Sylvester's sequence (English Wikipedia)

Analysis of information sources in references of the Wikipedia article "Sylvester's sequence" in English language version.

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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 & Underwood (1933), Salzer (1947), Soundararajan (2005), and Nathanson (2023). Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023. Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867. Rosenman, Martin; Underwood, F. (1933). "Problem 3536". American Mathematical Monthly. 40 (3): 180–181. doi:10.2307/2301036. JSTOR 2301036. Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339. Soundararajan, K. (2005). "Approximating 1 from below using n Egyptian fractions". arXiv:math.CA/0502247. Nathanson, Melvyn B. (January 2023). "Underapproximation by Egyptian fractions". Journal of Number Theory. 242: 208–234. arXiv:2202.00191. doi:10.1016/j.jnt.2022.07.005.
Boyer, Galicki & Kollár (2005). Boyer, Charles P.; Galicki, Krzysztof; Kollár, János (2005). "Einstein metrics on spheres". Annals of Mathematics. 162 (1): 557–580. arXiv:math.DG/0309408. doi:10.4007/annals.2005.162.557. MR 2178969. S2CID 13945306.
Galambos & Woeginger (1995); Brown (1979); Liang (1980). Galambos, Gábor; Woeginger, Gerhard J. (1995). "On-line bin packing — A restricted survey". Mathematical Methods of Operations Research. 42 (1): 25. doi:10.1007/BF01415672. MR 1346486. S2CID 26692460. Brown, D. J. (1979). A lower bound for on-line one-dimensional bin packing algorithms. Tech. Rep. R-864. Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign. Liang, Frank M. (1980). "A lower bound for on-line bin packing". Information Processing Letters. 10 (2): 76–79. doi:10.1016/S0020-0190(80)90077-0. MR 0564503.
Graham, Knuth & Patashnik (1989), formula 4.17, p. 109, and exercise 4.37, p. 147; see also Golomb (1963). Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989). Concrete Mathematics (2e ed.). Addison-Wesley. ISBN 978-0-201-55802-9. Golomb, Solomon W. (1963). "On certain nonlinear recurring sequences". American Mathematical Monthly. 70 (4): 403–405. doi:10.2307/2311857. JSTOR 2311857. MR 0148605.
Badea (1993). Badea, Catalin (1993). "A theorem on irrationality of infinite series and applications". Acta Arithmetica. 63 (4): 313–323. doi:10.4064/aa-63-4-313-323. MR 1218459.
Erdős & Graham (1980). Erdős, Paul; Graham, Ronald L. (1980). Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique, No. 28, Univ. de Genève. MR 0592420.
Guy & Nowakowski (1975). Guy, Richard; Nowakowski, Richard (1975). "Discovering primes with Euclid". Delta (Waukesha). 5 (2): 49–63. MR 0384675.
In their work, Seiden and Woeginger refer to Sylvester's sequence as "Salzer's sequence" after the work of Salzer (1947) on closest approximation. Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339.
Domaratzki et al. (2005). Domaratzki, Michael; Ellul, Keith; Shallit, Jeffrey; Wang, Ming-Wei (2005). "Non-uniqueness and radius of cyclic unary NFAs". International Journal of Foundations of Computer Science. 16 (5): 883–896. doi:10.1142/S0129054105003352. MR 2174328.

arxiv.org

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 & Underwood (1933), Salzer (1947), Soundararajan (2005), and Nathanson (2023). Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023. Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867. Rosenman, Martin; Underwood, F. (1933). "Problem 3536". American Mathematical Monthly. 40 (3): 180–181. doi:10.2307/2301036. JSTOR 2301036. Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339. Soundararajan, K. (2005). "Approximating 1 from below using n Egyptian fractions". arXiv:math.CA/0502247. Nathanson, Melvyn B. (January 2023). "Underapproximation by Egyptian fractions". Journal of Number Theory. 242: 208–234. arXiv:2202.00191. doi:10.1016/j.jnt.2022.07.005.
Boyer, Galicki & Kollár (2005). Boyer, Charles P.; Galicki, Krzysztof; Kollár, János (2005). "Einstein metrics on spheres". Annals of Mathematics. 162 (1): 557–580. arXiv:math.DG/0309408. doi:10.4007/annals.2005.162.557. MR 2178969. S2CID 13945306.
Nathanson (2023). Nathanson, Melvyn B. (January 2023). "Underapproximation by Egyptian fractions". Journal of Number Theory. 242: 208–234. arXiv:2202.00191. doi:10.1016/j.jnt.2022.07.005.
Jones (2006). Jones, Rafe (2006). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society. 78 (2): 523–544. arXiv:math.NT/0612415. Bibcode:2006math.....12415J. doi:10.1112/jlms/jdn034. S2CID 15310955.

books.google.com

Guy (2004). Guy, Richard K. (2004). "E24 Irrationality sequences". Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. p. 346. ISBN 0-387-20860-7. Zbl 1058.11001.

doi.org

Sylvester (1880). Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
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 & Underwood (1933), Salzer (1947), Soundararajan (2005), and Nathanson (2023). Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023. Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867. Rosenman, Martin; Underwood, F. (1933). "Problem 3536". American Mathematical Monthly. 40 (3): 180–181. doi:10.2307/2301036. JSTOR 2301036. Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339. Soundararajan, K. (2005). "Approximating 1 from below using n Egyptian fractions". arXiv:math.CA/0502247. Nathanson, Melvyn B. (January 2023). "Underapproximation by Egyptian fractions". Journal of Number Theory. 242: 208–234. arXiv:2202.00191. doi:10.1016/j.jnt.2022.07.005.
Boyer, Galicki & Kollár (2005). Boyer, Charles P.; Galicki, Krzysztof; Kollár, János (2005). "Einstein metrics on spheres". Annals of Mathematics. 162 (1): 557–580. arXiv:math.DG/0309408. doi:10.4007/annals.2005.162.557. MR 2178969. S2CID 13945306.
Galambos & Woeginger (1995); Brown (1979); Liang (1980). Galambos, Gábor; Woeginger, Gerhard J. (1995). "On-line bin packing — A restricted survey". Mathematical Methods of Operations Research. 42 (1): 25. doi:10.1007/BF01415672. MR 1346486. S2CID 26692460. Brown, D. J. (1979). A lower bound for on-line one-dimensional bin packing algorithms. Tech. Rep. R-864. Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign. Liang, Frank M. (1980). "A lower bound for on-line bin packing". Information Processing Letters. 10 (2): 76–79. doi:10.1016/S0020-0190(80)90077-0. MR 0564503.
A proof by induction is given by Sylvester (1880), p. 333. Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
Graham, Knuth & Patashnik (1989), formula 4.17, p. 109, and exercise 4.37, p. 147; see also Golomb (1963). Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989). Concrete Mathematics (2e ed.). Addison-Wesley. ISBN 978-0-201-55802-9. Golomb, Solomon W. (1963). "On certain nonlinear recurring sequences". American Mathematical Monthly. 70 (4): 403–405. doi:10.2307/2311857. JSTOR 2311857. MR 0148605.
This series is the starting point for Sylvester (1880) Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
Sylvester (1880), p. 334. Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
Nathanson (2023). Nathanson, Melvyn B. (January 2023). "Underapproximation by Egyptian fractions". Journal of Number Theory. 242: 208–234. arXiv:2202.00191. doi:10.1016/j.jnt.2022.07.005.
Badea (1993). Badea, Catalin (1993). "A theorem on irrationality of infinite series and applications". Acta Arithmetica. 63 (4): 313–323. doi:10.4064/aa-63-4-313-323. MR 1218459.
Jones (2006). Jones, Rafe (2006). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society. 78 (2): 523–544. arXiv:math.NT/0612415. Bibcode:2006math.....12415J. doi:10.1112/jlms/jdn034. S2CID 15310955.
Odoni (1985). Odoni, R. W. K. (1985). "On the prime divisors of the sequence w_n+1 =1+w₁⋯w_n". Journal of the London Mathematical Society. Series II. 32: 1–11. doi:10.1112/jlms/s2-32.1.1. Zbl 0574.10020.
In their work, Seiden and Woeginger refer to Sylvester's sequence as "Salzer's sequence" after the work of Salzer (1947) on closest approximation. Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339.
Domaratzki et al. (2005). Domaratzki, Michael; Ellul, Keith; Shallit, Jeffrey; Wang, Ming-Wei (2005). "Non-uniqueness and radius of cyclic unary NFAs". International Journal of Foundations of Computer Science. 16 (5): 883–896. doi:10.1142/S0129054105003352. MR 2174328.
Curtiss (1922); Miller (1919). Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023. Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867.

harvard.edu

ui.adsabs.harvard.edu

Jones (2006). Jones, Rafe (2006). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society. 78 (2): 523–544. arXiv:math.NT/0612415. Bibcode:2006math.....12415J. doi:10.1112/jlms/jdn034. S2CID 15310955.

jstor.org

Sylvester (1880). Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
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 & Underwood (1933), Salzer (1947), Soundararajan (2005), and Nathanson (2023). Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023. Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867. Rosenman, Martin; Underwood, F. (1933). "Problem 3536". American Mathematical Monthly. 40 (3): 180–181. doi:10.2307/2301036. JSTOR 2301036. Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339. Soundararajan, K. (2005). "Approximating 1 from below using n Egyptian fractions". arXiv:math.CA/0502247. Nathanson, Melvyn B. (January 2023). "Underapproximation by Egyptian fractions". Journal of Number Theory. 242: 208–234. arXiv:2202.00191. doi:10.1016/j.jnt.2022.07.005.
A proof by induction is given by Sylvester (1880), p. 333. Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
Graham, Knuth & Patashnik (1989), formula 4.17, p. 109, and exercise 4.37, p. 147; see also Golomb (1963). Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989). Concrete Mathematics (2e ed.). Addison-Wesley. ISBN 978-0-201-55802-9. Golomb, Solomon W. (1963). "On certain nonlinear recurring sequences". American Mathematical Monthly. 70 (4): 403–405. doi:10.2307/2311857. JSTOR 2311857. MR 0148605.
This series is the starting point for Sylvester (1880) Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
Sylvester (1880), p. 334. Sylvester, J. J. (1880). "On a point in the theory of vulgar fractions". American Journal of Mathematics. 3 (4): 332–335. doi:10.2307/2369261. JSTOR 2369261.
In their work, Seiden and Woeginger refer to Sylvester's sequence as "Salzer's sequence" after the work of Salzer (1947) on closest approximation. Salzer, H. E. (1947). "The approximation of numbers as sums of reciprocals". American Mathematical Monthly. 54 (3): 135–142. doi:10.2307/2305906. JSTOR 2305906. MR 0020339.
Curtiss (1922); Miller (1919). Curtiss, D. R. (1922). "On Kellogg's diophantine problem". American Mathematical Monthly. 29 (10): 380–387. doi:10.2307/2299023. JSTOR 2299023. Miller, G. A. (1919). "Groups possessing a small number of sets of conjugate operators". Transactions of the American Mathematical Society. 20 (3): 260–270. doi:10.2307/1988867. JSTOR 1988867.

kenta.blogspot.com

All prime factors p of Sylvester numbers s_n with p < 5×10⁷ and n ≤ 200 are listed by Vardi. Ken Takusagawa lists the factorizations up to s₉ and the factorization of s₁₀. The remaining factorizations are from a list of factorizations of Sylvester's sequence maintained by Jens Kruse Andersen. Retrieved 2014-06-13.

oeis.org

Sloane, N. J. A. (ed.). "Sequence A000058 (Sylvester's sequence)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
Sloane, N. J. A. (ed.). "Sequence A000215 (Fermat numbers)". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.

primerecords.dk

All prime factors p of Sylvester numbers s_n with p < 5×10⁷ and n ≤ 200 are listed by Vardi. Ken Takusagawa lists the factorizations up to s₉ and the factorization of s₁₀. The remaining factorizations are from a list of factorizations of Sylvester's sequence maintained by Jens Kruse Andersen. Retrieved 2014-06-13.

semanticscholar.org

api.semanticscholar.org

Boyer, Galicki & Kollár (2005). Boyer, Charles P.; Galicki, Krzysztof; Kollár, János (2005). "Einstein metrics on spheres". Annals of Mathematics. 162 (1): 557–580. arXiv:math.DG/0309408. doi:10.4007/annals.2005.162.557. MR 2178969. S2CID 13945306.
Galambos & Woeginger (1995); Brown (1979); Liang (1980). Galambos, Gábor; Woeginger, Gerhard J. (1995). "On-line bin packing — A restricted survey". Mathematical Methods of Operations Research. 42 (1): 25. doi:10.1007/BF01415672. MR 1346486. S2CID 26692460. Brown, D. J. (1979). A lower bound for on-line one-dimensional bin packing algorithms. Tech. Rep. R-864. Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign. Liang, Frank M. (1980). "A lower bound for on-line bin packing". Information Processing Letters. 10 (2): 76–79. doi:10.1016/S0020-0190(80)90077-0. MR 0564503.
Jones (2006). Jones, Rafe (2006). "The density of prime divisors in the arithmetic dynamics of quadratic polynomials". Journal of the London Mathematical Society. 78 (2): 523–544. arXiv:math.NT/0612415. Bibcode:2006math.....12415J. doi:10.1112/jlms/jdn034. S2CID 15310955.

umanitoba.ca

cs.umanitoba.ca

Domaratzki et al. (2005). Domaratzki, Michael; Ellul, Keith; Shallit, Jeffrey; Wang, Ming-Wei (2005). "Non-uniqueness and radius of cyclic unary NFAs". International Journal of Foundations of Computer Science. 16 (5): 883–896. doi:10.1142/S0129054105003352. MR 2174328.

unsw.edu.au

parabola.unsw.edu.au

Graham, Knuth & Patashnik (1989), research problem 4.65, p. 151; Vardi (1991); see also Chentouf (2020) Graham, Ronald; Knuth, Donald E.; Patashnik, Oren (1989). Concrete Mathematics (2e ed.). Addison-Wesley. ISBN 978-0-201-55802-9. Vardi, Ilan (1991). Computational Recreations in Mathematica. Addison-Wesley. pp. 82–89. ISBN 0-201-52989-0. Chentouf, A. Anas (2020). "On Sylvester's Sequence and some of its properties" (PDF). Parabola. 56 (2).

uqam.ca

lacim.uqam.ca

Badea (1995); Brown (1979). Badea, Catalin (1995). "On some criteria for irrationality for series of positive rationals: a survey" (PDF). Archived from the original (PDF) on 2008-09-11. Brown, D. J. (1979). A lower bound for on-line one-dimensional bin packing algorithms. Tech. Rep. R-864. Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign.

web.archive.org

Badea (1995); Brown (1979). Badea, Catalin (1995). "On some criteria for irrationality for series of positive rationals: a survey" (PDF). Archived from the original (PDF) on 2008-09-11. Brown, D. J. (1979). A lower bound for on-line one-dimensional bin packing algorithms. Tech. Rep. R-864. Coordinated Science Lab., Univ. of Illinois, Urbana-Champaign.

zbmath.org

Guy (2004). Guy, Richard K. (2004). "E24 Irrationality sequences". Unsolved Problems in Number Theory (3rd ed.). Springer-Verlag. p. 346. ISBN 0-387-20860-7. Zbl 1058.11001.
Odoni (1985). Odoni, R. W. K. (1985). "On the prime divisors of the sequence w_n+1 =1+w₁⋯w_n". Journal of the London Mathematical Society. Series II. 32: 1–11. doi:10.1112/jlms/s2-32.1.1. Zbl 0574.10020.