# Polygonal Numbers

Formalized Mathematics (2013)

- Volume: 21, Issue: 2, page 103-113
- ISSN: 1426-2630

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topAdam Grabowski. "Polygonal Numbers." Formalized Mathematics 21.2 (2013): 103-113. <http://eudml.org/doc/266886>.

@article{AdamGrabowski2013,

abstract = {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].},

author = {Adam Grabowski},

journal = {Formalized Mathematics},

keywords = {triangular number; polygonal number; reciprocals of triangular numbers},

language = {eng},

number = {2},

pages = {103-113},

title = {Polygonal Numbers},

url = {http://eudml.org/doc/266886},

volume = {21},

year = {2013},

}

TY - JOUR

AU - Adam Grabowski

TI - Polygonal Numbers

JO - Formalized Mathematics

PY - 2013

VL - 21

IS - 2

SP - 103

EP - 113

AB - 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].

LA - eng

KW - triangular number; polygonal number; reciprocals of triangular numbers

UR - http://eudml.org/doc/266886

ER -

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