Fermat’s Last Theorem

It is now 25 years since Andrew Wiles provided the proof of Fermat’s Last Theorem. A celebration of the event was held at the Isaac Newton Institute on 1 October 2018, with lectures (all now available online) by Sir Andrew Wiles, Jack Thorne and John Coates. The Wren Library provided a historical context for the day by displaying the first appearance of Fermat’s Last Theorem in print.

Pierre de Fermat famously wrote down his last theorem in the 1630s in the margin of a bilingual Greek and Latin edition of the Arithmetica of Diophantus of Alexandria. The copy with his annotation no longer survives, but Fermat’s son incorporated the conjecture into a new edition of Diophantus which he published in Toulouse after his father’s death, in 1670. Following the Diophantine proposition to divide a square into two other squares, Fermat’s observation reads:

Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos & generaliter nullam in infinitum ultra quadratum potestatem in duos eiusdem nominis fas est dividere cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet. It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvellous proof of this, which this margin is too narrow to contain.

The simple proposition took more than 350 years to prove, and became the subject of a bestselling book by Simon Singh in 1997.

Diophanti Alexandrini Arithmeticorvm libri sex, et De nvmeris mvltangvlis liber vnvs. Cvm commentariis C.G. Bacheti v.c. & obseruationibus D.P. de Fermat senatoris Tolosani (Toulouse: Bernardvs Bosc, 1670). Trinity College Library, T.17.21.

The diagram below is taken from a collection of Fermat’s mathematical writings compiled by his son Samuel de Fermat and published in 1679. This page shows the end of a long letter from Blaise Pascal to Fermat in which he discusses the problem of the division of a stake between two players whose game is interrupted before its close. The table shows the value of shares when two gamblers play, putting 256 pistoles at stake. ‘The numbers of the first line are always increasing. Those of the second do the same. Those of the third do the same. But after that, those of the fourth line diminish. Those of the fifth, etc. Which is strange.’

Varia opera mathematica D. Petri de Fermat, Senatoris Tolosani. Accesserunt selectæ quædam ejusdem epistolæ, vel ad ipsum à plerisque doctissimis viris Gallicè, Latinè, vel Italicè, de rebus ad mathematicas disciplinas, aut physicam pertinentibus scriptæ. (Toulouse: Johannes Pech, 1679). Trinity College Library, T.11.45.