2k. Lehmer's result produces a tighter bound for sets of small primes: (2k − 1) × max(3,(pk+1)/2). The number of consecutive pairs of integers that are smooth with respect to the first k primes are 1, 4, 10, 23, 40, 68, 108, 167, 241, 345, ... . The largest integer from all these pairs, for each k, is 2, 9, 81, 4375, 9801, 123201, 336141, 11859211, ... . OEIS also lists the number of pairs of this type where the larger of the two integers in the pair is square or triangular , as both types of pair arise frequently. Generalizations and applications Louis Mordell wrote about this result, saying that it "is very pretty, and there are many applications of it." In mathematics used Størmer's method to prove Catalan's conjecture on the nonexistence of consecutive perfect powers (other than 8,9) in the case where one of the two powers is a square. proved that every number x4 + 1, for x > 3, has a prime factor greater than or equal to 137. Størmer's theorem is an important part of his proof, in which he reduces the problem to the solution of 128 Pell equations. Several authors have extended Størmer's work by providing methods for listing the solutions to more general diophantine equations, or by providing more general divisibility criteria for the solutions to Pell equations. describe a computational procedure that, empirically, finds many but not all of the consecutive pairs of smooth numbers described by Størmer's theorem, and is much faster than using Pell's equation to find all solutions. In music theory In the musical practice of just intonation, musical intervals can be described as ratios between positive integers. More specifically, they can be described as ratios between members of the harmonic series. Any musical tone can be broken into its fundamental frequency and harmonic frequencies, which are integer multiples of the fundamental. This series is conjectured to be the basis of natural harmony and melody. The tonal complexity of ratios between these harmonics is said to get more complex with higher prime factors. To limit this tonal complexity, an interval is said to be n-limit when both its numerator and denominator are n-smooth. Furthermore, superparticular ratios are very important in just tuning theory as they represent ratios between adjacent members of the harmonic series. Størmer's theorem allows all possible superparticular ratios in a given limit to be