Fermat’s Last Theorem—It All Starts With a Little Confession

Cha Ji-an (Jang Na-ra) and Lee Hyun / David Lee (Seo In-guk) with purple flowers in 2015 Korean drama Hello Monster / I Remember You

The identities of Hello Monster‘s criminal masterminds are no real mysteries, but the show regales puzzle lovers all the same with ingenious cryptograms, including asterisk-based symbols that translate into flag semaphores coding for geographic coordinates of crime locations, with missing strokes of the asterisks representing flag positions, and crossword puzzles which black and white spaces encode binary numbers that lead to resident registration numbers of murder victims. There is also the male protagonist’s evaluation of the female protagonist: 15 plus alpha, 15 being the score he also gives to most of their colleagues and “plus alpha” being loan words used in Korea and Japan to mean “something extra.” When she requests an elaboration, he likens her worth to the exponents in Fermat’s Last Theorem—an unquantifiable value.

A 17th-century French lawyer who dabbled in mathematics in his spare time, Pierre de Fermat scribbled in a margin in his copy of the Greek text Arithmetica that there is no integer value of n greater than two for which an + bn = cand a, b and c are non-zero integers. There is no way one can, for instance, mash up two perfect cubes (i.e. cubes which lengths are whole numbers) and shape them into a larger perfect cube without leaving remaining material or hollow spaces (in other words, a3 + b3 ≠ c3). In contrast, the combinations of values for a, b and c that fit the equation when n is one (elementary addition) or two (the Pythagoras theorem, which says that the squared value of the hypotenuse of a right-angled triangle is the sum of the squares of the other sides) are infinite. Fermat claimed that he had “discovered a truly marvelous demonstration of this proposition” which the margin was too narrow to hold.

Cha Ji-an (Jang Na-ra) lying beside Lee Hyun / David Lee (Seo In-guk) in 2015 Korean drama Hello Monster / I Remember You

The lead couple of Hello Monster

For centuries afterwards, the theory, which simplicity belies the elusiveness of its proof, captivated mathematicians. A young French scholar, Sophie Germain, passed off herself as a man (in letters) to share her proof for values of n belonging to a certain class of prime numbers. Doctor-turned-mathematician Paul Wolfskehl, who was rumored to have aborted his suicide plan after reading a paper on Fermat’s theorem, bequeathed 100, 000 marks to whoever first proved the theory. Over in a New York subway station, someone taunted Fermat in a graffiti: “I have discovered a truly remarkable proof, but I can’t write it now because my train is coming.”

Finally, in 1985, German professor Gerhard Frey found that if there are numbers that fit the equation an + bn = cn for > 2, then a and b can make up an elliptic curve defined by the expression y2 = x(x − an)(x + bn). In general, elliptic curves, which are distinct from ellipses, take the form y2 = Ax+ Bx+ Cx + D, where A, B, C and D are integers or rational numbers.

Examples of elliptic curves

Two examples of elliptic curves (Released into the public domain by Tos; reference for cross-checking purposes: Brown, 2000)

Frey was of the opinion that the curve formed by Fermat’s equation would contradict a conjecture by two Japanese mathematicians (the “Taniyama–Shimura conjecture”) which said that all elliptic curves possess a property termed “modularity.” To appreciate this property, it may be helpful to revisit some pre-college mathematics: functions like sine are somewhat symmetrical as they can be transformed in ways that lead to the same results. Adding 2π to an angle θ in sin θ, for instance, produces the same outcome as sin θ. Modular forms are a special type of functions so symmetrical that an input variable can be transformed in infinite ways and yet the outcome stays constant.

Periodicity of the sine function as illustrated with a unit circle. 2π is the angle of a full rotation in radians, with 1 radian corresponding to the angle made by an arc of a circle that has the same length as the radius of that circle. Recall that the circumference is the product of 2π and the radius.

Periodicity of the sine function as illustrated with a unit circle. 2π is the angle of a full rotation in radians, with 1 radian corresponding to the angle made by an arc of a circle that has the same length as the radius of that circle. Recall that circumference is the product of 2π and the radius.

Periodicity of the sine function in wave form

Periodicity of the sine function in wave form

One year later, American scholar Kenneth Ribet proved Frey correct. Ribet’s proof relied on the idea that a line connecting two points on an elliptic curve can reach a third point on the curve and this third point can be reflected about the x-axis to correspond to a fourth point, which will be the sum of the first two points. There are finite sets of points which each contains the sum of any pair of points within itself. These sets of points constitute “finite groups,” and all the points in these groups must be modular for the curve to be modular. Ribet managed to demonstrate that a particular finite group of the hypothetical elliptic curve that would result from Fermat’s equation is not modular, which naturally means that the entire curve cannot be modular. What now remained was for someone to confirm that modularity is indeed a trait common to the full subset of elliptic curves this curve would belong to (i.e. establishing the validity of the Taniyama–Shimura conjecture in an application range covering Fermat’s case).

The idea behind Ribet's breakthrough

The background idea behind Ribet’s breakthrough: a line linking points I and II on an elliptic curve reaches point III, which is reflected over the x-axis to give point IV, the sum of I and II. (References: 1 | 2 ).

That someone is of course none other than Princeton academic Andrew Wiles. Wiles reduced the problem to a class number formula—a finite series that relates the size of two algebraic objects which appear unrelated. Here, one object would be a “deformation ring,” which Wiles associated with elliptic curves, and the other, a “Hecke ring,” which was connected to modular forms. By proving the formula and solving some miscellaneous issues, Wiles completed the puzzle in 1994 and Wolfskehl’s award found its owner. Not only that, researchers later used his methods to validate the whole conjecture, with the result that elliptic conundrums can now be solved by re-envisaging the conundrums as modular problems, which are easier to tackle. This bridge between previously disparate sub-disciplines also brought mathematicians one step closer toward a grand unified theory of mathematics.

It is a poignant thought, however, that while mathematics enthusiasts have taken more than three centuries to verify the non-existence of an entity throughout infinite space, all of us have only a lifetime each to demonstrate the kind of infinite love it symbolizes in the drama. No matter how boundless two people declare their mutual love to be, it perishes upon their deaths. The only way to break the mortality barrier is, perhaps, to mirror the mathematicians, the way they employ mirrors among different areas of the subject, by passing on the baton. Through the propagation of conditions that allow others to similarly experience love of such intensity in one form or another,  romantic love can survive death—as honestly engaged political love. We know the code word.

Purple symbol in opening of 2015 Korean drama Hello Monster / I Remember You

The Proof                                          The Drama

Making of the Pudding                    Drama Resources


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