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31 July 2019

The Mystery of Number 33 and the Diophantine Equations

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It is a number surrounded by a special mystique. For many years, 33 has fascinated the mathematical community by starring in one of the apparently simpler cases of a diophantine equation, but which is nevertheless pending resolution: it might seem easy to express the number 33 as the sum of the cubes of three whole numbers – that is, to find a solution for the equation a3 + b3+ c3 = 33 – but no one had yet succeeded since 1955 when mathematicians set out to solve this mathematical mystery.

At least that’s how it was until just a few months ago. Last April, the American mathematician Andrew Booker announced that 33 could be expressed by the sum of the cubes: (8866128975287528)3 + (-8778405442862239)3 + (-2736111468807040)3. His success was achieved through a brute force approach or method; in other words, he was assisted by a supercomputer that executed an algorithm designed by the mathematician for three solid weeks until this solution was found.

The American mathematician Andrew Booker. Credit: University of Bristol

But what’s so special about diophantine equations to arouse such fascination? Undoubtedly, one of the most powerful reasons is that they deal with some of the most basic and simple aspects of mathematics, those of integers and the most elementary algebraic operations. In fact, diophantine equations are defined as “polynomial equations that involve only sums, products and powers and in which both the coefficients and the only valid solutions are whole numbers.” In short, nothing less than the ABCs of mathematics.

A challenge on the table since the 3rd century

And yet, even with that presumed simplicity, these equations still manage to baffle the brightest minds capable of devising sophisticated proofs and tests with which to solve seemingly much more complex mathematical questions. This challenge has been on the table since the 3rd century AD, when the equations were enunciated by the Greek mathematician Diofante of Alexandria.

Pierre de Fermat again drew attention to the diophantine equations. Source: Wikimedia

After a few centuries of relative oblivion, the diophantine equations once again strongly demanded the attention of mathematicians from the 18th century onwards, following Pierre de Fermat’s enunciation of his famous last theorem —specifically, as an annotation in the margin of a copy of Diofante’s Arithmetica— and that in reality was still the supposed solution for a particular type of this sort of equation. Specifically, the note left by the French mathematician conjectured that for n greater than 2 there are no positive integers x, y and z that satisfy the equation: xn + yn = zn.

It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second, into two like powers. I have discovered a truly marvelous proof of this, which this margin is too narrow to contain.”

The search for the proof of this conjecture —and the legend that Fermat had found it, as he claimed in his annotation— stimulated the greatest mathematicians and propelled the development of a whole new field, that of number theory. It was not until 1995 that British mathematician Andrew Wiles successfully achieved the definitive proof.

In Hilbert’s list of problems to solve

Before that, in the year 1900 and within the framework of the Universal Congress of Mathematics held in Paris, the eminent David Hilbert included the resolution of the diophantine equations in his list of problems to be discerned by mathematicians in the years to come. Specifically, Hilbert pointed out the need to identify an algorithm that would make it possible to determine in a general way if any diophantine equation has a solution. It was a challenge that remains valid more than a century later, although with some nuances because in 1970 Russian mathematician Yuri Matiyasevich was able to demonstrate the impossibility of achieving a general algorithm for all diophantine equations. But this did not invalidate the search for an algorithm, for a general method, for each particular type of them.

David Hilbert included the resolution of the diophantine equations in his list of problems. Source: Open Logic

The case at hand, that of expressing any integer as the sum of three cubes, is one of the simplest of these equations. Yet the most that has been proved is that those integers that when divided by 9 yield a remainder of 4 or 5 have no solution. But what about the rest? All the others are called eligible. For years, it was conjectured that for some numbers there would be no solution. However, with each new discovery to the contrary, that is, with each particular solution identified, mathematical thinking has turned, so that the modern conjecture is that all eligible numbers have a solution.

That is where the true dimension of Booker’s achievement lies. A new result that reinforces and strengthens this intuition. As to the rest, after the resolution of number 33 only one apparently simple number resists among the first hundred integers: the number 42. And there are still eleven more without a solution among the first thousand integers.

Miguel Barral


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