ABSTRACT
In the late 1630s, Pierre de Fermat was reading Diophantus’s Arithmetica, which had recently received a new translation from Greek into Latin by Bachet. The Arithmetica consists of hundreds of algebraic problems covering linear, quadratic, and higher degree equations and uses algebraic techniques to find their solutions. Fermat must have been intrigued by Problem 8 from Book II, which asks how “to divide a square number into two squares.” Diophantus starts with a square, for example 16, and tries to write it in the form x2 +y2 with nonzero rational numbers x and y. To do so, he starts with an “obvious” solution, namely (x, y) = (4, 0), and uses it to find a nontrivial solution. Take, for example, a line with slope −2 through (4, 0), described parametrically by
x = 4− t, y = 2t
for a parameter t. Substitute this into x2 + y2 = 16:
16 = (4− t)2 + (2t)2 = 16− 8t+ 5t2.
