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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.

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

Application in music theory

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 note). 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.

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.

Formal statement of the theorem

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}...p_k^{e_k}mid e_iin{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

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 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 [1,max(3,(pk+1)/2)], 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

To find the ten consecutive pairs of {2,3,5}-smooth numbers 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

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).

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

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.

Notes

References

  • Cao, Zhen Fu (1991). "On the Diophantine equation (axm - 1)/(abx-1) = by2". Chinese Sci. Bull. 36 (4): 275–278. 1138803. }}
  • Chein, E. Z. (1976). "A note on the equation x2 = yq + 1". Proceedings of the American Mathematical Society 56 83–84. 0404133. }}
  • Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music". American Mathematical Monthly 79 1096–1100. 0313189.

| doi = 10.2307/2317424}}

  • Lehmer, D. H. (1964). "On a Problem of Størmer". Illinois Journal of Mathematics 8 57–79. 0158849. }}
  • Luo, Jia Gui (1991). "A generalization of the Störmer theorem and some applications". Sichuan Daxue Xuebao 28 (4): 469–474. 1148835. }}
  • Mabkhout, M. (1993). "Minoration de P(x4+1)". Rend. Sem. Fac. Sci. Univ. Cagliari 63 (2): 135–148. 1319302. }}
  • Mei, Han Fei; Sun, Sheng Fang (1997). "A further extension of Störmer's theorem". J. Jishou Univ. Nat. Sci. Ed. 18 (3): 42–44. 1490505. }}
  • Størmer, Carl (1897). "Quelques théorèmes sur l'équation de Pell x2 - Dy2 = ±1 et leurs applications". Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2):
  • Sun, Qi; Yuan, Ping Zhi (1989). "On the Diophantine equations (axn - 1)/(ax - 1) = y2 and (axn + 1)/(ax + 1) = y2". Sichuan Daxue Xuebao 26 20–24. 1059671. }}
  • Walker, D. T. (1967). "On the diophantine equation mX2 - nY2 = ±1". American Mathematical Monthly 74 504–513. 0211954. }}

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