In 1995, Andrew Wiles completed a proof of Fermat's Last Theorem. Although this was certainly a great mathematical feat, one shouldn't dismiss earlier attempts made by mathematicians and clever amateurs to solve the problem. In this book, aimed at amateurs curious about the history of the subject, the author restricts his attention exclusively to elementary methods that have produced rich results.
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From the reviews: MATHEMATICAL REVIEWS "The history of elementary approaches to Fermat is very rich indeed, and Ribenboim has arranged these approaches in a way that makes them accessible to interested readers without extensive mathematical backgrounds…both readable and fairly comprehensive. This book would likely be of great interest to an enthusiastic undergraduate with a basic knowledge of rings and fields. In addition to describing the history of one of the great problems in number theory, the book provides a gentle and well-motivated introduction to some important ideas in modern number theory…any reader who spends a few hours with this book is guaranteed to learn something new and interesting about Fermat’s last theorem."
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This chapter is devoted to the proof of special cases of Fermat's theorem: exponents 4, 3, 5, and 7. Read the first page Browse Sample Pages
Front Cover | Copyright | Table of Contents | Excerpt | Index
This chapter is devoted to the proof of special cases of Fermat's theorem: exponents 4, 3, 5, and 7. Read the first page Browse Sample Pages
Front Cover | Copyright | Table of Contents | Excerpt | Index
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2 of 2 people found the following review helpful
Format:Hardcover
If, like me, you were fascinated to hear that Fermat's so-called "last theorem" had been proven in 1995, then read Simon Singh or Amir Aczel's books popularizing the proof in outline, you probably wanted something more.
If, like me, you are a person who took some math in college, enjoys recreational mathematics books of the Douglas Hofstadter and Ian Stewart genre, and even sometimes picks up the odd number-theory book, you might consider yourself an "amateur."
If...if... this might seem like the book for you. I'd suggest that its not.
The mathematics in this book and its level of presentation was simply impenetrable by me. Not slow going... "no" going. That's frustrating to admit, but in a way fine, since it affirms of my admiration at a distance of the work that professional mathematicians do. I have seen many cited who state that Wiles' proof is simply beyond the ken of even 95% of working mathematicians. I believe this book must really be intended to serve some fraction of that group. Perhaps within the fold of mathematics these would consider themselves "amateurs". My two stars are offered only for them.
The book is simply not for the "lay" amateur. And Ribenboim's titling of it suggests that he does not even know that this lower caste, containing those of us who enjoy recreational mathematics and would describe ourselves as "amateurs", even exists. We know we exist as something mathematically distinct from the general population by the simple fact of the universally raised eyebrows that confront any mention of our interest in mathematics. Nevertheless, like any other species in a niche, we will have to continue to feed on a sparse supply of intellectual sustenance and learn to avoid the over-rich and indigestible fare of the higher forms.
Finally, if you haven't read Singh or Aczel I'd offer the former 5 stars and the latter 3 but recommend both. A truly fascinating story.
Format:Hardcover
Solid coverage of proofs relating to Fermat's Last Theorem up to Kummer's Theory. You will find proofs for n=2, n=3, n=4, n=5, and n=7. Requires solid background in Algebraic Number Theory. For example, you should already have a good understanding of the Quadratic Law of Reciprocity, Quadratic Fields, and Congruences. If you don't, I recommend Elementary Number Theory for Congruences and the Quadratic Law of Reciprocity and Stark's An Introduction to Number Theory for Quadratic Fields. I would also recommend starting out with Edward's Book on Fermat's Last Theorem which includes detailed coverage of Kummer's Theory.
Format:Hardcover
I find that this is a great book if you are an instructor or have a solid background in algebraic number theory. If you are unfamiliar with the Legendre Symbol, Gaussian integers, or the Law of Quadratic Reciprocity, you may wish to start out with a book such as Elementary Number Theory. If your are familiar with Algebraic Number Theory and wish to study in detail the Fermat Last Theorem proofs up to Kummer's Theory, this is a great book. I would recommend starting out with Edward's Book (Fermat's Last Theorem), for analysis of Euclid's proof of N=3. I found this very useful as an example of applications of Gaussian integers and Eisenstein integers. Ribenboim is one of the top experts about Fermat's Last Theorem and he is to praised for putting these beautiful proofs down. Even so, I would recommend purchasing other books to help explain this one. I found Stark's book very helpful in understanding Quadratic Fields.
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