# A brief 4,000 year history of Diophantine Equations

I filled in for a NY Math Circle class over the weekend. Since the topic was Primitive Pythagorean Triples, I had a blast. I also shared the following outline with the students. Each item is full of wonderful mathematics and anecdotes!

Plimpton 322, a Babylonian cuneiform tablet @ Columbia University. From 1900BCE – 1600BCE, and allegedly includes the Pythagorean triple (12709, 13500, 18541).

Pythagoras was born ca 580BCE on the island of Samos. Famous quote: “All is Number”. His proof of the theorem that bears his name involved cutting up a square of side a+b and rearranging the pieces.

Proclus (5th century CE) credits Pythagoras with the formula (2n+1, 2n^2 + 2n, 2n^2 + 2n + 1) of (necessarily) primitive pythagorean triples where the hypotenuse is 1 more than one of the sides. Proclus also credits Plato with the formula (2n, n^2 – 1, n^2 + 1) where the hypotenuse is 2 more than one of the sides. (For what n is this primitive?)

Euclid of Alexandria (born 365 BCE) is famous for the 13-book Elements. The Theorem of Pythagoras is Book I, Proposition 47. Analysis on Primitive Pythagorean Triples appears as Lemma 1 “To find two square numbers such that their sum is also a square” in Book X, just before Proposition 29.

By the way, Elisha Scott Loomis, an early 20th century mathematician, published The Pythagorean Proposition, which has 370 proofs of the theorem, and not a single one used trigonometry. This was republished in 1968 by the NCTM.

Diophantus of Alexandria (200CE – 298CE) is famous for his book Arithmetica. He sought integer (and perhaps rational too?) solutions to algebraic equations. The term Diophantine Equation typically refers to equations where we seek positive integer solutions.

Fermat (1608CE – 1665CE) wrote in the margin of his copy of Arithmetica that there are no integral values of x, y, z so that x^n + y^n = z^n if n > 2.

Andrew Wiles of Princeton University announced a proof in 1993 of Fermat’s Last Theorem, after working in secret for 7 years. An error was found in his proof, which was salvaged in 1994. Wiles’ proof was published in 1995.

Jumping back to 1900, David Hilbert asked mathematicians at the International Congress of Mathematicians to devise a method to determine whether a Diophantine equation has solutions. This is known as Hilbert’s 10th Problem.

Julia Robinson (1919-1985) was a Californian mathematician who worked on Hilbert’s 10th problem for decades, from the 1940s until final achieving a solution (in joint work with Martin Davis, Hilary Putnam, Yuri Matiyasevich and others) in 1970. In general, there is no such algorithm!

Mirroring the Hilbert Problems of 1900, the Clay Institute of Mathematics issued the Millennium Problems in 2000.