Fermat’s Last Theorem

I read the first 150 pages of it yesterday, I’ll complete it today – it’s a real pageturner. Even though it’s a book about mathematics (and mathematicians) I’ve yet to stumble upon anything which is harder to deal with than the stuff I encounter in my textbooks on a daily basis – it’s a very accessible book so far. Some quotes:

1. “Archimedes will be remembered when Aeschylus is forgotten, because languages die and mathematical ideas do not. ‘Immortality’ may be a silly word, but probably a mathematician has the best chance of whatever it may mean.” An introductionary quote by G. H. Hardy. I thought I’d post it also because of the recency of this discussion on this blog.

2. “It is true that I had hit the date of the displacement of the judges of Castres to Toulouse, where he [Fermat] is the Supreme Judge to the Sovereign Court of Parliament; and since then he has been occupied with capital cases of great importance, in which he has finished by imposing a sentence that has made a great stir; it concerned the condemnation of a priest, who had abused his functions, to be burned at the stake. This affair has just finished and the execution has followed.”

From a letter by the mathematician Sir Kenelm Digby to his collegue John Wallis. On the next page Singh drily remarks: “he devoted all his spare energy to mathematics and, when not sentencing priests to be burned at the stake, Fermat dedicated himself to his hobby.”

Another quote touching upon the same theme: “One story claims that a young student by the name of Hippasus was idly toying with the number √2, attempting to find the equivalent fraction. Eventually he came to realise that no such fraction existed, i.e. that √2 is an irrational number. [...] Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. [...] To his eternal shame he sentenced Hippasus to death by drowning.”

The above quotes are also relevant to the previous discussion – the point being of course that it’s not just ‘the dietary habits of Cleopatra’ that is forgotten over time. Aristotle at one point argued that the number zero should be outlawed. And these are only the things that we haven’t forgotten yet.

3. “The fame of Fermat’s Last Theorem comes solely from the sheer difficulty of proving it.”

4. The book explains a lot of mathematical concepts and deals with many interesting examples of stuff mathematicians have been working on during the ongoing development of (in particular) number theory. Some stuff I didn’t know, or didn’t know that I knew, or had forgotten:

“Friendly numbers are pairs of numbers such that each number is the sum of the divisors of the other number.” [...] “During the twentieth century mathematicians have extended the idea further and have searched for so-called ‘sociable’ numbers, three or more numbers which form a closed loop. For example, in the loop of 5 numbers (12,496; 14,288; 15,472; 14,536;14264) the divisors of the first number add up to the second, the divisors of the second add to the third, the divisors of the third add up to the fourth, the divisors of the fourth add up to the fifth, and the divisors of the fifth add up to the first.”

“Prime numbers are the numerical building blocks because all other numbers can be created by multiplying combinations of the prime numbers.” – I knew this at one point, but I’d forgotten. Basically, there are primes and then there’s everything else. At least as long as you only deal with natural numbers. It was somewhat important when it came to the proof because: “If one can prove Fermat’s Last Theorem for just the prime values of n, then the theorem is proved for all values of n.”

“All prime numbers (except 2) can be put into two categories; those which equal 4 n + 1 and those which equal 4 n – 1″

“26 is indeed the only number between a square and a cube.”

“The most significant and rarest of numbers are those whose divisors add up exactly to the number itself and these are the perfect numbers. The number 6 has the divisors 1, 2 and 3, and consequently it is a perfect number because 1 + 2 + 3 = 6. The next perfect number is 28″ [...] “Euclid discovered that perfect numbers are always the multiple of two numbers, one of which is a power of 2 and the other being the next power of 2 minus 1.”

“If you take any number and multiply it by 2, then the new number must be even. This is virtually the definition of an even number.” – you kinda knew this, and then again maybe you didn’t really. Also, “If you know that the square of a number is even, then the number itself must also be even.”

“The network formula shows an eternal relationsship between the three properties which describe any network:

V + R – L = 1,

where

V = the number of vertices (intersections) in the network,
L = the number of lines in the network,
R = the number of regions (enclosed areas) in the network.” (the book also contains the proof of this formula, created by Leonhard Euler.)

5. “Dear …………………,

Thank you for your manuscript on the proof of Fermat’s Last Theorem.
The first mistake is on :
Page ……… Line ………..
This invalidates the proof.

Professor E. M. Landau”

Landau was head of the mathematics department in Göttingen from 1909 and 1934, and it was his job to examine the entries submitted for the Wolfskehl Prize, a big prize offered by the estate of Paul Wolfskehl to anyone who proved Fermat’s Last Theorem. He printed out cards like the one above and gave them to his students to let them fill in the blanks. Because of the substantial amount of money on the line, there were a lot of faulty attemps submitted.

Maybe I’ll write another post on this book, maybe I won’t, but I have no problem recommending it. You can order the book here.

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