MATHEMATICAL FICTION:

a list compiled by Alex Kasman (College of Charleston)

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The Last Theorem (2008)
Arthur C. Clarke / Frederik Pohl
(click on names to see more mathematical fiction by the same author)
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Ranjit Subramanian, the protagonist in this science fiction novel, is a young Sri Lankan man who (re)discovers a short and elementary proof of Fermat's Last Theorem while enduring torture during an unjust imprisonment. The novel reads like a "classic" SF novel from the early latter half of the 20th Century, which is perhaps not surprising considering that its authors are two of the most famous authors from that period in the history of genre. The age of the authors does result in a few quaint anachronisms -- such as the beautiful wife who has an advanced degree of her own but gives up her career to raise the kids and make eggs for her brilliant husband -- but I personally enjoyed the opportunity to read one more new novel written in this classic style. However, the authors do a surprisingly bad job with the mathematics, and repeatedly (and unfairly) defame Andrew Wiles' actual proof of Fermat's Last Theorem.

The main underlying plot involves a plan by the great and advanced civilizations of the galaxy to destroy life on Earth before we can cause trouble for them. Throughout the novel, as other minor human plot lines are elaborated upon, we are reminded that the planned annihilation is getting closer and closer. Other features of the story include the building of a space elevator, sports in space, religion (primarily Hinduism) and atheism, terrorism and "extraordinary rendition", the United Nations and another international organization which "peacefully" destroys their opponents' electronics with a nuclear blast.

The primary mathematical content is the already mentioned "simple" proof of Fermat's Last Theorem (which I will discuss further below), but there are a few other features that deserve to be mentioned. Most interesting to me was the attempts of Subramanian to become a math professor. Since he was not an ideal student, and since all of his degrees are honorary ones that he received after becoming famous, he doesn't quite know how to do it. His initial failures at this and eventual success are an interesting glimpse of academia. Another mathematical subplot arises when Subramanian's youngest son who appears to be somewhat autistic impresses his father with his experimentation with Pentominos. The "Grand Galactics" are described as being interesting in mathematics, among other things, but not much is made of this as far as I could tell (which surprised me...did I miss something?). And, there is a cute scene in which Subramanian impresses a young girl named Ada (after Ada Lovelace) with some mathematical "tricks" including one that allows him to state the exact combinatorial possibilities of flipping an unknown number of coins.

Now, let me say a bit about Fermat's Last Theorem for those who may not know about this bit of real mathematical history. The claim that there are no positive integer solutions to

xn+yn=zn
when n is an integer greater than 2 goes by the name Fermat's Last Theorem (and was called so even before a proof was known, despite the fact that the term "theorem" is generally reserved for statements that have been proved mathematically).

Most people are familiar with the formula

x2+y2=z2
which is satisfied by the lengths of the sides of a right triangle. Many are also familiar with a common example of a right triangle which has sides of lengths 3, 4 and 5. So, as we see, there are positive, whole number solutions to this formula. But, what if the powers of two are replaced with higher powers, such as 3, 4, 957, or even higher? It does not seem at first that there is any reason to doubt that one could also find positive whole number solutions to those equations. However, in 1637, the amateur mathematician Pierre de Fermat wrote in the margin of a book he was reading that he had a mathematical proof that in fact there are no such solutions. Unfortunately, he did not tell us what this proof was. Moreover, for a very long time, mathematicians tried to prove this fact and were unable to find either a proof or a counter-example. During this period, the statement became quite famous and was frequently mentioned, not because it was particularly important but only because it seemed to be so challenging! Finally, in 1995, the mathematician Andrew Wiles presented his proof of the "last theorem" at a conference in Cambridge. Although a small problem was found with the original proof, with the help of his former student, Richard Taylor, the problem was corrected and the theorem was at last proved.

Unfortunately, the proof is not nearly as simple as the statement of the original conjecture. It depends on a lot of advanced mathematics, including the algebraic geometry of elliptic curves, which would certainly not have been among the tools available to Fermat. Most mathematicians now seem to doubt that Fermat had a proof at all, but I suppose some small minority might believe that a simpler, elementary proof is still out.

It is not unreasonable for the authors to suggest that a short proof still is out there waiting to be found. However, it is unreasonable of them to suggest that there is more wrong with Wiles' proof than the fact that Fermat couldn't possibly have come up with it. However, in the preface, in the text and in an appendix, they state (incorrectly) that the proof depends on steps that involve a computer and that it "cannot be read" by a human. I think they are simply repeating complaints that they have heard about some other high profile proofs and incorrectly applying them to FLT. (There is a proof of the Four Color Theorem which depends on the use of computers to verify a large number of cases which cannot be checked by hand. And the classification of finite simple groups is a research program that is supposedly completed, although the proof has not yet been collected all in one place...yet. But, neither of these problems apply to Wiles' proof of FLT!) Moreover, the appendix states that one of the authors (it doesn't say which) believes that FLT is formally undecideable...which means that this person must believe Wiles' proof is wrong. This is a pretty serious accusation, and not one that should be made lightly. Their description of Wiles' proof is so confused that I suspect they do not understand it themselves. (Really, the idea is not that complicated. What they should have said is this: it had already been proved that from numbers x, y, z satisfying the equation above with n>2 it is possible to create an elliptic curve over the rational numbers which is not modular. But, Wiles' proved that all elliptic curves over the rationals are modular. Consequently, there cannot be any such x, y and z!)

This is not the only mathematical confusion present in the book (but considering that it appears to be an insult to Wiles, I think it is the most serious problem). The lecture Subramanian gives on the infinitude of primes is -- at best -- unclear. I might think that they simply did not explain it well, except that their clear misconceptions everywhere suggest to me instead that they don't understand this simple proof either. Also, their description of trapdoor codes completely misses the point, which is that given the number N that is a product of two very large primes it is possible to encrypt a message but not possible to decrypt the message unless the factors of the number are known. (They suggest that the encryption could be achieved by adding the number N to the signal...which would be silly because someone could then just subtract the same number to get the signal back!) For my description of how these codes really work, see here.

This novel was fun to read. It made me feel nostalgiac about the classic years of science fiction while still being current enough to seem new. However, I am troubled by the authors' apparent misconceptions, and so surprised that very little happens with the math that I think I must have missed something. (Did you read this book and see an explicit connection made anywhere between FLT and the Grand Galactics? If so, let me know. Maybe this key plot point somehow got lost between the two authors?)

Contributed by Alex

This is one more example of a mathematician proving an important result in prison. A short discussion of this topic was begun in my description of Ghost Dancer.

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(Note: This is just one work of mathematical fiction from the list. To see the entire list or to see more works of mathematical fiction, return to the Homepage.)

Works Similar to The Last Theorem
According to my `secret formula', the following works of mathematical fiction are similar to this one:
  1. Fermat's Lost Theorem by Jerry Oltion
  2. The Light of Other Days by Arthur C. Clarke / Stephen Baxter
  3. Rama II by Arthur C. Clarke / Gentry Lee
  4. The Humans: A Novel by Matt Haig
  5. The Girl Who Played with Fire by Stieg Larsson
  6. Fermat's Best Theorem by Janet Kagan
  7. The Atrocity Archives by Charles Stross
  8. The Gold at Starbow's End (aka Starburst / aka Alpha Aleph) by Frederik Pohl
  9. Stamping Butterflies by Jon Courtenay Grimwood
  10. Oh, Brother by Stanley Hart
Ratings for The Last Theorem:
RatingsHave you seen/read this work of mathematical fiction? Then click here to enter your own votes on its mathematical content and literary quality or send me comments to post on this Webpage.
Mathematical Content:
3/5 (2 votes)
..
Literary Quality:
2.5/5 (2 votes)
..

Categories:
GenreScience Fiction, Adventure/Espionage,
MotifAcademia, Aliens, War, Autism, Romance, Religion,
TopicAlgebra/Arithmetic/Number Theory, Real Mathematics, Fictional Mathematics,
MediumNovels,

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(Maintained by Alex Kasman, College of Charleston)