The Number Theory Revival

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After the work of Diophantus, number theory in Europe languished for about 1000 years. In Asia there was significant progress, as we saw in Chapter 5, on topics such as Pell’s equation. The first signs of reawakening in Europe came in the 14th century, when Levi ben Gershon found formulas for the numbers of permutations and combinations, using rudimentary induction proofs. Interest in number theory gathered pace with the rediscovery of Diophantus by Bombelli, and the publication of a new edition by Bachet de M´eziriac (1621). It was this book that inspired Fermat and launched number theory as a modern mathematical discipline. Fermat mastered and extended the techniques of Diophantus, such as the chord and tangent method for finding rational points on cubic curves. He also shifted the emphasis from rational solutions to integer solutions. He proved “Fermat’s little theorem” that $n^p - n$ is divisible by p for any prime p, and claimed “Fermat’s last theorem” that $x^n + y^n = z^n$ has no positive integer solutions when n> 2. We know that Fermat had a proof of his “last theorem” for n = 4, but he seems to have been mistaken in thinking that he could prove it for arbitrary n. The proof now known uses highly sophisticated ideas, not conceivable in the 17th century. Nevertheless, it is strangely appropriate that the modern proof reduces Fermat’s last theorem to a problem about cubic curves.