Polite number

A Young diagram representing visually a polite expansion 15 = 4 + 5 + 6

In number theory, a polite number is a positive integer that can be written as the sum of two or more consecutive positive integers. A positive integer which is not polite is called impolite.[1][2] The impolite numbers are exactly the powers of two, and the polite numbers are the natural numbers that are not powers of two.

Polite numbers have also been called staircase numbers because the Young diagrams which represent graphically the partitions of a polite number into consecutive integers (in the French notation of drawing these diagrams) resemble staircases.[3][4][5] If all numbers in the sum are strictly greater than one, the numbers so formed are also called trapezoidal numbers because they represent patterns of points arranged in a trapezoid.[6][7][8][9][10][11][12]

The problem of representing numbers as sums of consecutive integers and of counting the number of representations of this type has been studied by Sylvester,[13] Mason,[14][15] Leveque,[16] and many other more recent authors.[1][2][17][18][19][20][21][22][23] The polite numbers describe the possible numbers of sides of the Reinhardt polygons.[24]

  1. ^ a b Adams, Ken (March 1993), "How polite is x?", The Mathematical Gazette, 77 (478): 79–80, doi:10.2307/3619263, JSTOR 3619263, S2CID 171530924.
  2. ^ a b Griggs, Terry S. (December 1991), "Impolite Numbers", The Mathematical Gazette, 75 (474): 442–443, doi:10.2307/3618630, JSTOR 3618630, S2CID 171681914.
  3. ^ Mason, John; Burton, Leone; Stacey, Kaye (1982), Thinking Mathematically, Addison-Wesley, ISBN 978-0-201-10238-3.
  4. ^ Stacey, K.; Groves, S. (1985), Strategies for Problem Solving, Melbourne: Latitude.
  5. ^ Stacey, K.; Scott, N. (2000), "Orientation to deep structure when trying examples: a key to successful problem solving", in Carillo, J.; Contreras, L. C. (eds.), Resolucion de Problemas en los Albores del Siglo XXI: Una vision Internacional desde Multiples Perspectivas y Niveles Educativos (PDF), Huelva, Spain: Hergue, pp. 119–147, archived from the original (PDF) on 2008-07-26.
  6. ^ Gamer, Carlton; Roeder, David W.; Watkins, John J. (1985), "Trapezoidal numbers", Mathematics Magazine, 58 (2): 108–110, doi:10.2307/2689901, JSTOR 2689901.
  7. ^ Jean, Charles-É. (March 1991), "Les nombres trapézoïdaux" (French), Bulletin de l'AMQ: 6–11.
  8. ^ Haggard, Paul W.; Morales, Kelly L. (1993), "Discovering relationships and patterns by exploring trapezoidal numbers", International Journal of Mathematical Education in Science and Technology, 24 (1): 85–90, doi:10.1080/0020739930240111.
  9. ^ Feinberg-McBrian, Carol (1996), "The case of trapezoidal numbers", Mathematics Teacher, 89 (1): 16–24, doi:10.5951/MT.89.1.0016.
  10. ^ Smith, Jim (1997), "Trapezoidal numbers", Mathematics in School, 5: 42.
  11. ^ Verhoeff, T. (1999), "Rectangular and trapezoidal arrangements", Journal of Integer Sequences, 2: 16, Bibcode:1999JIntS...2...16V, Article 99.1.6.
  12. ^ Jones, Chris; Lord, Nick (1999), "Characterising non-trapezoidal numbers", The Mathematical Gazette, 83 (497): 262–263, doi:10.2307/3619053, JSTOR 3619053, S2CID 125545112.
  13. ^ Sylvester, J. J.; Franklin, F (1882), "A constructive theory of partitions, arranged in three acts, an interact and an exodion", American Journal of Mathematics, 5 (1): 251–330, doi:10.2307/2369545, JSTOR 2369545. In The collected mathematical papers of James Joseph Sylvester (December 1904), H. F. Baker, ed. Sylvester defines the class of a partition into distinct integers as the number of blocks of consecutive integers in the partition, so in his notation a polite partition is of first class.
  14. ^ Mason, T. E. (1911), "On the representations of a number as a sum of consecutive integers", Proceedings of the Indiana Academy of Science: 273–274.
  15. ^ Mason, Thomas E. (1912), "On the representation of an integer as the sum of consecutive integers", American Mathematical Monthly, 19 (3): 46–50, doi:10.2307/2972423, JSTOR 2972423, MR 1517654.
  16. ^ Leveque, W. J. (1950), "On representations as a sum of consecutive integers", Canadian Journal of Mathematics, 2: 399–405, doi:10.4153/CJM-1950-036-3, MR 0038368, S2CID 124093945,
  17. ^ Pong, Wai Yan (2007), "Sums of consecutive integers", College Math. J., 38 (2): 119–123, arXiv:math/0701149, Bibcode:2007math......1149P, doi:10.1080/07468342.2007.11922226, MR 2293915, S2CID 14169613.
  18. ^ Britt, Michael J. C.; Fradin, Lillie; Philips, Kathy; Feldman, Dima; Cooper, Leon N. (2005), "On sums of consecutive integers", Quart. Appl. Math., 63 (4): 791–792, doi:10.1090/S0033-569X-05-00991-1, MR 2187932.
  19. ^ Frenzen, C. L. (1997), "Proof without words: sums of consecutive positive integers", Math. Mag., 70 (4): 294, doi:10.1080/0025570X.1997.11996560, JSTOR 2690871, MR 1573264.
  20. ^ Guy, Robert (1982), "Sums of consecutive integers" (PDF), Fibonacci Quarterly, 20 (1): 36–38, doi:10.1080/00150517.1982.12430026, Zbl 0475.10014.
  21. ^ Apostol, Tom M. (2003), "Sums of consecutive positive integers", The Mathematical Gazette, 87 (508): 98–101, doi:10.1017/S002555720017216X, JSTOR 3620570, S2CID 125202845.
  22. ^ Prielipp, Robert W.; Kuenzi, Norbert J. (1975), "Sums of consecutive positive integers", Mathematics Teacher, 68 (1): 18–21, doi:10.5951/MT.68.1.0018.
  23. ^ Parker, John (1998), "Sums of consecutive integers", Mathematics in School, 27 (2): 8–11.
  24. ^ Mossinghoff, Michael J. (2011), "Enumerating isodiametric and isoperimetric polygons", Journal of Combinatorial Theory, Series A, 118 (6): 1801–1815, doi:10.1016/j.jcta.2011.03.004, MR 2793611