Folha: 'The harmony of perfect powers'
Eugène Charles Catalan
A positive integer is called a perfect power if it is a power (square, cube, etc.) of another positive integer. In other words, perfect powers are numbers of the form m a where both the base, m, and the exponent, a, are integers greater than 1. In 1844, the mathematician Eugène Charles Catalan stated that among all perfect powers, the only ones that are consecutive integers are 8 = 2³ and 9 = 3² . This conjecture would challenge mathematicians worldwide for over a century and a half.
Catalan (1814–1894) was born in the beautiful city of Bruges, now part of Belgium, but at the time under Dutch rule. As a child, he moved with his family to Paris, where, in 1833, he enrolled in the renowned École Polytechnique and met the excellent mathematician Joseph Liouville (1809–1882). At the end of the following year, he and most of his colleagues, leftists and republicans, were expelled by the conservative monarchical government of the time. But, with Liouville's help, Catalan was able to resume his studies a few months later, eventually graduating in mathematics in 1841.
As he became a professor and researcher, he remained politically active, eventually being elected to parliament. However, today he is remembered primarily for his work in mathematics, especially the famous conjecture. In 1865, he returned to Belgium as a professor at the University of Liège, the city where he died.
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The problem of the conjecture is even older: in 1343, the Jewish thinker Levi ben Gershon (1288–1344), better known as Gersonides, had already proven that among all numbers of the form 2⁰ or 3⁰ , the only ones that are consecutive integers are 8 = 2³ and 9 = 3² . In other words, Gersonides showed that the conjecture is true if we restrict ourselves to powers of base 2 or 3. But removing this restriction, considering all other bases as well, makes the question much more difficult.
In 1976, the Dutchman Robert Tijdman found a value N such that Catalan's conjecture holds whenever the base or the exponent is greater than N. Theoretically, from then on it would only be necessary to test each of the numbers less than or equal to N, which are finite in number. But Tijdman's N is colossally large, so the number of cases is too large for any computer to test.