Størmer's theorem


In number theory, Størmer's theorem, named after Carl Størmer, gives a finite bound on the number of consecutive pairs of smooth numbers that exist, for a given degree of smoothness, and provides a method for finding all such pairs using Pell equations. It follows from the Thue–Siegel–Roth theorem that there are only a finite number of pairs of this type, but Størmer gave a procedure for finding them all.

Statement

If one chooses a finite set of prime numbers then the -smooth numbers are defined as the set of integers
that can be generated by products of numbers in. Then Størmer's theorem states that, for every choice of, there are only finitely many pairs of consecutive -smooth numbers. Further, it gives a method of finding them all using Pell equations.

The procedure

Størmer's original procedure involves solving a set of roughly 3k Pell equations, in each one finding only the smallest solution. A simplified version of the procedure, due to D. H. Lehmer, is described below; it solves fewer equations but finds more solutions in each equation.
Let P be the given set of primes, and define a number to be P-smooth if all its prime factors belong to P. Assume p1 = 2; otherwise there could be no consecutive P-smooth numbers, because all P-smooth numbers would be odd. Lehmer's method involves solving the Pell equation
for each P-smooth square-free number q other than 2. Each such number q is generated as a product of a subset of P, so there are 2k − 1 Pell equations to solve. For each such equation, let xi, yi be the generated solutions, for i in the range from 1 to max, where pk is the largest of the primes in P.
Then, as Lehmer shows, all consecutive pairs of P-smooth numbers are of the form /2, /2. Thus one can find all such pairs by testing the numbers of this form for P-smoothness.

Example

To find the ten consecutive pairs of -smooth numbers let P =. There are seven P-smooth squarefree numbers q : 1, 3, 5, 6, 10, 15, and 30, each of which leads to a Pell equation. The number of solutions per Pell equation required by Lehmer's method is max = 3, so this method generates three solutions to each Pell equation, as follows.
Størmer's original result can be used to show that the number of consecutive pairs of integers that are smooth with respect to a set of k primes is at most 3k − 2k. Lehmer's result produces a tighter bound for sets of small primes: × max.
The number of consecutive pairs of integers that are smooth with respect to the first k primes are
The largest integer from all these pairs, for each k, is
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

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 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 found. For example, in the 3-limit, the only possible superparticular ratios are 2/1, 3/2, 4/3, and 9/8. That is, the only pairs of consecutive integers that have only powers of two and three in their prime factorizations are,,, and. If this is extended to the 5-limit, six additional superparticular ratios are available: 5/4, 6/5, 10/9, 16/15, 25/24, and 81/80. All are musically meaningful.