as Plimpton 322 goes a step
further. Dating from the period between 1900 B.C. and 1600 B.C., the
tablet has columns of numbers that apparently represent what are now
called PYTHAGOREAN TRIPLES.
The whole numbers a, b, and c are a Pythagorean triple if a and b
are the lengths of two sides of a right triangle with hypotenuse c,
so a2 + b2 = c2.
In general, for any number k, the corresponding Pythagorean triple
is a = 2k + 1, b = 2k(k + 1), and c = b + 1. For example, when k =
1, a = 3, b = 4, and c = 5. When k = 2, a = 5, b = 12, and c = 13.
The Babylonians used a sexagesimal, or base 60, number system. The
Plimpton tablet has several columns of numbers, written in cuneiform
script. The following table shows the numbers in two of the columns
written in decimal notation. One apparent error is corrected (4825
replaces 11521 in the second row).
119
169
3367
4825
4601
6649
12709
18541
65
97
319
481
2291
3541
799
1249
Mathematics historian Howard Eves has conjectured that each pair of
numbers represents two of the three members of a Pythagorean triple,
corresponding to one side and the hypotenuse of a right triangle.
The numbers also fit the following formula for finding Pythagorean
triples: a = 2uv, b = u2 - v2, and c = u2 + v2, where u and v are
relatively prime, one number is odd while the other is even, and u
is greater than v. For example, when u = 12 and v = 5, b = 119 and c
= 169 (as given in the first row of the table) and a must be 120.
It's straightforward to extend the Pythagorean formula to right
triangles in three and higher dimensions. For example, for a
rectangular box that is a units long, b units wide, and c units
high, the diagonal d obeys the following relationship: d2 = a2 + b2
+ c2. Moreover, you can look for analogous relationships for
triangles on the surface of a sphere, on the hyperbolic plane, and
in other spaces.
Generalizing the Pythagorean equation for triangles with integer
sides to powers greater than 2 leads to Fermat's last theorem and
the so-called ABC conjecture.
THE AMAZING ABC CONJECTURE
In number theory, straightforward, reasonable questions are
remarkably easy to ask; yet many of these questions are surprisingly
difficult or even impossible to answer.
Fermat's last theorem, for instance, involves an equation of the
form x^n + y^n = z^n. More than 300 years ago, Pierre de Fermat
(1601-1665) conjectured that the equation has no solution if x, y,
and z are all positive integers and n is a whole number greater than
2. Andrew J. Wiles of Princeton University finally proved Fermat's
conjecture in 1994.
In order to prove the theorem, Wiles had to draw on and extend
several ideas at the core of modern mathematics. In particular, he
tackled the Shimura-Taniyama-Weil conjecture, which provides links
between the branches of mathematics known as algebraic geometry and
complex analysis.
That conjecture dates back to 1955,
when the late Yutaka Taniyama published it in Japanese as a research
problem. Goro Shimura of Princeton and Andre Weil of the Institute
for Advanced Study provided key insights in formulating the
conjecture, which proposes a special kind of equivalence between the
mathematics of objects called elliptic curves and the mathematics of
certain motions in space.
The equation of Fermat's last theorem is one example of a type known
as a Diophantine equation -- an algebraic expression of several
variables whose solutions are required to be rational numbers
(either whole numbers or fractions, which are ratios of whole
numbers). These equations are named for the mathematician Diophantus
of Alexandria, who discussed such problems in his book Arithmetica.
In fact, it was in the margin of a page of a Latin translation of
Arithmetica that Fermat first set down the proposition that came to
be known as Fermat's last theorem. He had studied the book closely,
making marginal notes in his copy. After Fermat's death, his son
published a new edition of Arithmetica that included the notes in an
appendix.
Interestingly, the Wiles proof of Fermat's last theorem was a
by-product of his deep inroads into proving the Shimura-Taniyama-Weil
conjecture. Now, the Wiles effort could help point the way to a
general theory of three-variable Diophantine equations.
Historically, mathematicians have always had to state and solve such
problems on a case-by-case basis. An overarching theory would
represent a tremendous advance.
The key element appears to be a problem termed the ABC conjecture,
which was formulated in the mid-1980s by Joseph Oesterle of the
University of Paris VI and David W. Maser of the Mathematics
Institute of the University of Basel in Switzerland. That conjecture
offers a new way of expressing Diophantine problems, in effect
translating an infinite number of Diophantine equations (including
the equation of Fermat's last theorem) into a single mathematical
statement.
Like many problems in number theory, the ABC conjecture can be
stated in relatively simple, understandable terms. It incorporates
the concept of a square-free number: an integer that is not
divisible by the square of any number. For instance, 15 and 17 are
square free, but 16 and 18 are not.
The square-free part of an integer n is defined to be the largest
square-free number that can be formed by multiplying the prime
factors of n. That quantity is denoted sqp(n). Thus, for n = 15, the
prime factors are 5 and 3, and 3 x 5 = 15, a square-free number. So
sqp(15) = 15. On the other hand, for n = 16, the prime factors are
all 2, which means that sqp(16) = 2. Similarly, sqp(17) = 17 and
sqp(18) = 6.
In general, if n is square free, the square-free part of n is just
n. Otherwise, sqp(n) represents what's left over after all the
factors that create a square have been eliminated. In other words,
sqp(n) is the product of the distinct prime numbers that divide n.
So sqp(9) = sqp(3 x 3) = 3; sqp(1400) = sqp(2 x 2 x 2 x 5 x 5 x 7) =
2 x 5 x 7 = 70.
With these preliminaries out of the way, mathematician Dorian
Goldfeld of Columbia University describes the ABC conjecture in the
following terms: The problem deals with pairs of numbers that have
no factors in common. Suppose A and B are two such numbers and that
C is their sum. For example, if A = 3 and B = 7, then C = 3 + 7 =
10. Now, consider the square-free part of the product A x B x C:
sqp(ABC) = sqp(3 x 7 x 10) = 210.
For most choices of A and B, sqp(ABC) is greater than C, as in the
example above. In other words, sqp(ABC)/C is larger than 1. Once in
a while, however, that isn't true. For instance, if A is 1 and B is
8, then C = 1 + 8 = 9, sqp(ABC) = sqp(1 x 8 x 9) = sqp(1 x 2 x 2 x 2
x 3 x 3) = 1 x 2 x 3 = 6, and sqp(ABC)/C = 6/9 = 2/3. Similarly, if
A is 3 and B is 125, the ratio is 15/64, and if A is 1 and B is 512,
the ratio is 2/9.
Masser proved that the ratio sqp(ABC)/C can get arbitrarily small.
In other words, if you name any number greater than zero, no matter
how small, you can find integers A and B for which sqp(ABC)/C is
smaller than that number.
In contrast, the ABC conjecture states that [sqp(ABC)]^n/C does
reach a minimum value if n is any number greater than 1 -- even a
number such as 1.0000000000001, which is just barely larger than 1.
The tiny change in the expression makes a vast difference in its
mathematical behavior.
Astonishingly, a proof of the ABC conjecture would provide a way of
establishing Fermat's last theorem in less than a page of
mathematical reasoning. Indeed, many famous conjectures and theorems
in number theory would follow immediately from the ABC conjecture,
sometimes in just a few lines.
"The ABC conjecture is amazingly simple compared to the deep
questions in number theory," says Andrew J. Granville of the
University of Georgia in Athens. "This strange conjecture turns out
to be equivalent to all the main problems. It's at the center of
everything that's been going on."
"Nowadays, if you're working on a problem in number theory, you
often think about whether the problem follows from the ABC
conjecture," he adds.
"The ABC conjecture is the most
important unsolved problem in Diophantine analysis," Goldfeld writes
in Math Horizons. "It is more than utilitarian; to mathematicians it
is also a thing of beauty. Seeing so many Diophantine problems
unexpectedly encapsulated into a single equation drives home the
feeling that all the sub disciplines of mathematics are aspects of a
single underlying unity, and that at its heart lie pure language and
simple expressibility."
“Though more than 2,500 years old, Veljan concludes, "this
'folklore' theorem remains eternally youthful, as many people
continue to find new interpretations, generalizations, analogues,
proofs, and applications.”
BIBLIOGRAPHY
October issue of Mathematics Magazine.2000
The History of Mathematics, Roger Cooke of the University of Vermont
Ahmed, A. 1999. Extension of Pythagorean triples. Mathematics
Enrichment
Beardon, A. 1997. Pythagorean triples. Mathematics Enrichment.
Beardon, T., and B. Hardy. 1998. Picturing Pythagorean triples.
Mathematics Enrichment
Veljan, D. 2000. The 2500-year-old Pythagorean theorem. Mathematics
Magazine 73(October):260.
