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2013 | 21 | 2 | 103-113

Tytuł artykułu

Polygonal Numbers

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Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
In the article the formal characterization of triangular numbers (famous from [15] and words “EYPHKA! num = Δ+Δ+Δ”) [17] is given. Our primary aim was to formalize one of the items (#42) from Wiedijk’s Top 100 Mathematical Theorems list [33], namely that the sequence of sums of reciprocals of triangular numbers converges to 2. This Mizar representation was written in 2007. As the Mizar language evolved and attributes with arguments were implemented, we decided to extend these lines and we characterized polygonal numbers. We formalized centered polygonal numbers, the connection between triangular and square numbers, and also some equalities involving Mersenne primes and perfect numbers. We gave also explicit formula to obtain from the polygonal number its ordinal index. Also selected congruences modulo 10 were enumerated. Our work basically covers the Wikipedia item for triangular numbers and the Online Encyclopedia of Integer Sequences (http://oeis.org/A000217). An interesting related result [16] could be the proof of Lagrange’s four-square theorem or Fermat’s polygonal number theorem [32].

Wydawca

Rocznik

Tom

21

Numer

2

Strony

103-113

Opis fizyczny

Daty

wydano
2013-06-01

Twórcy

  • Institute of Informatics University of Białystok Akademicka 2, 15-267 Białystok Poland

Bibliografia

  • [1] Kenichi Arai and Hiroyuki Okazaki. Properties of primes and multiplicative group of a field. Formalized Mathematics, 17(2):151-155, 2009. doi:10.2478/v10037-009-0017-7.[Crossref]
  • [2] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
  • [3] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
  • [4] Grzegorz Bancerek and Krzysztof Hryniewiecki. Segments of natural numbers and finite sequences. Formalized Mathematics, 1(1):107-114, 1990.
  • [5] Czesław Bylinski. Finite sequences and tuples of elements of a non-empty sets. Formalized Mathematics, 1(3):529-536, 1990.
  • [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
  • [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
  • [8] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
  • [9] Czesław Bylinski. The sum and product of finite sequences of real numbers. Formalized Mathematics, 1(4):661-668, 1990.
  • [10] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
  • [11] Agata Darmochwał. Finite sets. Formalized Mathematics, 1(1):165-167, 1990.
  • [12] Yuzhong Ding and Xiquan Liang. Solving roots of polynomial equation of degree 2 and 3 with complex coefficients. Formalized Mathematics, 12(2):85-92, 2004.
  • [13] Yoshinori Fujisawa, Yasushi Fuwa, and Hidetaka Shimizu. Public-key cryptography and Pepin’s test for the primality of Fermat numbers. Formalized Mathematics, 7(2):317-321, 1998.
  • [14] Yuichi Futa, Hiroyuki Okazaki, Daichi Mizushima, and Yasunari Shidama. Operations of points on elliptic curve in projective coordinates. Formalized Mathematics, 20(1):87-95, 2012. doi:10.2478/v10037-012-0012-2.[Crossref]
  • [15] Carl Friedrich Gauss. Disquisitiones Arithmeticae. Springer, New York, 1986. English translation.
  • [16] Richard K. Guy. Every number is expressible as a sum of how many polygonal numbers? American Mathematical Monthly, 101:169-172, 1994.
  • [17] Thomas L. Heath. A History of Greek Mathematics: From Thales to Euclid, Vol. I. Courier Dover Publications, 1921.
  • [18] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
  • [19] Jarosław Kotowicz. Real sequences and basic operations on them. Formalized Mathematics, 1(2):269-272, 1990.
  • [20] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
  • [21] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887-890, 1990.
  • [22] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829-832, 1990.
  • [23] Robert Milewski. Natural numbers. Formalized Mathematics, 7(1):19-22, 1998.
  • [24] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
  • [25] Konrad Raczkowski and Andrzej Nedzusiak. Series. Formalized Mathematics, 2(4):449-452, 1991.
  • [26] Marco Riccardi. The perfect number theorem and Wilson’s theorem. Formalized Mathematics, 17(2):123-128, 2009. doi:10.2478/v10037-009-0013-y.[Crossref]
  • [27] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335-338, 1997.
  • [28] Andrzej Trybulec. On the sets inhabited by numbers. Formalized Mathematics, 11(4): 341-347, 2003.
  • [29] Andrzej Trybulec and Czesław Bylinski. Some properties of real numbers. Formalized Mathematics, 1(3):445-449, 1990.
  • [30] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
  • [31] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
  • [32] André Weil. Number Theory. An Approach through History from Hammurapi to Legendre. Birkh¨auser, Boston, Mass., 1983.
  • [33] Freek Wiedijk. Formalizing 100 theorems.
  • [34] Freek Wiedijk. Pythagorean triples. Formalized Mathematics, 9(4):809-812, 2001.
  • [35] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

Typ dokumentu

Bibliografia

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Identyfikator YADDA

bwmeta1.element.doi-10_2478_forma-2013-0012
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