@Article{NEWTON07.MIZ,
  author          = {Ziobro, Rafa{\l}},
  title           = {Pascal's Triangle and {L}ucas's Theorem},
  year            = {2024},
  month = {December},
  day = {24},
  ADDRESS1 = {Department of Carbohydrate Technology, University of Agriculture in Krakow},
  abstract        = {The use of Pascal's triangle as an object of mathematical studies has occurred
   much earlier than its name may suggest, but still gathers a lot of interest due to its relevance
   to scientific studies and teaching \cite {Lapis2012} \cite {Smoluk_2017}.
   The article illustrates the creation of the Triangle in Mizar using finite sequences, and shows
   similar constructions developed more recently \cite {Anggoro_2010}.  Additionally it provides some simple
   proofs about the divisibility of factorials and binomial coefficients,
   and the Lucas's theorem \cite {granville1997arithmetic},
   in the form sometimes reffered to as Anton's Lemmma \cite{Bes_1997},
   recently formalized for Isabelle/HOL \cite{edmonds_2023}.},
  acknowledgement = {Ad Maiorem Dei Gloriam},
  keywords        = {arithmetic triangle; binomial coefficient; },
  msc2010         = {11Y55 11A67 03B35},
  section1        = {Preliminaries},
  section2        = {Divisibility of Factorials},
  section3        = {Lucas's Theorem},
  section4        = {Pascal's Triangle and Other Triangles},
}


@Article{Bes_1997,
  author    = {Bés, Alexis},
  journal   = {Annals of Pure and Applied Logic},
  title     = {On {P}ascal triangles modulo a prime power},
  year      = {1997},
  issn      = {0168-0072},
  month     = dec,
  number    = {1},
  pages     = {17--35},
  volume    = {89},
  doi       = {10.1016/s0168-0072(97)85376-6},
  publisher = {Elsevier BV},
}

@InProceedings{granville1997arithmetic,
  author       = {Granville, Andrew},
  booktitle    = {Organic Mathematics: Proceedings of the Organic Mathematics Workshop.},
  title        = {Arithmetic properties of binomial coefficients. I. Binomial coefficients modulo prime powers},
  year         = {1997},
  address      = {Burnaby, BC},
  editor       = {Jonathan M. Borwein},
  month        = dec,
  organization = {American Mathematical Soc.},
  pages        = {253--276},
  series       = {CMS conference proceedings},
  volume       = {20},
  isbn         = {9780821806685},
  lccn         = {97000179},
  url          = {https://books.google.pl/books?id=bEgtaaU0QRoC},
}

@Article{Anggoro_2010,
  author    = {Anggoro, Alif and Liu, Eddy and Tulloch, Angus},
  journal   = {The College Mathematics Journal},
  title     = {The Rascal Triangle},
  year      = {2010},
  issn      = {1931-1346},
  month     = nov,
  number    = {5},
  pages     = {393--395},
  volume    = {41},
  doi       = {10.4169/074683410x521991},
  publisher = {Informa UK Limited},
}

@Article{Lapis2012,
  author       = {Lapis, Włodzimierz},
  journal      = {Investigationes Linguisticae},
  title        = {Dystynktywność ciągów},
  year         = {2012},
  month        = {mar.},
  pages        = {58–71},
  volume       = {25},
  doi          = {10.14746/il.2012.25.4},
  url          = {https://pressto.amu.edu.pl/index.php/il/article/view/9760},
}

@Comment{jabref-meta: databaseType:bibtex;}
