“Arithmetic” by Diophantus of Alexandria, a 3rd century Greek mathematician, is one of the most influential and fruitful works in mathematics. It originally consisted of 13 “books” (chapters), but only six of these were known in 1621, when the French mathematician and poet Claude Bachet (1581–1638) published an annotated Latin translation (four more chapters were discovered in 1968, in the a temple library in Iran). Bachet’s translation played a unique role in the history of number theory.
It was in its margins that Fermat wrote, in 1637, that he knew how to prove that the equation xn+ andn= zn has no positive integer solutions when n is an integer greater than 2. This famous statement became known as “Fermat’s Last Theorem”, but it was only proved in 1993/94 by the Englishman Andrew Wiles.
Elsewhere in the book, Bachet points out that Diophantus seemed to think that every positive integer can be written as a sum of four perfect squares, that is, four numbers of the form a2 where a is an integer (for example, 42 = 22+22+32+52). Bachet writes that he has checked this fact for all integers up to 325, and that he would like to see proof that it is always true.
Fermat read carefully and found proof. At least that’s what he told in several letters written in the 1630s to 1650s. In fact, he went further, stating that every integer is the sum of three triangular numbers, four square numbers, five pentagonal numbers, and so on. For a change, Fermat did not publish the reasoning, but historians believe that in this case he really knew how to prove this beautiful result.
Euler became interested in the question of the four squares, starting in 1730, obtaining partial advances. But the complete solution was not reached by Lagrange until 1772 (nowadays, the result is called Lagrange’s theorem). The following year, Euler published a work in which he congratulated his French colleague and presented another solution.
The question of the three triangular numbers was proved by Gauss on July 10, 1796. We know the exact date because he noted in his diary: “Eureka! num = Δ + Δ + Δ”.
Fermat’s general statement (for triangular, square, pentagonal numbers, etc.) was finally resolved by Cauchy in 1813. But the story was far from over. I will return to her in the next few weeks.
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