Problems in number theory can usually be stated quite simply. Even a toddler might know that 9 minus 8 equals 1. Most primary school children know that 9 = 3 × 3 and that 8 = 2 × 2 × 2. Finally, most secondary school children know that 9 equals 3² and 8 equals 2³. This allows us to see another way of writing the equation 9 − 8 = 1—namely, 3² − 2³ = 1. Is it possible to formulate probing questions about such a simple and innocent equation? As it turns out, the answer is an unequivocal yes. As unbelievable as it may sound, this innocent-looking equation has been the source of puzzlement for one and a half centuries.

In 1844 the mathematics periodical Crelle’s Journal published a query by the Belgian mathematician Eugène Charles Catalan. It asked whether apart from the numbers 2 and 3 there exist integers x, y, u, and v, all greater than 1, that provide a solution to the equation x^u − y^v = 1 (just as 2 and 3 do in the equation 3² − 2³ = 1). Catalan suggested that there were none but could not provide proof for this.

Deceptively simple as this conjecture might seem, its solution is all the more complex. That u and v would have to be primary numbers was realized fairly soon, but after that there was no progress on the question for 158 years. Only in the spring of 2002 did something happen. The mathematician Preda Mihailescu, of the University of Paderborn in Germany, found the key to this conjecture.

How did he do it? For the Rumanian-born mathematician it all began in Switzerland, at Zurich’s venerable Eidgenössische Technische Hochschule. This renowned institution was where Mihailescu acquired the necessary mathematical tools for his later discoveries. But just be-

fore he put the finishing touches to his doctoral thesis, he decided to switch from university to industry. He would return to academia only later, at which point he decided to embark on a second doctoral thesis. The topic was prime numbers, and this one Mihailescu actually finished. But it was while working for a high-tech firm as an expert on fingerprints that Mihailescu encountered the so-called Catalan conjecture for the first time.

In the early 14th century, more than 500 years before Catalan formulated the conjecture in Crelle’s Journal, Levi Ben Gerson, a Jewish scholar known as Leo Hebraeus—who lived in Catalonia of all places—mentioned a variant of this problem. The rabbi proved that 8 and 9 are the only powers of 2 and 3 that differ by 1. Four centuries later Leonhad Euler demonstrated that the conjecture was true if the powers—u and v in the formula—were limited to the integers 2 and 3. And then it all went quiet again until 1976, the year in which the next step toward progress was made.

Based on the seminal work of the Cambridge mathematician Alan Baker, the Dutchman Robert Tijdeman, of the University of Leiden in the Netherlands, was able to prove that there could only be a finite number of solutions, should any exist at all, to the above equation. In the very same year it was demonstrated that in this case the powers would have to be smaller than 10¹¹⁰.

Even though this is an astronomically large number—a 1 followed by 110 zeros—the result opened the flood-gates. From then on it was just a question of lowering this upper limit on the potential solutions to a manageable number and then testing Catalan’s equation running through all exponents. Maurice Mignotte, of the University Louis Pasteur in Strasbourg, France, was the first to lower the bar. In 1999 he demonstrated that the exponents of potential solutions had to be less than 10¹⁶. At the same time it was already proven that they had to be larger than 10⁷. The range had thus been significantly reduced but was still far too large for a computer-assisted solution to the problem.

It was time for Mihailescu’s first strike. While day-dreaming on a boring train ride to Zurich after a conference in Paris, an idea suddenly hit him: The exponents in Catalan’s equation had to be Wieferich pairs, two numbers that can be divided by each other in a somewhat complicated manner. Wieferich pairs are very rare, and so far only six of them have been found. Thus the hunt for possible solutions to Catalan’s equation could be limited to Wieferich pairs that, furthermore, had to be smaller than 10¹⁶. With one stroke the problem had become amenable to computer verification. A project was launched whereby Internet users could put the idle times of their PCs to work, hunting for Wieferich pairs and testing them on Catalan’s equation. But the search progressed very slowly, and in 2001 the project was aborted. By this time the lower limit had at least been raised to 10⁸, but testing even this reduced range of numbers—10⁸ to 10¹⁶—would take years.

Now was the time for Mihailescu’s second strike. He recalled an obscure subject called the “theory of cyclotomic fields” that had been developed by the German mathematician Eduard Kummer (1810–1893) in a futile attempt to prove Fermat’s conjecture. Thus, a century later, Mihailescu was able to draw on Kummer’s spadework to plug the last hole in the proof of Catalan’s conjecture.

How does one feel after solving an age-old, world-renowned problem? It was no real high, Mihailescu recounts. After having been convinced on half a dozen earlier occasions that he had achieved his goal, only to find a gaping hole shortly thereafter, he had become very cautious. With time he just slowly slithered into the certitude that he had found success at last. So he showed the proof to his colleague Mignotte, who had spent half a lifetime on the problem. The next morning Mignotte informed him that he thought the proof was correct. They did not rejoice, but they were very happy.