Størmer's theorem

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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.[1]

Statement[edit]

Formally, the theorem states that, if one chooses a finite set P = {p1, ... pk} of prime numbers and considers the set of integers

S = \left\{p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}\mid e_i\in\{0,1,2,\ldots\}\right\}

that can be generated by products of numbers in P, then there are only finitely many pairs of consecutive numbers in S. Further, it gives a method of finding them all using Pell equations.

The procedure[edit]

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,[2] 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 can be no consecutive P-smooth numbers. Lehmer's method involves solving the Pell equation

x^2-2qy^2 = 1\

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(3, (pk + 1)/2) (inclusive), 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 (xi − 1)/2, (xi + 1)/2. Thus one can find all such pairs by testing the numbers of this form for P-smoothness.

Example[edit]

To find the ten consecutive pairs of {2,3,5}-smooth numbers (in music theory, giving the superparticular ratios for just tuning) let P = {2,3,5}. There are seven P-smooth squarefree numbers q (omitting the eighth P-smooth squarefree number, 2): 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, (5 + 1)/2) = 3, so this method generates three solutions to each Pell equation, as follows.

  • For q = 1, the first three solutions to the Pell equation x2 − 2y2 = 1 are (3,2), (17,12), and (99,70). Thus, for each of the three values xi = 3, 17, and 99, Lehmer's method tests the pair (xi − 1)/2, (xi + 1)/2 for smoothness; the three pairs to be tested are (1,2), (8,9), and (49,50). Both (1,2) and (8,9) are pairs of consecutive P-smooth numbers, but (49,50) is not, as 49 has 7 as a prime factor.
  • For q = 3, the first three solutions to the Pell equation x2 − 6y2 = 1 are (5,2), (49,20), and (485,198). From the three values xi = 5, 49, and 485 Lehmer's method forms the three candidate pairs of consecutive numbers (xi − 1)/2, (xi + 1)/2: (3,2), (25,24), and (243,242). Of these, (3,2) and (25,24) are pairs of consecutive P-smooth numbers but (243,242) is not.
  • For q = 5, the first three solutions to the Pell equation x2 − 10y2 = 1 are (19,6), (721,228), and (27379,8658). The Pell solution (19,6) leads to the pair of consecutive P-smooth numbers (9,10); the other two solutions to the Pell equation do not lead to P-smooth pairs.
  • For q = 6, the first three solutions to the Pell equation x2 − 12y2 = 1 are (7,2), (97,28), and (1351,390). The Pell solution (7,2) leads to the pair of consecutive P-smooth numbers (3,4).
  • For q = 10, the first three solutions to the Pell equation x2 − 20y2 = 1 are (9,2), (161,36), and (2889,646). The Pell solution (9,2) leads to the pair of consecutive P-smooth numbers (4,5) and the Pell solution (161,36) leads to the pair of consecutive P-smooth numbers (80,81).
  • For q = 15, the first three solutions to the Pell equation x2 − 30y2 = 1 are (11,2), (241,44), and (5291,966). The Pell solution (11,2) leads to the pair of consecutive P-smooth numbers (5,6).
  • For q = 30, the first three solutions to the Pell equation x2 − 60y2 = 1 are (31,4), (1921,248), and (119071,15372). The Pell solution (31,4) leads to the pair of consecutive P-smooth numbers (15,16).

Counting solutions[edit]

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: (2k − 1) × max(3,(pk+1)/2).[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, ... (sequence A002071 in OEIS).

The largest integer from all these pairs, for each k, is

2, 9, 81, 4375, 9801, 123201, 336141, 11859211, ... (sequence A117581 in OEIS).

OEIS also lists the number of pairs of this type where the larger of the two integers in the pair is square (sequence A117582 in OEIS) or triangular (sequence A117583 in OEIS), as both types of pair arise frequently.

Generalizations and applications[edit]

Louis Mordell wrote about this result, saying that it "is very pretty, and there are many applications of it."[3]

In mathematics[edit]

Chein (1976) 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.

Mabkhout (1993) 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.[4]

In music theory[edit]

In the theory of musical tuning, musical tones can be described as integer multiples of a fundamental frequency, and the multiples generated by products of small prime numbers are of particular importance: in Pythagorean tuning, only tones corresponding to integer multiples of the form 2i × 3j are allowed, while in just tuning, only the tones corresponding to numbers of the form 2i × 3j × 5k are allowed, where i, j, and k may range over any non-negative integer value. The difference between one tone and another forms a musical interval that can be measured by the ratio between the two corresponding integers, and in music the superparticular ratios between consecutive integers are of particular importance.

Størmer's theorem implies that, for Pythagorean tuning, the only possible superparticular ratios are 2/1 (the octave), 3/2 (the perfect fifth), 4/3 (the perfect fourth), and 9/8 (the whole step). That is, the only pairs of consecutive integers that have only powers of two and three in their prime factorizations are (1,2), (2,3), (3,4), and (8,9). For just tuning, six additional superparticular ratios are available: 5/4, 6/5, 10/9, 16/15, 25/24, and 81/80; all are musically meaningful.[5]

Some modern musical theorists have developed p-limit musical tuning systems for primes p larger than 5; Størmer's theorem applies as well in these cases, and describes how to calculate the set of possible superparticular ratios for these systems.

Notes[edit]

  1. ^ Størmer (1897).
  2. ^ a b Lehmer (1964).
  3. ^ As quoted by Chapman (1958).
  4. ^ In particular see Cao (1991), Luo (1991), Mei & Sun (1997), Sun & Yuan (1989), and Walker (1967).
  5. ^ Halsey & Hewitt (1972) give a direct proof for this case, avoiding Størmer's more general method.

References[edit]