The plot of “In the Shadow of Genius” sounds like a typical “Simpsons” episode: The anti-hero of the popular US animated series, Homer Simpson, struggles with a mid-life crisis. As he realizes with disappointment, he hasn’t achieved anything worth mentioning in his life. So Homer decides to emulate the famous inventor Thomas Edison and tries to develop technical innovations, which of course ends in disaster. But if you follow the episode, which was first broadcast in 1998, you will be in for a surprise – at least if you are familiar with mathematics.
Because if you take a closer look, a special detail in one scene stands out: Homer stands – in the style of a nerdy professor – thoughtfully with glasses at a blackboard covered in scribbles. In addition to the obligatory donuts, which are not only Homer’s favorite food but also play a major role in the area of topology, there is a seemingly harmless equation: 3987 12 4365 12 = 4472 12 . If you type it into a calculator, it turns out to be correct. The amazing thing is that it contradicts one of the most established theorems in mathematics, Fermat’s great theorem.
This dates back to the 17th century and looks simple at first glance: it states that the equation x n y n = z n has no integer, positive solutions x, y and z if n is greater than two. If one chooses n = 1, then the equation is always satisfied: no matter how one chooses the values for x and y, z will always be a positive integer result, for example: 3 6 = 9. Even Homer, the one in the series is often portrayed as stupid, one trusts this insight.
For n = 2 it gets a bit trickier because the equation becomes quadratic: x 2 y 2 = z 2 . When x and y are integers, z does not necessarily have to be the same, e.g. for x = 1 and y = 2 the formula is 1 2 2 2 = 5 – and 5 is not a square number. That is, although there is a solution for z (the square root of 5), it is not an integer. Nevertheless, there are exceptions for which the quadratic equation has a suitable solution, for example: 4 2 3 2 = 25 = 5 2 .
This can be interpreted geometrically, in the spirit of Pythagoras, whose famous formula students like Lisa and Bart Simpson encounter in middle school: If x 2 y 2 = z 2 have integer solutions x, y and z, then there are right triangles , whose side lengths x, y and z also have integer values. And as it turns out, there are infinitely many of them.
As soon as one considers the equation for n = 3, however, one surprisingly no longer finds a single integer solution for x 3 y 3 = z 3 . This means that a cube with integer side lengths z cannot be split into two other cubes that also have integer side lengths ( x and y ). The same applies to all other values of n .
The French scholar Pierre de Fermat (1607-1665) recognized this early on – and claimed in a side note that he could also prove it. In a book by the ancient scientist Diophantus of Alexandria, he noted in Latin: “I have discovered a truly marvelous proof of this, but this margin is too narrow to contain it.” It was not the first time that Fermat had done so. In fact, he left numerous similar clues elsewhere. Other experts have been able to prove all of this.
Convinced that this proof was also easy to find, a number of mathematicians, including well-known figures such as Leonhard Euler and Ernst Eduard Kummer, tried it – and failed. Because, as is usual in the abstract subject, a problem is not necessarily easy to solve just because it is easy to formulate.
In fact, it took more than 350 years for the riddle to be cracked. The stroke of genius was achieved by Andrew Wiles in 1994, who revealed the secret of Fermat’s great theorem. His impressive work made waves: He developed novel methods that led to further groundbreaking discoveries in the field. For this he was honored with the Abel Prize in 2016, one of the highest honors in mathematics.
For the proof you have to leave the algebra you know from school and go into more complex mathematical areas. In 1984 Gerhard Frey conjectured that from the solutions x, y and z of the equation x n y n = z n for n > 2 one could construct a strange kind of curve: an elliptic curve, for which there is no representation as a modular form, however— a highly symmetric function that exists in the realm of complex numbers (with roots of negative numbers).
However, another conjecture states that every elliptic curve can be represented as a modular form. After Ken Ribet proved Frey’s hypothesis in 1986, the second remained open: it had to be shown that every elliptic curve has an associated modular form. In the mid-1990s, Wiles succeeded in closing this gap as well, thereby proving Fermat’s big theorem.
However, one question remains unanswered: more than three centuries ago, Fermat could not have known about the mathematical relationships that Wiles used in his publication. Elliptical curves and modular forms were not known at that time. Was the scholar joking with the marginal note? Or had he just thought he had found proof and miscalculated? There is a third possibility: there may be a much simpler method of proof that no one has found yet.
Nobody seriously doubts that Wiles’ approach is correct. Many experts have checked his technical essay, especially since some of his techniques are used again and again to reveal other mathematical relationships. This reduces the likelihood that an error could have crept in somewhere.
But how is it that in the popular TV series, Homer Simpson casually scribbles on a chalkboard an equation that appears to disprove Fermat’s great theorem? After all, 3987 12 4365 12 = 4472 12 represents an integer solution of the equation x n y n = z n for n = 12 – and it really shouldn’t exist.
Fortunately, the mystery can be solved quickly. Calculating the twelfth power of a four-digit number results in an enormous value consisting of 43 digits. Ordinary pocket calculators cannot handle this, their display usually only has ten digits, which is why they round the numerical values up or down. However, if you use a more accurate calculator or computer program, you will find that the results do not agree exactly. For example, 3987 12 4365 12 = 4472.0000000070576171875 12 is a better approximation of the actual solution, which is much more complicated.
In reality, therefore, there is no positive integer z that solves the equation 3987 12 4365 12 = z 12 . So the real problem was not with Fermat or Wiles, but with the limited resolution of conventional pocket calculators.
Of course, Homer’s formula shouldn’t really have surprised real mathematicians, since rounding errors are nothing new in electronic devices. The experts were more surprised that such content appeared at all in the “Simpsons”, a series that actually has nothing to do with mathematics. This panel picture did not come about by chance, because it takes a lot of background knowledge to find this almost-solution.
Another surprise might be that many of the writers in the TV series are computer scientists, mathematicians or physicists, including David X. Cohen, who is responsible for the Fermat joke. He had written a computer program especially for this purpose, which spit out the almost solution. It may not have been mere coincidence that he chose Fermat’s big theorem: in fact, as a student, Cohen attended lectures by Ken Ribet, who had done the preparatory work for Wiles’ proof by proving Frey’s conjecture.
“In the Shadow of Genius” is therefore by no means the only episode in which the authors of “The Simpsons” have discreetly included nerd jokes. In his book “Homer’s Last Theorem”, the mathematician Simon Singh presents many other entertaining examples Producers probably wanted to increase general interest in the unpopular subject?In any case, it invites you to take a closer look at a cozy TV evening in the future – you might make a mathematical discovery in the process.
Originally Posted by “Does Homer Simpson Really Disprove One of History’s Greatest Mathematicians?” comes from Spektrum.de.