The 2500-year-old Pythagorean Theorem

Pythagoras of Samos was a Greek philosopher responsible for important developments in mathematics, astronomy and the theory of music. He left Samos because of the tyrant who ruled there and went to southern Italy about 532 BC. He founded a philosophical and religious school in Croton that had many followers.
PYTHAGOREAN THEOREM
Although the theorem now known as Pythagoras's theorem was known to the Babylonians 1000 years earlier he may have been the first to prove it.
It's the fact that
“THE SQUARE OF THE HYPOTENUSE OF A RIGHT TRIANGLE IS EQUAL TO THE SUM OF THE SQUARES OF THE TWO ADJACENT SIDES”.
According to one legend, Pythagoras (c.580 B.C.-c.500 B.C.) discovered the theorem while waiting to see Polycrates, the tyrannical ruler of the city of Samos. Cooling his heels in a palace hall, Pythagoras spent the time pondering the floor's square tiling. He imagined how a diagonal line cutting across a square would divide the square into two right triangles. He noted that the area of a square erected over the diagonal is double the area of the square erected on an adjacent side. In other words, the square on the hypotenuse is equal to the sum of the squares on the triangle's two legs. Pythagoras came to believe that the same relationship would hold when the legs have unequal lengths.
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.
"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.
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