Did Pythagoras discover his theorem?
M.V.N. Murthy, The Institute of Mathematical Sciences, Chennai

The theorem
All of us study Pythagoras' theorem in our school. It is actually very simple. Consider a right-angled triangle, that is, a triangle that has one right angle, or an angle measuring 90o. The sides that bound this angle are perpendicular to each other. The side opposite to or facing the right angle is the longest of all the sides and is called the hypotenuse. If the two shorter sides measure lengths a and b, while the hypotenuse has a length c, then Pythagoras' theorem can be written as an algebraic equation: a^2+b^2 = c^2. In words, in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of the two other sides.
The set of numbers (a,b,c) given by the equation above are usually called Pythagorean triples. Given any two numbers (p,q) the triple is easily generated by constructing the set of three numbers with the following identification a=p^2-q^2, b=2p^2q^2, c=p^2+q^2. You can see that a2 + b2 = (p2-q2)2 + (2p2 q2)2 or, expanding the terms using (x-y)2 = x2+y2-2xy, a2+b2=(p4+q4-2p2 q2) +(4p2 q2), or simplifying, a2+b2=p4+q4+2p2 q2 = (p2+q2)2 = c2.
The theorem is named after the Greek mathematician Pythagoras who is traditionally believed to have stated this theorem. He is supposed to have lived about 2,500 years ago during 570 BC--495 BC.
The theorem can be proved in many different ways and is possibly the most "proved" theorem in Mathematics. Because of its popularity even outside mathematics it has attracted attention among artistes, writers, musicians, etc.
There were others
But was Pythagoras the first one to discover this theorem? While he did discover the theorem, it appears he was not the first one to do so. The popular reference to his name is debatable since the theorem has been discoverd many times, by different people at very different times and places. In fact it is the most {\it discovered} theorem, in one form or the other.
The Babylonians
First of all there appears a statement of a problem posed by the Babylonians, more than 1000 years before Pythagoras, some time between 2000 BC and 1786 BC. This stated problem has a solution that happens to be the set of numbers (6,8,10). Now (6,8,10) belong to a set of numbers called Pythagorean triples as we defined earlier. That is, these are numbers such that the sum of the squares of the first two numbers is the square of the third number: 6^2+8^2=36+64=100=10^2.
You can find several examples of such number, the simplest being (3,4,5). A Mesopotamian tablet, called Plimpton 322, contains a table with many entries probably related to these triples (see figure sourced from http://en.wikipedia.org/wiki/Pythagorean_theorem). This tablet was written around 1800 BC. Such tablets were produced by Babylonians by writing on wet clay with a stylus. It has fifteen rows of four columns of numbers with the last one just numbering the rows from 1 to 15. The 11th row of the tablet, for example, is interpreted as containing the Pythagorean triple (3,4,5). It is not clear what they did with these numbers but nevertheless it is an algebraic discovery of the Pythagoras theorem since there is no reference to geometry here.

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Sexagesimals
The tablet is not easy to read since it was written in a type of cuneiform script. Also, the Babylonians did not use our modern decimal system but used a sexagesimal notation for numbers! What is that?! It is a numerical system with 60 as base instead of 10. Think of 60 minutes in an hour, so 70 minutes is one hour and 10 minutes or 1:10.
However, it is also a little confusing since they also counted 60 as six 10s! So, they had a Y shaped symbol for one and a < shaped symbol for 10. So Y equals 1, YY equals 2, < equals 10, <Y equals 11, etc., upto <<<<<YYYYYYYYY equals 59. Of course they saved space by grouping the symbols close together. Adding to the confusion is the fact that they had no symbol for zero, but this will not trouble us here!
So, for instance, the 11th row has, as its entries (leaving out the first one which is complicated to explain), 45, 1:15, 11. This refers to a triangle with short side a=3/4 and hypotenuse c=5/4. So 3/4 is written as 45 (since 3/4 of 1 hour is 45 minutes) and 5/4 is written as 1:15. The last column simply says that this is the 11th row.
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The Indians
Some time around the 8th century BC, the theorem appears in the Indian text Baudhayana Sulba Sutra. Baudhayana was an Indian mathematician whose statement of the Pythagorean theorem, in translation reads: "A rope stretched along the length of the diagonal produces an area which the vertical and horizontal sides make together." The word Sutra literally means a rope but is now intrepreted to mean rules.
Here again the Pythagorean triples appear algebraically along with a geometrical proof for an isosceles triangle. In fact, Baudhayana is credited with calculating the value of pi (the ratio of the circumference to diameter of a circle) before Pythagoras. He also addressed the problem of finding a circle whose area is the same as that of a square. He also found the square root of 2 to five decimal places by calculating the diagonal of a square of unit sides: "The measure is to be increased by a third and by a fourth decreased by the 34th. That is its diagonal approximately". That is, the square root of two equals 1 + 1/3 + 1/(3x4) - 1/(3x4x34) = 577/408=1.414216.
Another mathematician, Aapastambha appears to have provided a more general proof of the Pythagorean theorem.
The Chinese
Not to be left behind, Chinese mathematicians also seem to have discovered independently the theorem. A mathematical text, Zhou Bi Suan Jing , from around the 1st century BC records a geometrical proof of Pythagoras' theorem. The contents of the text must have been known much earlier. The proof is simple and easy to understand from the figure given in the Chinese text sourced from http://en.wikipedia.org/wiki/Pythagorean\_theorem.
The figure contains a large square which is divided into four triangles and a smaller square. The four triangles are congruent with the sides being identical but with different orientation. Let us assume the large square has a side with length c. The two sides of the triangle are a, b where b is the larger than a. By construction the small square in the centre has sides given by b-a. The area of the larger square is the sum of the areas of the four triangles and the small square. Therefore c^2 = (b-a)^2 + 4 x(ab)2=a^2+b^2, which constitutes a geometric proof of Pythagoras' theorem.
Modern Times
As you may have noticed, given two positive integers (p,q) we can always construct the Pythagorean triple (a,b,c) with integer values satisfying the theorem. Can one generalise this? The answer came in the form of Fermat's Last Theorem which states that there are no three positive integers (a,b,c) that satisfy the equation a^n+b^n=c^n for any integer n greater than 2. This theorem was first conjectured (guessed at) by Pierre de Fermat in 1637 who famously said that he had the proof but could not fit it in the margin of his notes! It remained the most famous unsolved problem until it was finally proved in 1995 by British mathematician Andrew Wiles.