Fermat’s Last Theorem. The story of a riddle that confounded the world’s greatest minds for 358 years. Foreword by John Lynch.

*(English)*Zbl 0930.00001
London: Fourth Estate Ltd. xxii, 362 p. (1997).

The author directed the excellent BBC Horizon program “Fermat’s Last Theorem”, transcript at http://www.bbc.co.uk/horizon/fermattran.shtml ; the video has been available from the AMS. This, the book of the film – so to speak – is surely of noticeably less value to mathematicians. There’s one important proviso: the present book has proved highly popular in the general community. We mathematicians might be well advised to learn from it just how one might communicate with the real world. That said, the mathematical content of the book is vanishingly small; the attempt to explain Frey’s contribution gets it wrong (not just wrong in the constipated sense that some ‘ı’ is undotted; but because ‘i ’s are randomly replaced by ‘t ’s). Also, a warning: The American edition (entitled Fermat’s Enigma) omits various pieces that might amuse mathematicians. The British (and European) edition is the one commented on here.

So, what does one get? First, some excellent pieces from the original program. Wiles’ description of the nature of mathematical discovery: “Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room and it’s dark, completely dark, one stumbles around bumping into the furniture and then gradually you learn where each piece of furniture is, and finally after six months or so you find the light switch, you turn it on, suddenly it’s all illuminated, you can see exactly where you were.” is a highlight. But the book does not quite convey the charm of the program generally.

We do get some fairly turgid quasi-history, about the Pythagoreans, the development of ‘Fermat’s riddle’, and the like. I suggest that E. T. Bell ’s ‘The Last Theorem’ (MAA Spectrum, Washington 1990; Zbl 0706.11001) remains more fun for mathematicians (and is at worst only little more inaccurate). Interlaced with this is the new recent part of the Fermat legend. For real people that’s done well enough, I suppose, but I would have hoped that mathematicians might prefer touches of the real thing as in the reviewer’s ‘Notes on Fermat’s Last Theorem’ (Wiley-Interscience, 1996; see Zbl 0882.11001); or better yet, Andrew Granville’s review at Am. Math. Mon. 106, No. 2, 177-181 (1999). The BBC Horizon program is reviewed in extenso at Notices Am. Math. Soc. 44, No. 1, 26-28 (1997); and the present book at ibid. 44, No. 10, 1304-1306 (1997). Mathematicians really wanting details of the proof of Fermat’s Last Theorem should turn to the report of the Boston, 1995 workshop (Gary Cornell, Joseph Silverman, and Glenn Stevens (eds.), Modular Forms and Fermat’s Last Theorem, New York, NY: Springer (1997; Zbl 0878.11004)). Mathematicians hoping for a friendly introduction to that report might turn to the reviewer’s book already cited. [Granville writes about ‘Notes on Fermat’s Last Theorem’: “At the very least, if you have a clever undergraduate student, bored by upper division calculus and ready for something a little more poignant, get her to read this book and let her first experience of research level mathematics be provoking, inspiring and fun.”]

The British edition of ‘Fermat’s Enigma’ contains at pages 310-318 (in the context of musings on what might replace Fermat’s Last Theorem as the great enigma of mathematics) a somewhat gratuitous discussion of Kepler’s sphere-packing problem. Those of us who attended Neil Sloane’s delightful lecture at the Berlin ICM, 1998, had the gratification of hearing it announced from the audience that the matter had been settled. In this context it seems clear that it is the Super Fermat Conjecture (I call it the Generalized Fermat Conjecture in my book), to the effect that \(x^r+y^s=z^t\) has no solutions in nonzero relatively prime integers \(x\), \(y\) and \(z\) if all of the integers \(r\), \(s\) and \(t\) exceed \(2\), that remains the most inspiring replacement for the FLT. Already there’s a new Wolfskehl, the Texan banker Andrew Beal, offering a substantial monetary reward for settling the matter.

But, no. There’s no marvellous forgotten amateur proof of the Super FC, just as there surely was no such proof of the FLT. Fermat realised quickly that his marginal remark was mistaken, as is evidenced by his only ever boasting about a proof in the case of exponent \(n=3\) (and reporting his proof for \(n=4\)). Besides, Fermat’s ingenious constructions on elliptic curves in those cases are likely to remain incomprehensible to amateurs.

In summary: Fermat’s Enigma is a fine book with more than enough good in it both to justify its bestseller status and to warrant it being a mathematician’s gift of choice to friends wondering just what it is that exercises some mathematicians. But Fermat’s Enigma is about mathematics and its history; it is not itself mathematics. If those friends are themselves mathematicians, one can already do better. That said, I reluctantly endorse the suggestion that Fermat’s Last Theorem is still awaiting a first rate exposition, whether for the general community or for the practioner in the mathematical sciences.

For a review of the American edition (New York: Walker and Company 1997) see Zbl 0913.01003 and for the German edition (München: Carl Hanser 1998) see Zbl 0930.00002 below.

So, what does one get? First, some excellent pieces from the original program. Wiles’ description of the nature of mathematical discovery: “Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the first room and it’s dark, completely dark, one stumbles around bumping into the furniture and then gradually you learn where each piece of furniture is, and finally after six months or so you find the light switch, you turn it on, suddenly it’s all illuminated, you can see exactly where you were.” is a highlight. But the book does not quite convey the charm of the program generally.

We do get some fairly turgid quasi-history, about the Pythagoreans, the development of ‘Fermat’s riddle’, and the like. I suggest that E. T. Bell ’s ‘The Last Theorem’ (MAA Spectrum, Washington 1990; Zbl 0706.11001) remains more fun for mathematicians (and is at worst only little more inaccurate). Interlaced with this is the new recent part of the Fermat legend. For real people that’s done well enough, I suppose, but I would have hoped that mathematicians might prefer touches of the real thing as in the reviewer’s ‘Notes on Fermat’s Last Theorem’ (Wiley-Interscience, 1996; see Zbl 0882.11001); or better yet, Andrew Granville’s review at Am. Math. Mon. 106, No. 2, 177-181 (1999). The BBC Horizon program is reviewed in extenso at Notices Am. Math. Soc. 44, No. 1, 26-28 (1997); and the present book at ibid. 44, No. 10, 1304-1306 (1997). Mathematicians really wanting details of the proof of Fermat’s Last Theorem should turn to the report of the Boston, 1995 workshop (Gary Cornell, Joseph Silverman, and Glenn Stevens (eds.), Modular Forms and Fermat’s Last Theorem, New York, NY: Springer (1997; Zbl 0878.11004)). Mathematicians hoping for a friendly introduction to that report might turn to the reviewer’s book already cited. [Granville writes about ‘Notes on Fermat’s Last Theorem’: “At the very least, if you have a clever undergraduate student, bored by upper division calculus and ready for something a little more poignant, get her to read this book and let her first experience of research level mathematics be provoking, inspiring and fun.”]

The British edition of ‘Fermat’s Enigma’ contains at pages 310-318 (in the context of musings on what might replace Fermat’s Last Theorem as the great enigma of mathematics) a somewhat gratuitous discussion of Kepler’s sphere-packing problem. Those of us who attended Neil Sloane’s delightful lecture at the Berlin ICM, 1998, had the gratification of hearing it announced from the audience that the matter had been settled. In this context it seems clear that it is the Super Fermat Conjecture (I call it the Generalized Fermat Conjecture in my book), to the effect that \(x^r+y^s=z^t\) has no solutions in nonzero relatively prime integers \(x\), \(y\) and \(z\) if all of the integers \(r\), \(s\) and \(t\) exceed \(2\), that remains the most inspiring replacement for the FLT. Already there’s a new Wolfskehl, the Texan banker Andrew Beal, offering a substantial monetary reward for settling the matter.

But, no. There’s no marvellous forgotten amateur proof of the Super FC, just as there surely was no such proof of the FLT. Fermat realised quickly that his marginal remark was mistaken, as is evidenced by his only ever boasting about a proof in the case of exponent \(n=3\) (and reporting his proof for \(n=4\)). Besides, Fermat’s ingenious constructions on elliptic curves in those cases are likely to remain incomprehensible to amateurs.

In summary: Fermat’s Enigma is a fine book with more than enough good in it both to justify its bestseller status and to warrant it being a mathematician’s gift of choice to friends wondering just what it is that exercises some mathematicians. But Fermat’s Enigma is about mathematics and its history; it is not itself mathematics. If those friends are themselves mathematicians, one can already do better. That said, I reluctantly endorse the suggestion that Fermat’s Last Theorem is still awaiting a first rate exposition, whether for the general community or for the practioner in the mathematical sciences.

For a review of the American edition (New York: Walker and Company 1997) see Zbl 0913.01003 and for the German edition (München: Carl Hanser 1998) see Zbl 0930.00002 below.

Reviewer: A.J.van der Poorten (North Ryde)