How to Solve a Diophantine Equation

These notes represent an extended version of a talk I gave for the participants of the IMO 2009 and other interested people. We introduce diophantine equations and show evidence that it can be hard to solve them. Then we demonstrate how one can solve a specific equation related to numbers occurring several times in Pascal’s Triangle with state-of-the-art methods. 1. Diophantine Equations The topic of this text is Diophantine Equations. A diophantine equation is an equation of the form F (x1, x2, . . . , xn) = 0 , where F is a polynomial with integer coefficients, and one asks for solutions in integers (or rational numbers, depending on the problem). They are named after Diophantos of Alexandria on whom not much is known with any certainty. Most likely he lived around 300 AD. He wrote the Arithmetika, a text consisting of 13 books, a number of which have been preserved. In this text, he explains through many examples ways of solving certain kinds of equations like the above in rational numbers. Diophantos was also one of the first to introduce symbolic notation for the powers of an indeterminate. To give you a flavor of this kind of question, let me show you some examples. Ideally, you should cover up the part of the page below the equation and try to find a solution for yourself before you read on. The first equation is x + y + z = 29 , an equation in three unknowns, to be solved in (not necessarily positive) integers. I trust it did not take you very long to come up with a solution like (x, y, z) = (3, 1, 1) or maybe (4,−3,−2). Now let us look at x + y + z = 30 . Try to solve it for a while before you look up a solution in this footnote. This solution is the smallest and was found by computer search in July 1999 and published in 2007 [1]. This already indicates that it may be quite hard to find a solution to a given diophantine equation. Now consider