Prime proof zeros in on crucial numbers

Fermat’s last theorem is just one of many examples of innocent-looking problems that can long stymie even the most astute mathematicians. It took about 350 years to prove Fermat’s tantalizing conjecture.

Now, Preda Mihailescu of the Swiss Federal Institute of Technology in Zurich has proved a theorem that is likely to lead to a solution of Catalan’s conjecture, another venerable problem involving relationships among whole numbers. He describes his result in a paper to be published in the Journal of Number Theory.

“This is a very important contribution,” says mathematician Andrew Granville of the University of Georgia in Athens. Mihailescu’s work probably puts the resolution of Catalan’s problem into the foreseeable future, he notes.

Named for Belgian mathematician Eugène Charles Catalan, the conjecture concerns powers of whole numbers. For example, the sequence of all squares and cubes of whole numbers greater than 1 begins with the integers 4, 8, 9, 16, 25, 27, and 36. In this sequence, 8 (the cube of 2) and 9 (the square of 3) are not only powers but also consecutive whole numbers.

In 1844, Catalan asserted that among powers of whole numbers, the only pair of consecutive numbers that arises is 8 and 9. Since then, Catalan’s conjecture has posed a challenge to number theorists akin to that provided by Fermat’s last theorem (SN: 11/5/94, p. 295).

Solving Catalan’s problem amounts to a search for whole number solutions to the equation xpyq = 1, where x, y, p, and q are all greater than 1. The conjecture suggests that there is only one such solution: 32 – 23 = 1.

In a major step toward resolving Catalan’s conjecture, Robert Tijdeman of the University of Leiden in the Netherlands showed in 1976 that even if it is not true, there is a finite rather than an infinite number of solutions to the equation. In effect, each of the exponents p and q must be less than a certain value.

Last year, Maurice Mignotte of the Université Louis Pasteur in Strasbourg, France, demonstrated that p had to be less than 7.15 – 1011 and q less than 7.78 – 1016. Meanwhile, computations showed that no consecutive powers other than 8 and 9 occur below 107.

In the latest advance, Mihailescu proved that, if additional solutions to the equation exist, the exponents p and q are a pair of what are known as double Wieferich primes. These pairs obey the following relationship: p(q – 1) must leave a remainder of 1 when divided by q2, and q(p – 1) must leave a remainder of 1 when divided by p2. The pair of prime numbers 2 and 1,093 fits this relationship.

Only six examples of double Wieferich primes have been identified so far. All of these pairs are below the range specified by the computations addressing Catalan’s conjecture. A major collaborative computational effort has now been mounted to find additional double Wieferich primes, but mathematicians are betting that a theoretical approach to proving Catalan’s conjecture will beat out the computers.

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