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Issues in Science and Technology Librarianship |
Summer 2013
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DOI:10.5062/F4833PZK |

Information Literacy Librarian

The Pennsylvania State University Libraries

University Park, Pennsylvania

gnm1@psu.edu

In May 1995, the major mathematical journalAnnals of Mathematicspublished two articles together proving Fermat's Last Theorem, a mathematical problem that has frustrated mathematicians for over 350 years. This article will cover the short modern bibliographic record leading to its solution.

It looks remarkably simple. An algebraic problem: x^{n} + y^{n} = z^{n} with x, y, z, and n as natural numbers (i.e., 1, 2, 3 ... to infinity). When the exponent n = 1, there are numerous natural numbers that solve this equation. Any natural number to the exponent n=1 is that natural number.

Examples: 1^{1} + 2^{1} = 3^{1} or 10^{1} + 15^{1} = 25^{1} or 100^{1} + 1000^{1} = 1100^{1}

When the exponent is 2, there are several natural numbers that solve this equation.

Examples: 3^{2} + 4^{2} = 5^{2} or 5^{2} + 12^{2} = 13^{2}

This is also known as Pythagoras's theorem. In a right triangle, the sum of the squares of both sides is equal to the square of the other side (i.e., hypotenuse).

However, the troubles began with the exponent n = 3. No one seemed to find a natural number solution to make this equation true. The troubles continued when people tried to find natural number solutions when n = 4, or n = 5, or n = 6, and onward toward infinity. In fact, no one could find any solutions to this equation when n > 2. With an infinite amount of natural numbers, it is impossible to test n for all natural numbers. What was needed was a "proof": a series of logical statements, one leading directly to the next with no gaps in the reasoning, that ended with an undeniable conclusion. In this case, that there were no natural number solutions for this algebraic equation.

Enter Pierre de Fermat, a 17th century French lawyer and amateur mathematician. He read a book titled *Arithmetica* written by Diophantus, a 3rd century C.E. Greek mathematician, and Fermat came up with the conjecture (i.e., believed true but not proven) that there are no whole number solutions to this problem when n > 2.

In 1637, Fermat wrote in Latin the following in his copy of *Arithmetica:* "Cubum autem in duos cubos, aut quadrato-quadratum in duos quadrato-quadratos, et 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" In translation, "It is impossible for a cube to be the sum of two cubes, a fourth power to be the sum of two fourth powers, or in general for any number that is a power greater than the second to be the sum of two like powers. I have discovered a truly marvelous demonstration of this proposition that this margin is too narrow to contain" (WolframMathWorld 2013).

Unfortunately, Pierre de Fermat did not write out his proof anywhere and it went with him to his grave. Over the years, many of Fermat's other theorems were proven correct by mathematicians, except this theorem. Known as Fermat's Last Theorem because it alone survived unsolved, it befuddled mathematicians for 358 years until May 1995, when it was finally solved by then Princeton University Professor Andrew Wiles with help from Cambridge University Professor Richard Taylor.

There are many mathematical proofs, both proved and unproved, yet Fermat's Last Theorem has captured the imagination of the public because of its simplicity. Unlike other still unsolved mathematical theories, Fermat's Last Theorem is understandable even by children, yet it has a long history of eluding many famous mathematicians.

The purpose of this bibliographic essay is not to explain "how" to solve Fermat's Last Theorem or why it was so difficult to solve. The final published proof was over one hundred pages of very esoteric and arcane mathematical formulas that can take a world-class mathematician months to read. This essay is also not an attempt to give a complete historical overview on the solving of this remarkable feat. This essay will instead offer resources, both primary and secondary, that can be used for anyone interested in the modern history of this puzzling theorem from obscurity to world-wide adulation over its solution.

This bibliographic essay will be in two sections. First, it will focus on those mathematicians whose work lead Andrew Wiles to solve this proof. Second, it will cover the public uproar that occurred in 1993 when Wiles announced to the world the solution to Fermat's Last Theorem at Cambridge University, and also the problem with the proof upon peer-reviewing and its final resolution in 1995.

The modern journey began with two mathematicians named Yutaka Taniyama and Goro Shimura and the strange world of elliptic curves and modular forms. This paper will not try to explain these very arcane yet important concepts of advanced algebraic theory, and for our purposes we can think, as mathematician Barry Mazur of Harvard University, says of them as "...two completely different worlds"(Singh 1997, p.190). Mathematicians worked in the world of elliptic curves or in the world of modular forms, but rarely did they attempt to connect both of them together. In the 1950s, Taniyama and Shimura came up with a novel idea that every elliptic curve was a modular form, and this was known as the "Taniyama-Shimuira Conjecture." It was a conjecture because it was not proven that for all elliptic curves, there was a definite, corresponding modular form.

Many mathematicians believed that the Taniyama-Shimura Conjecture was unsolvable because there is an infinite number of elliptic curves and an infinite number of modular forms, and there was no method yet devised to group them all, count them all, and pair them up.

In 1985, German mathematician Gerhard Frey presented another conjecture, known as the Epsilon Conjecture. The conjecture stated that if there was a natural number solution to x^{n} + y^{n} = z^{n} with x, y, z, and n as natural numbers (i.e., 1, 2, 3 ... to infinity), it would create a "weird" elliptic curve, but most importantly, that curve was not modular. If the Epsilon Conjecture could be proven to be correct, then by a logical conclusion, Fermat's Last Theorem could also be correct. To reiterate the argument by contradiction:

- Taniyama-Shimura Conjecture says all elliptic curves are modular.
- The Epsilon Conjecture says a natural number solution to x
^{n}+ y^{n}= z^{n}with n > 2 creates an elliptic curve that is not modular. - Thus: Fermat was correct that there are no natural number solution to x
^{n}+ y^{n}= z^{n}with n > 2, and Fermat's Last Theorem is proved.

Now someone had to prove the Epsilon Conjecture. Enter mathematician Kenneth Ribet at the University of California. In 1986, he was working on a possible solution to the Epsilon Conjecture, and in casual conversation with Barry Mazur mentioned his problems to develop the full strength of the proof. Mazur said, "You've already done it! All you have to do is add some gamma-zero of (M) structure and just run through your argument and its works." Ribet later comment to Mazur's suggestion was "How did I not see it? I was completely astonished because it has never occurred to me to add the extra gamma-zero of (M) structure, simple as it sounds" (Singh 1997, p.201).

With the Epsilon Conjecture proven and now known as the Ribet Theorem, all that was needed was proving the Taniyama-Shimura Conjecture.

Two papers leading up to the Epsilon Conjecture.

Serre, Jean-Pierre.1987. Sur les représentations modulaires de degré 2 de Gal(Q/Q).Duke Mathematical Journal54(1):179-229. DOI: 10.1215/S0012-7094-87-05413-5

Frey, Gerhard.1986. Links between stable elliptic curves and certain Diophantine equations.Annales Universitatis Saraviensis, Series Mathematicae1:1-40.

Two of Ribet's papers about the proving of the Epsilon Conjecture (now the Ribet Theorem) and its implications to solving Fermat's last theorem via the Taniyama-Shimura Conjecture.

Ribet, Kenneth A.1990. On modular representations of Gal(Q/Q) arising from modular forms.Inventiones Mathematicae100:431-476.

Ribet, Kenneth A.1990. From the Taniyama-Shimura conjecture to Fermat's last theorem.Annales de la faculté des sciences de Toulouse 5e série. tome11(1): 116-139.

Andrew Wiles was a Princeton University mathematics professor in 1986, and he was stunned to hear that the Epsilon Conjecture had been proven. It was his boyhood dream to solve Fermat's Last Theorem, however it meant he had to solve one of the most difficult mathematical conjectures that many mathematicians thought unsolvable. For seven years, from 1986 to 1993, he labored in secret to devise the methods to prove the Taniyama-Shimura Conjecture, utilizing the techniques of many mathematicians to join the worlds of elliptic curves and modular forms. In June 1993, he presented his findings at a mathematical conference at the Sir Isaac Newton Institute at Cambridge University in England. He named his three lectures "Modular Forms, Elliptic Curves, and Galois Representations," not mentioning Fermat's Last Theorem, and at the very end of his third lecture, he stated before a packed room that the Taniyama-Shimura Conjecture had be proven and thus so was Fermat's Last Theorem, ending with the words," I think I'll stop here" (Singh 1997, p.249).

The reaction by the press was intense for the proving of a relatively obscure mathematical theorem. The range of news articles written after the announcement shows the world's interest of this solution.^{1}

Demonstrating a proof before a throng of mathematicians caught up in the elation of the moment was not sufficient. Before the proof was accepted, it had to be peer-reviewed by other mathematicians and then published in a scholarly journal. It that time, the proof was over 200 pages long, and due to its size and importance, it was sectioned out to six different reviewers. Over the next few months, mathematicians read the proof line-by-line, and slowly it became evident that there was a "gap" -- an error in the work that did not make the proof complete. The news of this gap leaked into the press, and Andrew Wiles had to redo his work after all that adulation from his colleagues and the world.

The gap proved insurmountable and Wiles had to take a different approach to his problem. He called on the help of Richard Taylor, mathematics professor at Cambridge University, and together they labored to repair the proof.

There were notices in the mathematical literature that progress was being made despite the acknowledgement of the gap.

On October 25, 1994, Wiles and Taylor released two manuscripts to be peer reviewed. The first offered the proof of the Taniyama-Shimura Conjecture, and the other explained a technique used in the proof. After being peer reviewed, both articles were accepted and published in the *Annals of Mathematics.*

After the announcement of the proof, the mathematical news journals followed the peer-reviewing of the paper, the discovery of the gap, and the progress to its solution.

Ribet, Kenneth.1993. Wiles proves Taniyama's conjecture; Fermat's last theorem follows.The Journal of the Notices of the American Mathematical Society.40(6):575-6.

Cox, David A.1994. Introduction to Fermat's last theorem.The American Mathematical Monthly.101(1):3-14.

Gouvea, Fernando Q.1994. A marvelous proof.The American Mathematical Monthly.101(3):203-22.

Jackson, Allyn.1994. Update on proof of Fermat's last theorem: gap appears in proof but experts laud Wiles's accomplishment.The Journal of the Notices of the American Mathematical Society.41(3):185-6.

Rubin, K., and Silverberg, A.1994. A report on Wiles' Cambridge lectures.Bulletin (New Series) of the American Mathematical Society.31(1):15-38.

The following news blurb announced that on October 25, 1994, two manuscripts were released.

Jackson, Allyn.1995. Another step toward Fermat.The Journal of the Notices of the American Mathematical Society.42(1):48.

The following two articles proving Fermat's Last Theorem were published in May 1995. The Wiles paper contains the proof solving Fermat's Last Theorem. The Taylor and Wiles paper describes a step necessary to solving the theorem.

Wiles, Andrew.1995. Modular elliptic curves and Fermat's last theorem.Annals of Mathematics, Second Series.142(3):443-551.

Taylor, Richard, and Wiles, Andrew.1995. Ring-theoretic properties of certain Hecke algebras.Annals of Mathematics, Second Series.142(3):553-572.

It is quite obvious that Andrew Wiles's one hundred page proof is not Fermat's. The Wiles solution certainly could not fit in a hundred thin margins in Diophantus's *Arithmetica,* and also that much of the mathematical concepts such as elliptic curves and modular forms used to solve this problem were derived in the twentieth century, unknown to Fermat and his contemporary mathematicians. If indeed Fermat had a proof, it is still out there.

Sir Isaac Newton wrote, "If I had seen further it is by standing on the shoulders of giants" (Oxford Dictionary of Philosophy 2008). The documentary *The Proof* (Lynch and Singh 1997) ends with the listing of no less than 22 mathematicians over four centuries who have contributed to Wiles's accomplishment; proof that mathematics, like every other major scientific field of study, demands the hard and diligent work of all who came before.

^{1} Articles are chronological from June 24th to August 2nd, 1993.

- Kolata G. (June 24, 1993). At last, shout of 'eureka!" in age-old math mystery.
*New York Times* - Kolata G. (June 25, 1993). 350 years later, math conundrum bites the dust.
*International Herald Tribune* - Augereau J-F. (June 25, 1993). a
^{n}+ b^{n}= c^{n}, le theoreme de Fermat enfin resolu?*Le Monde* - Smith M. (June 25, 1993). 350-year-old mathematics problem may be solved.
*The Toronto Star* - (no byline). (June 28, 1993). Mystery solved.
*USA Today* - Cipra B. (July 2, 1993). Fermat's last theorem finally yields.
*Science* - Lemonick, M.D. (July 5, 1993). Fini to Fermat's last theorem.
*Time Magazine* - Peterson I. (July 3, 1993). Curvy path leads to Fermat's last theorem.
*Science News* - McDonald K. (July 7, 1993). x
^{n}+ y^{n}= z^{n}: Princeton professor appears to have proven Fermat's last theorem.*The Chronicle of Higher Education* - Czerny R. (July 8, 1993). My days were numbered: a fond farewell to Fermat's theorem.
*The Ottawa Citizen* - Kalbfleisch, J. (July 21, 1993). Cracking a 350-year-old conumdrum; how a mathematician solved Fermat's lasting puzzle.
*The Gazette*(Montreal, Quebec) - Schwartz, J. (August 2, 1993). This equation figures to answer a 17th-century puzzle.
*The Washington Post.*

**Lynch J, Singh S.** 1997. *The Proof.* [video documentary]. South Burlington (VT): WGBH Boston Video.

**Newton, Isaac.** 2008. In Blackburn S, editor. *The Oxford Dictionary of Philosophy.* Oxford (UK): Oxford University Press. 2nd rev. ed. [Internet].[Accessed June 10, 2013]. Available from http://www.oxfordreference.com/view/10.1093/acref/9780199541430.001.0001/acref-9780199541430-e-2165

**Singh S.** 1997. *Fermat's Enigma: the Quest to Solve the World's Greatest Mathematical Problem.* New York: Walker.

**WolframMathWorld.** 2013. Fermat's Last Theorem. [Internet]. (Updated June 5, 2013). [Accessed June 10, 2013]. Available from: http://mathworld.wolfram.com/FermatsLastTheorem.html

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