- Sylvester's sequence
In
number theory , Sylvester's sequence is asequence ofinteger s in which each member of the sequence is the product of the previous members, plus one. The first few terms of the sequence are::2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443 OEIS|id=A000058.Sylvester's sequence is named after
James Joseph Sylvester , who first investigated it in 1880. Its values grow doubly exponentially, and the sum of its reciprocals forms a series ofunit fraction s that converges to 1 more rapidly than any other series of unit fractions with the same sum. The recurrence by which it is defined allows the numbers in the sequence to be factored more easily than other numbers of the same magnitude, but, due to the rapid growth of the sequence, complete prime factorizations are known only for a few of its members. Values derived from this sequence have also been used to construct finiteEgyptian fraction representations of 1, SasakianEinstein manifold s, and hard instances foronline algorithm s.Formal definitions
Formally, Sylvester's sequence can be defined by the formula:The product of an empty set is 1, so "s"0 = 2.
Alternatively, one may define the sequence by the recurrence: with "s"0 = 2.It is straightforward to show by induction that this is equivalent to the other definition.
Connection with Egyptian fractions
The
unit fraction s formed by the reciprocals of the values in Sylvester's sequence generate an infinite series::The partial sums of this series have a simple form,:as may be proved by induction. Clearly this identity is true for "j" = 0, as both sides are zero. For larger "j", expanding the left side of the identity using the induction hypothesis produces:as was to be proved. Since this sequence of partial sums ("s""j"-2)/("s""j"-1) converges to one, the overall series forms an infiniteEgyptian fraction representation of the number one::One can find finite Egyptian fraction representations of one, of any length, by truncating this series and subtracting one from the last denominator::The sum of the first "k" terms of the infinite series provides the closest possible underestimate of 1 by any "k"-term Egyptian fraction. [This claim is commonly attributed to Curtiss (1922), but Miller (1919) appears to be making the same statement in an earlier paper. See also Rosenman (1933), Salzer (1947), and Soundararajan (2005).] For example, the first four terms add to 1805/1806, and therefore any Egyptian fraction for a number in the open interval (1805/1806,1) requires at least five terms.It is possible to interpret the Sylvester sequence as the result of a
greedy algorithm for Egyptian fractions , that at each step chooses the smallest possible denominator that makes the partial sum of the series be less than one. Alternatively, the terms of the sequence after the first can be viewed as the denominators of theodd greedy expansion of 1/2.Closed form formula and asymptotics
The Sylvester numbers grow doubly exponentially as a function of "n". Specifically, it can be shown that
:
for a number "E" that is approximately 1.264. [Graham, Knuth, and Patashnik (1989) set this as an exercise; see also Golomb (1963).] This formula has the effect of the following
algorithm ::s0 is the nearest integer to E2; s1 is the nearest integer to E4; s2 is the nearest integer to E8; for s"n", take E2, square it "n" more times, and take the nearest integer. This would only be a practical algorithm if we had a better way of calculating E to the requisite number of places than calculating s"n" and taking its repeated square root.The double-exponential growth of the Sylvester sequence is unsurprising if one compares it to the sequence of
Fermat number s "F""n"; the Fermat numbers are usually defined by a doubly exponential formula, , but they can also be defined by a product formula very similar to that defining Sylvester's sequence::
Uniqueness of quickly growing series with rational sums
As Sylvester himself observed, Sylvester's sequence seems to be unique in having such quickly growing values, while simultaneously having a series of reciprocals that converges to a rational number.
To make this more precise, it follows from results of Badea (1993) that, if a sequence of integers grows quickly enough that:and if the series:converges to a rational number "A", then, for all "n" after some point, this sequence must be defined by the same recurrence:that can be used to define Sylvester's sequence.
Erdős (1980) conjectured that, in results of this type, the inequality bounding the growth of the sequence could be replaced by a weaker condition,:Badea (1995) surveys progress related to this conjecture; see also Brown.
Divisibility and factorizations
If "i" < "j", it follows from the definition that "s""j" ≡ 1 (mod "s""i"). Therefore, every two numbers in Sylvester's sequence are
relatively prime . The sequence can be used to prove that there are infinitely manyprime number s, as any prime can divide at most one number in the sequence.Some effort has been expended in an attempt to factor the numbers in Sylvester's sequence, but much remains unknown about their factorization. For instance, it is not known if all numbers in the sequence are squarefree, although all the known terms are.
As Vardi (1991) describes, it is easy to determine which Sylvester number (if any) a given prime "p" divides: simply compute the recurrence defining the numbers modulo "p" until finding either a number that is congruent to zero (mod "p") or finding a repeated modulus. Via this technique he found that 1166 out of the first three million primes are divisors of Sylvester numbers, [This appears to be a typo, as Andersen finds 1167 prime divisors in this range.] and that none of these primes has a square that divides a Sylvester number. A general result of Jones (2006) implies that the set of prime factors of Sylvester numbers has density zero in the set of all primes.
The following table shows known factorizations of these numbers, (except the first four, which are all prime): [All prime factors "p" of Sylvester numbers "s""n" with "p" < 5×107 and "n" ≤ 200 are listed by Vardi. Ken Takusagawa lists the [http://kenta.blogspot.com/2006/01/factoring-sylvesters-sequence.html factorizations up to "s"9] and the [http://kenta.blogspot.com/2006/04/sylvester-10th-factored.html factorization of "s"10] . The remaining factorizations are from [http://hjem.get2net.dk/jka/math/sylvester-factors.htm a list of factorizations of Sylvester's sequence] maintained by Jens Kruse Andersen, retrieved October 2, 2007.]
As is customary, P"n" and C"n" denote prime and composite numbers "n" digits long.
Applications
Boyer et al. (2005) use the properties of Sylvester's sequence to define large numbers of Sasakian
Einstein manifold s having thedifferential topology of odd-dimensional spheres orexotic sphere s. They show that the number of distinct Sasakian Einstein metrics on atopological sphere of dimension 2"n" − 1 is at least proportional to "s""n" and hence has double exponential growth with "n".As Galambos and Woeginger (1995) describe, Brown (1979) and Liang (1980) used values derived from Sylvester's sequence to construct lower bound examples for online
bin packing algorithm s. Seiden and Woeginger (2005) similarly use the sequence to lower bound the performance of a two-dimensional cutting stock algorithm. [In their work, Seiden and Woeginger refer to Sylvester's sequence as "Salzer's sequence" after Salzer's (1947) paper on closest approximation.]Znám's problem concerns sets of numbers such that each number in the set divides but is not equal to the product of all the other numbers, plus one. Without the inequality requirement, the values in Sylvester's sequence would solve the problem; with that requirement, it has other solutions derived from recurrences similar to the one defining Sylvester's sequence. Solutions to Znám's problem have applications to the classification of surface singularities (Brenton and Hill 1988) and to the theory ofnondeterministic finite automata (Domaratzki et al. 2005).Curtiss (1922) describes an application of the closest approximations to one by "k"-term sums of unit fractions, in lower-bounding the number of divisors of any
perfect number , and Miller (1919) uses the same property to lower bound the size of certain groups.See also
*
Cahen's constant
*Primary pseudoperfect number Footnotes
References
* cite journal
author = Badea, Catalin
title = A theorem on irrationality of infinite series and applications
year = 1993
journal =Acta Arithmetica
volume = 63
pages = 313–323
id = MathSciNet | id = 1218459* cite web
author = Badea, Catalin
title = On some criteria for irrationality for series of positive rationals: a survey
year = 1995
url = http://www.lacim.uqam.ca/~plouffe/OEIS/citations/caen.pdf* cite journal
author = Boyer, Charles P.; Galicki, Krzysztof; Kollár, János
title = Einstein metrics on spheres
journal =Annals of Mathematics
volume = 162
issue = 1
year = 2005
pages = 557–580
id = arxiv | archive = math.DG | id = 0309408 MathSciNet | id = 2178969
url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.annm/1134073591* cite journal
author = Brenton, Lawrence and Hill, Richard
title = On the Diophantine equation 1=Σ1/"n""i" + 1/Π"n""i" and a class of homologically trivial complex surface singularities
journal =Pacific Journal of Mathematics
url = http://projecteuclid.org/Dienst/UI/1.0/Summarize/euclid.pjm/1102689567
volume = 133
issue = 1
year = 1988
pages = 41–67
id = MathSciNet | id = 0936356* cite paper
author = Brown, D. J.
title = A lower bound for on-line one-dimensional bin packing algorithms
version = Tech. Rep. R-864
date = 1979
publisher = Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign* cite journal
author = Curtiss, D. R.
title = On Kellogg's diophantine problem
journal =American Mathematical Monthly
url = http://www.jstor.org/stable/2299023
volume = 29
year = 1922
pages = 380–387* cite journal
author = Domaratzki, Michael; Ellul, Keith; Shallit, Jeffrey; Wang, Ming-Wei
title = Non-uniqueness and radius of cyclic unary NFAs
journal = International Journal of Foundations of Computer Science
volume = 16
issue = 5
pages = 883–896
year = 2005
url = http://www.cs.umanitoba.ca/~mdomarat/pubs/DESW_dcfs.ps
id = MathSciNet | id = 2174328
doi = 10.1142/S0129054105003352* cite book
author = Erdős, Paul and Graham, Ronald L.
year = 1980
title = Old and new problems and results in combinatorial number theory
publisher = Monographies de L'Enseignement Mathématique, No. 28, Univ. de Genève
id = MathSciNet | id = 0592420* cite journal
author = Galambos, Gábor; Woeginger, Gerhard J.
title = On-line bin packing — A restricted survey
journal = Mathematical Methods of Operations Research
volume = 42
issue = 1
year = 1995
doi = 10.1007/BF01415672
id = MathSciNet | id = 1346486* cite journal
author = Golomb, Solomon W.
authorlink = Solomon Golomb
title = On certain nonlinear recurring sequences
year = 1963
journal =American Mathematical Monthly
volume = 70
issue = 4
pages = 403–405
url = http://www.jstor.org/stable/2311857
id = MathSciNet | id = 0148605* cite book
author = Graham, R.; Knuth, D. E.; Patashnik, O.
title = Concrete Mathematics
edition = 2nd ed.
year = 1989
publisher = Addison-Wesley
pages = Exercise 4.37
id = ISBN 0-201-55802-5* cite journal
author = Jones, Rafe
title = The density of prime divisors in the arithmetic dynamics of quadratic polynomials
year = 2006
id = arxiv | archive = math.NT | id = 0612415* cite journal
author = Liang, Frank M.
title = A lower bound for on-line bin packing
year = 1980
journal = Information Processing Letters
volume = 10
issue = 2
pages = 76–79
id = MathSciNet | id = 0564503
doi = 10.1016/S0020-0190(80)90077-0* cite journal
author = Miller, G. A.
title = Groups possessing a small number of sets of conjugate operators
journal =Transactions of the American Mathematical Society
year = 1919
volume = 20
issue = 3
pages = 260–270
url = http://www.jstor.org/stable/1988867* cite journal
author = Rosenman, Martin
title = Problem 3536
journal =American Mathematical Monthly
volume = 40
issue = 3
year = 1933
pages = 180
url = http://www.jstor.org/stable/2301036* cite journal
author = Salzer, H. E.
journal =American Mathematical Monthly
year = 1947
title = The approximation of numbers as sums of reciprocals
volume = 54
issue = 3
pages = 135–142
url = http://www.jstor.org/stable/2305906
id = MathSciNet | id = 0020339* cite journal
author = Seiden, Steven S.; Woeginger, Gerhard J.
title = The two-dimensional cutting stock problem revisited
journal = Mathematical Programming
volume = 102
issue = 3
year = 2005
pages = 519–530
doi = 10.1007/s10107-004-0548-1
id = MathSciNet | id = 2136225* cite journal
author = Soundararajan, K.
year = 2005
title = Approximating 1 from below using "n" Egyptian fractions
id = arxiv|archive = math.CA|id = 0502247* cite journal
author = Sylvester, J. J.
authorlink = J. J. Sylvester
title = On a point in the theory of vulgar fractions
url = http://www.jstor.org/stable/2369261
journal =American Journal of Mathematics
volume = 3
issue = 4
year = 1880
pages = 332–335* cite book
author = Vardi, Ilan
title = Computational Recreations in Mathematica
year = 1991
publisher = Addison-Wesley
pages = 82–89
id = ISBN 0-201-52989-0External links
* [http://www.mathpages.com/home/kmath455.htm Irrationality of Quadratic Sums] , from K. S. Brown's MathPages.
*
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