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.^[1]

Statement

Formally, the theorem states that, if one chooses a finite set P = {p₁, ... p_k} 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

Størmer's original procedure involves solving a set of roughly 3^k 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 2^k − 1 Pell equations to solve. For each such equation, let x_i, y_i be the generated solutions, for i in the range from 1 to max(3, (p_k + 1)/2) (inclusive), where p_k is the largest of the primes in P.

Then, as Lehmer shows, all consecutive pairs of P-smooth numbers are of the form (x_i − 1)/2, (x_i + 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 (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 x_i = 3, 17, and 99, Lehmer's method tests the pair (x_i − 1)/2, (x_i + 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 x_i = 5, 49, and 485 Lehmer's method forms the three candidate pairs of consecutive numbers (x_i − 1)/2, (x_i + 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 3^k − 2^k. Lehmer's result produces a tighter bound for sets of small primes: (2^k − 1) × max(3,(p_k+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

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

In mathematics

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

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.^[5] Furthermore, superparticular ratios are very important in just tuning theory as they represent ratios between adjacent members of the harmonic series.^[6]

Størmer's theorem allows us to evaluate all possible superparticular ratios in a given limit. For example, in the 3-limit (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). If we extend this to the 5-limit, six additional superparticular ratios are available: 5/4 (the major third), 6/5 (the minor third), 10/9 (the minor tone), 16/15 (the minor second), 25/24 (the minor semitone), and 81/80 (the syntonic comma); all are musically meaningful.

Notes

1. Størmer (1897).
2. 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. Harry Partch, Genesis of a Music: An Account of a Creative Work, Its Roots, and Its Fulfillments, second edition, enlarged (New York: Da Capo Press, 1974), p. 73. ISBN 0-306-71597-X; ISBN 0-306-80106-X (pbk reprint, 1979).
6. Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music". American Mathematical Monthly (Mathematical Association of America) 79 (10): 1096–1100. doi:10.2307/2317424. JSTOR 2317424. MR 0313189.

References

• Cao, Zhen Fu (1991). "On the Diophantine equation (ax^m - 1)/(abx-1) = by²". Chinese Sci. Bull. 36 (4): 275–278. MR 1138803.
• Chapman, Sydney (1958). "Fredrik Carl Mulertz Stormer, 1874-1957". Biographical Memoirs of Fellows of the Royal Society 4: 257–279. doi:10.1098/rsbm.1958.0021. JSTOR 769515.
• Chein, E. Z. (1976). "A note on the equation x² = y^q + 1". Proceedings of the American Mathematical Society 56 (1): 83–84. doi:10.2307/2041579. JSTOR 2041579. MR 0404133.
• Halsey, G. D.; Hewitt, Edwin (1972). "More on the superparticular ratios in music". American Mathematical Monthly 79 (10): 1096–1100. doi:10.2307/2317424. JSTOR 2317424. MR 0313189.
• Lehmer, D. H. (1964). "On a Problem of Størmer". Illinois Journal of Mathematics 8: 57–79. MR 0158849.
• Luo, Jia Gui (1991). "A generalization of the Störmer theorem and some applications". Sichuan Daxue Xuebao 28 (4): 469–474. MR 1148835.
• Mabkhout, M. (1993). "Minoration de P(x⁴+1)". Rend. Sem. Fac. Sci. Univ. Cagliari 63 (2): 135–148. MR 1319302.
• Mei, Han Fei; Sun, Sheng Fang (1997). "A further extension of Störmer's theorem [Chinese]". Journal of Jishou University (Natural Science Edition) 18 (3): 42–44. MR 1490505.
• Størmer, Carl (1897). "Quelques théorèmes sur l'équation de Pell $x^2 - Dy^2 = \pm1$ et leurs applications". Skrifter Videnskabs-selskabet (Christiania), Mat.-Naturv. Kl. I (2). delete character in |title= at position 43 (help)
• Sun, Qi; Yuan, Ping Zhi (1989). "On the Diophantine equations $(ax^n - 1)/(ax - 1) = y^2$ and $(ax^n + 1)/(ax + 1) = y^2$". Sichuan Daxue Xuebao 26: 20–24. MR 1059671. delete character in |title= at position 30 (help)
• Walker, D. T. (1967). "On the diophantine equation mX² - nY2 = ±1". American Mathematical Monthly 74 (5): 504–513. doi:10.2307/2314877. JSTOR 2314877. MR 0211954.