Innocent-looking problems involving whole numbers can stymie even the most astute mathematicians. As in the case of Fermat’s last theorem, centuries of effort may go into proving such tantalizing, deceptively simple conjectures in number theory (SN: 11/5/94, p. 295).
Now, Preda Mihailescu of the University of Paderborn in Germany finally may have the key to a venerable problem known as Catalan’s conjecture, which concerns powers of whole numbers.
Consider the sequence of all squares and cubes of whole numbers greater than 1, a sequence that 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, Belgian mathematician Eugène Charles Catalan asserted that, among all powers of whole numbers, the only pair of consecutive integers is 8 and 9. Solving Catalan’s problem amounts to a search for whole-number solutions to the equation xp – yq = 1, where x, y, p, and q are all greater than 1. The conjecture proposes that there is only one such solution: 32 – 23 = 1.
A breakthrough in solving the problem occurred in 1976 when Robert Tijdeman of the University of Leiden in the Netherlands showed that there is a finite rather than an infinite number of solutions to the equation. In 2000, Mihailescu proved that if additional solutions to the equation exist, the pair of exponents must be of a rare type known as double Wieferich primes (SN: 12/2/00, p. 357). A prime is a whole number evenly divisible only by itself and 1.
Mihailescu continued to work on the problem, and he apparently cracked it earlier this year. He has now sent a draft of his purported proof of Catalan’s conjecture to several mathematicians for checking.
It isn’t certain yet that Mihailescu’s proof will hold up, but there are encouraging signs. Yuri F. Bilu of the University of Bordeaux I in Talence, France, has analyzed Mihailescu’s work and written a favorable commentary outlining the proof’s main steps. Mihailescu was scheduled to present the proof publicly for the first time on May 24 at a Canadian Number Theory Association meeting in Montreal.