Progress and Plimpton 322

I have now worked through the rest of the first chapter of 'Elements of Number Theory' and some further topics were introduced, namely, Diophantine Equations, Pythagorean Triples, The Diophantus chord method and Gaussian integers.

Pythagorean triples are natural numbers (x,y,z) which are solutions of the Pythagorean equation
$$x^{2}+y^{2}=z^{2}$$

A Babylonian clay tablet called Plimpton 322 (this is its museum catalogue number) shows that such numbers were known about 3,800 years ago - a whole 1,300 years before Pythagoras! As presented in Stillwell's book, there are 15 pairs of real numbers which can be interpreted as values of y and z since z²-y² is a perfect square. Further investigation of these numbers (exercises 1.8.2 and 1.8.3 of this book) shows that six of these triples can be constructed from other smaller triples. If (a₁,b₁,c₁) and (a₂,b₂,c₂) are Pythagorean triples, then so too is (a₁a₂-b₁b₂, a₁b₂+b₁a₂,c₁c₂). This also means that a triple (a,b,c) can also generate triple (a²-b², 2ab, c²). Clearly, if we are looking for triples of this type, then the component z must not be prime.

For example, for the triple in Plimpton 322 that is (4800,4601,6649), we have 6649=61x109 (both 61 and 109 are prime), so we can have c₁=61 and c₂=109. There are two triples with these z values (60,11,61) and (91,60,109) and indeed these two triples do generate (4800,4601,6649) as you can check. Another example is the triple (240,161,289) and this can be generated from the single triple (15, 8,17) since 17²=289. Note that these two examples from Plimpton 322 are what are termed primitive triples, since their x, y and z terms do not have a common prime divisor.

It all goes to show that the Babylonians knew a thing or two about maths if they were aware of these complicated relationships.