References analysis of the Wikipedia article "Regular number" in the English version

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Pomerance (1995). Pomerance, Carl (1995), "The role of smooth numbers in number-theoretic algorithms", Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Basel: Birkhäuser, pp. 411–422, MR 1403941.

Sachs (1947). Sachs, A. J. (1947), "Babylonian mathematical texts. I. Reciprocals of regular sexagesimal numbers", Journal of Cuneiform Studies, 1 (3), The American Schools of Oriental Research: 219–240, doi:10.2307/1359434, JSTOR 1359434, MR 0022180, S2CID 163783242.

Halsey & Hewitt (1972) note that this follows from Størmer's theorem (Størmer 1897), and provide a proof for this case; see also Silver (1971). Halsey, G. D.; Hewitt, Edwin (1972), "More on the superparticular ratios in music", American Mathematical Monthly, 79 (10), Mathematical Association of America: 1096–1100, doi:10.2307/2317424, JSTOR 2317424, MR 0313189. 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). Silver, A. L. Leigh (1971), "Musimatics or the nun's fiddle", American Mathematical Monthly, 78 (4), Mathematical Association of America: 351–357, doi:10.2307/2316896, JSTOR 2316896.

Heninger, Rains & Sloane (2006). Heninger, Nadia; Rains, E. M.; Sloane, N. J. A. (2006), "On the integrality of

n

th roots of generating functions", Journal of Combinatorial Theory, Series A, 113 (8): 1732–1745, arXiv:math.NT/0509316, doi:10.1016/j.jcta.2006.03.018, MR 2269551, S2CID 15913795}.

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Knuth (1972). Knuth, D. E. (1972), "Ancient Babylonian algorithms" (PDF), Communications of the ACM, 15 (7): 671–677, doi:10.1145/361454.361514, S2CID 7829945. A correction appears in CACM 19(2), 1976, stating that the tablet does not contain all 231 of the numbers of interest. The article (corrected) with a brief addendum appears in Selected Papers on Computer Science, CSLI Lecture Notes 59, Cambridge Univ. Press, 1996, pp. 185–203, but without the Appendix that was included in the original paper.

Fowler & Robson (1998). Fowler, David; Robson, Eleanor (1998), "Square Root Approximations in Old Babylonian Mathematics: YBC 7289 in Context" (PDF), Historia Mathematica, 25 (4): 366–378, doi:10.1006/hmat.1998.2209, page 375.

Clarke (1877). Clarke, A. R. (January 1877), "Just intonation", Nature, 15 (377): 253, Bibcode:1877Natur..15..253C, doi:10.1038/015253b0.

Honingh & Bod (2005). Honingh, Aline; Bod, Rens (2005), "Convexity and the well-formedness of musical objects", Journal of New Music Research, 34 (3): 293–303, doi:10.1080/09298210500280612, S2CID 16321292.

Kopiez (2003). Kopiez, Reinhard (2003), "Intonation of harmonic intervals: adaptability of expert musicians to equal temperament and just intonation", Music Perception, 20 (4): 383–410, doi:10.1525/mp.2003.20.4.383

Wolf (2003). Wolf, Daniel James (March 2003), "Alternative tunings, alternative tonalities", Contemporary Music Review, 22 (1–2): 3–14, doi:10.1080/0749446032000134715, S2CID 191457676

x 2 - Dy 2 = \pm1

Howard & Longair (1982). Howard, Deborah; Longair, Malcolm (May 1982), "Harmonic proportion and Palladio's Quattro Libri", Journal of the Society of Architectural Historians, 41 (2): 116–143, doi:10.2307/989675, JSTOR 989675

See, e.g., Hemmendinger (1988) or Yuen (1992). Hemmendinger, David (1988), "The "Hamming problem" in Prolog", ACM SIGPLAN Notices, 23 (4): 81–86, doi:10.1145/44326.44335, S2CID 28906392. Yuen, C. K. (1992), "Hamming numbers, lazy evaluation, and eager disposal", ACM SIGPLAN Notices, 27 (8): 71–75, doi:10.1145/142137.142151, S2CID 18283005.

Gingerich (1965). Gingerich, Owen (1965), "Eleven-digit regular sexagesimals and their reciprocals", Transactions of the American Philosophical Society, 55 (8), American Philosophical Society: 3–38, doi:10.2307/1006080, JSTOR 1006080.

Heninger, Rains & Sloane (2006). Heninger, Nadia; Rains, E. M.; Sloane, N. J. A. (2006), "On the integrality of

n

th roots of generating functions", Journal of Combinatorial Theory, Series A, 113 (8): 1732–1745, arXiv:math.NT/0509316, doi:10.1016/j.jcta.2006.03.018, MR 2269551, S2CID 15913795}.

Temperton (1992). Temperton, Clive (1992), "A generalized prime factor FFT algorithm for any

N = 2 p 3 q 5 r

", SIAM Journal on Scientific and Statistical Computing, 13 (3): 676–686, doi:10.1137/0913039, S2CID 14764894.

Barton (1908); McClain (1974). Barton, George A. (1908), "On the Babylonian origin of Plato's nuptial number", Journal of the American Oriental Society, 29, American Oriental Society: 210–219, doi:10.2307/592627, JSTOR 592627. McClain, Ernest G. (1974), "Musical "Marriages" in Plato's "Republic"", Journal of Music Theory, 18 (2), Duke University Press: 242–272, JSTOR 843638.

Veller, Nowak & Davis (2015). Veller, Carl; Nowak, Martin A.; Davis, Charles C. (May 2015), "Extended flowering intervals of bamboos evolved by discrete multiplication", Ecology Letters, 18 (7): 653–659, Bibcode:2015EcolL..18..653V, doi:10.1111/ele.12442, PMID 25963600

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Berndt & Rankin (1995). Berndt, Bruce C.; Rankin, Robert Alexander, eds. (1995), Ramanujan: letters and commentary, History of mathematics, vol. 9, American Mathematical Society, p. 23, Bibcode:1995rlc..book.....B, ISBN 978-0-8218-0470-4.

Clarke (1877). Clarke, A. R. (January 1877), "Just intonation", Nature, 15 (377): 253, Bibcode:1877Natur..15..253C, doi:10.1038/015253b0.

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Asmussen (2001), for instance, states that "within any piece of tonal music" all intervals must be ratios of regular numbers, echoing similar statements by much earlier writers such as Habens (1889). In the modern music theory literature this assertion is often attributed to Longuet-Higgins (1962), who used a graphical arrangement closely related to the tonnetz to organize 5-limit pitches. Asmussen, Robert (2001), Periodicity of sinusoidal frequencies as a basis for the analysis of Baroque and Classical harmony: a computer based study (PDF), Ph.D. thesis, University of Leeds, archived from the original (PDF) on 2016-04-24, retrieved 2007-03-15. Habens, Rev. W. J. (1889), "On the musical scale", Proceedings of the Musical Association, 16, Royal Musical Association: 16th Session, p. 1, JSTOR 765355. Longuet-Higgins, H. C. (1962), "Letter to a musical friend", Music Review (August): 244–248.

x 2 - Dy 2 = \pm1

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Wolf (2003). Wolf, Daniel James (March 2003), "Alternative tunings, alternative tonalities", Contemporary Music Review, 22 (1–2): 3–14, doi:10.1080/0749446032000134715, S2CID 191457676

Heninger, Rains & Sloane (2006). Heninger, Nadia; Rains, E. M.; Sloane, N. J. A. (2006), "On the integrality of

n

th roots of generating functions", Journal of Combinatorial Theory, Series A, 113 (8): 1732–1745, arXiv:math.NT/0509316, doi:10.1016/j.jcta.2006.03.018, MR 2269551, S2CID 15913795}.

Temperton (1992). Temperton, Clive (1992), "A generalized prime factor FFT algorithm for any

N = 2 p 3 q 5 r

", SIAM Journal on Scientific and Statistical Computing, 13 (3): 676–686, doi:10.1137/0913039, S2CID 14764894.

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Regular number (English Wikipedia)

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