Sobre os números P-ádicos: aspectos históricos, matemáticos e epistemológicos
DOI:
10.47976/RBHM2024v24n481-32Palavras-chave:
Números p-ádicos, História da Matemática, Intuição, Topologia dos números p-ádicosResumo
Reconhecidamente, o processo de engendramento evolutivo matemático que concorreu para a evolução e a determinação de bases sólidas para a noção de número real constitui um emblemático divisor de águas para a Matemática no século XIX. Assim, a partir de um conhecimento atual de vários métodos e formas de determinação e do completamento do corpo dos números racionais, a despeito do método empregado, podemos indicar a natureza de um corpo arquimediano e completo que nominamos por conjunto dos números reais. Não obstante, o presente trabalho apresenta uma discussão de elementos históricos, matemáticos e epistemológicos sobre uma outra classe de números que representa, também, um completamento do conjunto dos números racionais e que se denota por , onde é um número primo. Dessa forma, os números p-ádicos são apresentados, tendo em vista proporcionar ao leitor uma vital compreensão do seu papel epistemológico que concorre para um entendimemo diferenciado e substancialmente ampliado para a noção abstrata e basilar em Matemática do que conhecemos ordinariamente por número.
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Referências
ALVES, Francisco, R. V. (2017). Fórmula de de moivre, ou de binet ou de lamé: demonstrações e generalidades sobre a sequência generalizada de fibonacci – SGF. Revista Brasileira de História da Matemática, v, 17, nº 33, 1 – 16.
AMICE, Ivete. (1975). Le nombres p-adique. Paris: Presses Universitaire. Disponível em: https://www-fourier.ujf-grenoble.fr/~panchish/GDL16/amice_les-nombres-p-adiques%20%5B2755599%5D.pdf
AÇIKGÖZ, Mehmet; ASLAN,Nurgül; Köşkeroğlu, Nurten e Aracı, Serkan. (2009). p-Adic Approach to Linear 2-Normed Spaces. Mathematica Moravica, v. 13, nº 2, 7 – 21. Disponível em: http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2009/1450-59320902007A.pdf
ADERBURG, Drew. (2002). A Mathematical Seduction. Math Horizons, 9:3, 12-15. Disponível em: https://web.williams.edu/Mathematics/eburger/BurgerMathHorizons.pdf
BELAIR, Luc. (2012). Panorama of p-adic model theory, Annales des Sciences Mathematiques du Quebec, nº 36, p. 43-75, 2012. Disponível em: http://www.logique.jussieu.fr/modnet/Publications/Introductory%20Notes%20and%20surveys/Belair.pdf
BOREVICH, Z. I.; SHAFAREVICH, I. R. (1966). Number Theory. New York: Academic Press.
BURGER, Edward, B. (2000). Exploring the number jungle: a journey into diophantine analysis. New York: Americal Mathematical Society.
CĂLIN, Mureşan Alexe. (2006). Non-Archimedian Fields. Topological Properties of Zp, Qp (p-adics Numbers). Matematică - Informatică – Fizică, v. 63, nº 2, 43 – 48. Disponível em: http://bmif.unde.ro/docs/20062/7%20MuresanA.pdf
CARVALHO, Maria Pires de; LOURENÇO, João Nuno P. (2015). Convergência de séries p-ádicas. Boletim da Sociedade Portuguesa de Matemática, 1 – 30, Disponível em: https://repositorio-aberto.up.pt/bitstream/10216/90725/2/104694.pdf
CHINEA, C. (2000). Matemática. ¿qué son los números p-adicos?. carlos s. chinea, octubr. Divulgación de la matemática en la red. 1 – 5. Disponível em: https://casanchi.com/casanchi_2000/19_padicos01.pdf
CORNELISSEN, Gunther; KATO, Fumiharu. (2005). The p-adic icosaedron. Notes in American Mathematical Association - AMS. v. 52, nº 7, 720 – 727. Disponível em: https://www.ams.org/notices/200507/fea-cornelissen.pdf
COHEN, Henri. (2007). Number Theory Volume I: Tools and Diophantine Equations. New York: Springer.
CORRY, Leo. (2004). Modern Algebra and the Rise of Mathematical Structures, Basel and Boston, Birkhäuser.
CUOCO, Albert. A. Visualizing the p-adic integers. American Mathematical Monthly. v. 98, nº 4, 355 – 364. Disponível em: https://www.jstor.org/stable/pdf/2323809.pdf?refreqid=excelsior%3Ada3401d345ac7e62ccac8f6378fb2cfc
CROMPTON, Catherine (2007) Some Geometry of the p-adic rationals. Rose-Hulman Undergraduate Mathematics Journal: v. 8 : Iss. 1 , Article 2. 1 – 13. Disponível em: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?referer=https://www.google.com.br/&httpsredir=1&article=1183&context=rhumj
DELAHAYE. Jean-Paul. (2001). Les nombres infinit vers la gauche. Pour la Science, nº 279, 100 – 104. Disponível em: http://www.lifl.fr/~jdelahay/pls/081.pdf
EHRLICH. Philip. (2006). The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes. The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Archive History in Exact Sciences. V. 60, 1 – 121. Disponível em: http://prima.lnu.edu.ua/faculty/mechmat/Departments/MFAUKR/attachments/erlich.pdf
FERREIRA, Jamil. (2010). A construção dos números. Textos Universitários. Rio de Janeiro: SBM.
GOLDBLATT, Robert. (1998). Lectures on the Hyprreeals numbers: an introduction to non standard analisys. New York: Springer.
GOUVÊA, F. Q. (1997). p-adic numbers. New York: Springer.
GUSMÃO, Italo, B. (2016). Números p-ádicos. Dissertação de Mestrado Profissional PROFMAT, João Pessoa: UFPB.. Disponível em: https://repositorio.ufpb.br/jspui/bitstream/tede/9337/2/arquivototal.pdf
GAUTSCHI, Walter. Alessandro M. (2002). Ostrowski (1893–1986): la sua vita e le opere. Boll. Docenti Matem. v. 45, 9–19. Disponível em: https://www.cs.purdue.edu/homes/wxg/AMOital.pdf
HASSE, Helmut. (1980). Number Theory. Springer: New York.
HEFEZ, Abramo. (2013). Um curso de Àlgebra, v. 1, Rio de Janeiro: SBM.
HOLLY, Jan E. (2001). Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields. The mathematical association of America. October, 721 – 728. Disponível em: https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf
HUAMAN, Ronald M. (2015). Raıces p-adicas de la unidad. (Tesis de Maestrıa). Pontificia Universidad del Perú: Lima. Disponível em: http://tesis.pucp.edu.pe/repositorio/bitstream/handle/123456789/6415/MAS_HUAMAN_RONALD_PADICAS.pdf;sequence=1
LAGES, Elon. (2010). Curso de Análise, v. 1. Rio de Janeiro: SBM.
LAPIDUS, Michel L.; HUNG. Lu. (2011). The Geometry of p-Adic Fractal Strings: A Comparative Survey. Contemporary Mathematics, v. 551, 163 – 206. Disponível em: http://www.math.ucr.edu/~lapidus/papers/ContMath/GeometrypAdicStringsSurvey10893.pdf
LAPIDUS, Michel L.; HUNG. Lu. (2008). Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and Applications. v. 4, p. 1 – 10. Disponível em: https://pdfs.semanticscholar.org/6dee/45208234f48fe156d174d4ad56b17363d2ef.pdf
LEGUAY, Mathieu; HENRI, Joseph. (1992). Distance p-adique : une distance qui n'est pas habituelle. MATh.en.JEANS” au Palais de la Découverte. 33 – 36. Disponível em: http://mathenjeans.free.fr/amej/edition/actes/actespdf/92033036.pdf
MARCOS, José. E. (2006). The algebraic closure of a p-adic number field is a complete topological field. Mathematica Slovaca,v. 56, No. 3, p. 317—331. Disponível em: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1010.7259&rep=rep1&type=pdf
MASIAS, Henry Zorrilla. (2011). Complecion no arquimedeana. Pro Mathematica, v. 25, 49-50. Disponível em: http://revistas.pucp.edu.pe/index.php/promathematica/article/viewFile/2666/2610
NARICI, Lawrence; BECKENSTEIN, Edward. (1981). Strange Terrain--Nonarchimedean Spaces. The American Mathematical Monthly, Vol. 88, No. 9 (Nov., 1981), pp. 667-676. Disponível em: https://www.jstor.org/stable/pdf/2320670.pdf
OSTROWSKI, A. (1916). Uber einige L¨osungen der Funktionalgleichung ¨ φ(x)φ(y) = φ(xy)”, Acta Mathematica (2nd ed.), nº 41, no. 1, 271–284. Disponível em: https://projecteuclid.org/euclid.acta/1485887472
KATO, Kazuya; KUROKAWA, Nobushige; SAITO, Takeshi, (2000). Number theory 1: Fermat’s dream, Translations of Mathematical Monographs, vol. 186, American Mathematical Society, Providence, RI.
KATOK, S. (2007). p-adic analysis compared with real. Student mathematical library, American Mathematical Soc.
KOBLITZ, Neal. (1984). p-adic Numbers, p-adic Analysis, and Zeta- Functions. New York: Springer.
KHRENNIKOV, Andrei; KOTOVICHM, Nikolay. (2017). Image Segmentation with the Aid of the p-Adic Metrics. In: TONI, Bourama. New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. 143 – 155.
KUMAR, Ashish; RANI, Mamta; CHUGH, Renu. (2013). New 5-adic Cantor sets and fractal string. SpringerPlus, 1 – 7. Disponível em: https://link.springer.com/content/pdf/10.1186%2F2193-1801-2-654.pdf
LEWIS, Robert Y. (2019). A formal proof of Hensel’s lemma over the p-adic integers. CPP’19, January p. 14–15, Lisbon, PT. Disponível em: http://robertylewis.com/padics/padics.pdf
NATARAJAN, P N; RANGANATHAN, K. N. (2000). A Geometry in which all Triangles are Isosceles An Introduction to Non-Archimedean Analysis. Resonance, October, 32 – 42. Disponível em: https://www.ias.ac.in/article/fulltext/reso/005/10/0032-0042
NEUKIRCH, J. (1990). The p-Adic numbers. EBBINGHAUS, H.-D. Numbers. New York: Springer. 155 – 179.
OLIVEIRA, Graciano. (2009). O corpo dos p-ádicos. Gazeta Matemática, Dezembro, 7 – 18. Disponível em: http://gazeta.spm.pt/getArtigo?gid=258
ORBEGOSO, Jorge Luis Rojas. Completitud y clausura algebraica de campos P-ádicos. Universidad nacional mayor de san marcos. Perú: Lima, 2016. Disponível em: http://cybertesis.unmsm.edu.pe/bitstream/handle/cybertesis/6315/Rojas_oj.pdf?sequence=1
PITANEN, Matti. (2015). How Imagination Could Be Realized p-Adically?. Journal of Consciousness Exploration & Research, v 6, nº 6, 354 – 356. Disponível em: https://jcer.com/index.php/jcj/article/viewFile/468/488
REDDY, B. Surender; SHANKARAIAH, D. (2013). On i-cauchy sequences in p-adic linear 2-normed spaces. International Journal of Pure and Applied Mathematics. v. 89, nº 4, 483 – 496. Disponivel em: https://ijpam.eu/contents/2013-89-4/4/4.pdf
RIBENBOIM, Paulo. (1999). The Theory of Classical Valuations. New York: Springer.
ROBERT, Alain. (1996). Qu´est-qe que le nombres p- ádique? Societé des enseignants neuchâtelois des Sciences. Bulletin, setembre, 5 – 12. Disponível em: http://www.sens-neuchatel.ch/bulletin/anciens-no-pdf/BULL18.PDF
ROZIKOV, U. A. (2013). What are p-Adic Numbers? What are They Used for?. Asia Pacific Mathematics Newsletter. v. 3, nº 4, 1 – 6, October. Disponível em: http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf
SCHIKHOF W. H. (1984). Ultrametric Calculus: An introduction to p-adic Analysis. Cambridge: University Press.
SCHLICHENMAIER, Martin. (2007). An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Springer: New York.
SILVERMANM, Joseph H. (2013). What is the p-adic Mandelbrot Set?. Notices of the AMS, setember, v. 60, nº 8, 1048 – 1050. Disponível em: http://www.ams.org/notices/201308/201308-full-issue.pdf
STEUDING, Jorn. (2002). The world of p-adic numbers and p-adic functions. Faculty of Physics and Mathematics. nº 5, 90 – 107. Disponível em: http://siauliaims.su.lt/pdfai/2002/STEUD-02.pdf
TEITELBAUM. Jeremy. (1995). The Geometry of p-adic Symmetric Spaces. Notice in American Mathematical Monthly. v. 42, nº 10, 1120 – 1126. Disponível em: https://www.ams.org/notices/199510/teitelbaum.pdf
TELLER, Jacek. (2012). Newton polygons on p-adic number fields. (dissertation in Arts and Mathematics). East Caroline University. Disponível em: http://thescholarship.ecu.edu/bitstream/handle/10342/3848/Teller_ecu_0600M_10676.pdf?sequence=1
TORRES, Sergio Carrillo; AMAYA, Carlos Hurtado. (2001). Una introduccion a los numeros p-adicos. Memorias XVIII encuentro de geometrıa y VI de aritmetica, 359 – 369. Disponível em: http://funes.uniandes.edu.co/5615/1/CarrilloUnaintroduccionGeometr%C3%ADa2008.PDF
VACCON. Tristan. (2015). Précision p-adique. (thésis de doctorat). Rennes: Université de Rennes. Disponível em: https://tel.archives-ouvertes.fr/tel-01205269v2/document
ALVES, Francisco, R. V. (2017). Fórmula de de moivre, ou de binet ou de lamé: demonstrações e generalidades sobre a sequência generalizada de fibonacci – SGF. Revista Brasileira de História da Matemática, v, 17, nº 33, 1 – 16.
AMICE, Ivete. (1975). Le nombres p-adique. Paris: Presses Universitaire. Disponível em: https://www-fourier.ujf-grenoble.fr/~panchish/GDL16/amice_les-nombres-p-adiques%20%5B2755599%5D.pdf
AÇIKGÖZ, Mehmet; ASLAN,Nurgül; Köşkeroğlu, Nurten e Aracı, Serkan. (2009). p-Adic Approach to Linear 2-Normed Spaces. Mathematica Moravica, v. 13, nº 2, 7 – 21. Disponível em: http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2009/1450-59320902007A.pdf
ADERBURG, Drew. (2002). A Mathematical Seduction. Math Horizons, 9:3, 12-15. Disponível em: https://web.williams.edu/Mathematics/eburger/BurgerMathHorizons.pdf
BELAIR, Luc. (2012). Panorama of p-adic model theory, Annales des Sciences Mathematiques du Quebec, nº 36, p. 43-75, 2012. Disponível em: http://www.logique.jussieu.fr/modnet/Publications/Introductory%20Notes%20and%20surveys/Belair.pdf
BOREVICH, Z. I.; SHAFAREVICH, I. R. (1966). Number Theory. New York: Academic Press.
BURGER, Edward, B. (2000). Exploring the number jungle: a journey into diophantine analysis. New York: Americal Mathematical Society.
CĂLIN, Mureşan Alexe. (2006). Non-Archimedian Fields. Topological Properties of Zp, Qp (p-adics Numbers). Matematică - Informatică – Fizică, v. 63, nº 2, 43 – 48. Disponível em: http://bmif.unde.ro/docs/20062/7%20MuresanA.pdf
CARVALHO, Maria Pires de; LOURENÇO, João Nuno P. (2015). Convergência de séries p-ádicas. Boletim da Sociedade Portuguesa de Matemática, 1 – 30, Disponível em: https://repositorio-aberto.up.pt/bitstream/10216/90725/2/104694.pdf
CHINEA, C. (2000). Matemática. ¿qué son los números p-adicos?. carlos s. chinea, octubr. Divulgación de la matemática en la red. 1 – 5. Disponível em: https://casanchi.com/casanchi_2000/19_padicos01.pdf
CORNELISSEN, Gunther; KATO, Fumiharu. (2005). The p-adic icosaedron. Notes in American Mathematical Association - AMS. v. 52, nº 7, 720 – 727. Disponível em: https://www.ams.org/notices/200507/fea-cornelissen.pdf
COHEN, Henri. (2007). Number Theory Volume I: Tools and Diophantine Equations. New York: Springer.
CORRY, Leo. (2004). Modern Algebra and the Rise of Mathematical Structures, Basel and Boston, Birkhäuser.
CUOCO, Albert. A. Visualizing the p-adic integers. American Mathematical Monthly. v. 98, nº 4, 355 – 364. Disponível em: https://www.jstor.org/stable/pdf/2323809.pdf?refreqid=excelsior%3Ada3401d345ac7e62ccac8f6378fb2cfc
CROMPTON, Catherine (2007) Some Geometry of the p-adic rationals. Rose-Hulman Undergraduate Mathematics Journal: v. 8 : Iss. 1 , Article 2. 1 – 13. Disponível em: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?referer=https://www.google.com.br/&httpsredir=1&article=1183&context=rhumj
DELAHAYE. Jean-Paul. (2001). Les nombres infinit vers la gauche. Pour la Science, nº 279, 100 – 104. Disponível em: http://www.lifl.fr/~jdelahay/pls/081.pdf
EHRLICH. Philip. (2006). The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes. The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Archive History in Exact Sciences. V. 60, 1 – 121. Disponível em: http://prima.lnu.edu.ua/faculty/mechmat/Departments/MFAUKR/attachments/erlich.pdf
FERREIRA, Jamil. (2010). A construção dos números. Textos Universitários. Rio de Janeiro: SBM.
GOLDBLATT, Robert. (1998). Lectures on the Hyprreeals numbers: an introduction to non standard analisys. New York: Springer.
GOUVÊA, F. Q. (1997). p-adic numbers. New York: Springer.
GUSMÃO, Italo, B. (2016). Números p-ádicos. Dissertação de Mestrado Profissional PROFMAT, João Pessoa: UFPB.. Disponível em: https://repositorio.ufpb.br/jspui/bitstream/tede/9337/2/arquivototal.pdf
GAUTSCHI, Walter. Alessandro M. (2002). Ostrowski (1893–1986): la sua vita e le opere. Boll. Docenti Matem. v. 45, 9–19. Disponível em: https://www.cs.purdue.edu/homes/wxg/AMOital.pdf
HASSE, Helmut. (1980). Number Theory. Springer: New York.
HEFEZ, Abramo. (2013). Um curso de Àlgebra, v. 1, Rio de Janeiro: SBM.
HOLLY, Jan E. (2001). Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields. The mathematical association of America. October, 721 – 728. Disponível em: https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf
HUAMAN, Ronald M. (2015). Raıces p-adicas de la unidad. (Tesis de Maestrıa). Pontificia Universidad del Perú: Lima. Disponível em: http://tesis.pucp.edu.pe/repositorio/bitstream/handle/123456789/6415/MAS_HUAMAN_RONALD_PADICAS.pdf;sequence=1
LAGES, Elon. (2010). Curso de Análise, v. 1. Rio de Janeiro: SBM.
LAPIDUS, Michel L.; HUNG. Lu. (2011). The Geometry of p-Adic Fractal Strings: A Comparative Survey. Contemporary Mathematics, v. 551, 163 – 206. Disponível em: http://www.math.ucr.edu/~lapidus/papers/ContMath/GeometrypAdicStringsSurvey10893.pdf
LAPIDUS, Michel L.; HUNG. Lu. (2008). Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and Applications. v. 4, p. 1 – 10. Disponível em: https://pdfs.semanticscholar.org/6dee/45208234f48fe156d174d4ad56b17363d2ef.pdf
LEGUAY, Mathieu; HENRI, Joseph. (1992). Distance p-adique : une distance qui n'est pas habituelle. MATh.en.JEANS” au Palais de la Découverte. 33 – 36. Disponível em: http://mathenjeans.free.fr/amej/edition/actes/actespdf/92033036.pdf
MARCOS, José. E. (2006). The algebraic closure of a p-adic number field is a complete topological field. Mathematica Slovaca,v. 56, No. 3, p. 317—331. Disponível em: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1010.7259&rep=rep1&type=pdf
MASIAS, Henry Zorrilla. (2011). Complecion no arquimedeana. Pro Mathematica, v. 25, 49-50. Disponível em: http://revistas.pucp.edu.pe/index.php/promathematica/article/viewFile/2666/2610
NARICI, Lawrence; BECKENSTEIN, Edward. (1981). Strange Terrain--Nonarchimedean Spaces. The American Mathematical Monthly, Vol. 88, No. 9 (Nov., 1981), pp. 667-676. Disponível em: https://www.jstor.org/stable/pdf/2320670.pdf
OSTROWSKI, A. (1916). Uber einige L¨osungen der Funktionalgleichung ¨ φ(x)φ(y) = φ(xy)”, Acta Mathematica (2nd ed.), nº 41, no. 1, 271–284. Disponível em: https://projecteuclid.org/euclid.acta/1485887472
KATO, Kazuya; KUROKAWA, Nobushige; SAITO, Takeshi, (2000). Number theory 1: Fermat’s dream, Translations of Mathematical Monographs, vol. 186, American Mathematical Society, Providence, RI.
KATOK, S. (2007). p-adic analysis compared with real. Student mathematical library, American Mathematical Soc.
KOBLITZ, Neal. (1984). p-adic Numbers, p-adic Analysis, and Zeta- Functions. New York: Springer.
KHRENNIKOV, Andrei; KOTOVICHM, Nikolay. (2017). Image Segmentation with the Aid of the p-Adic Metrics. In: TONI, Bourama. New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. 143 – 155.
KUMAR, Ashish; RANI, Mamta; CHUGH, Renu. (2013). New 5-adic Cantor sets and fractal string. SpringerPlus, 1 – 7. Disponível em: https://link.springer.com/content/pdf/10.1186%2F2193-1801-2-654.pdf
LEWIS, Robert Y. (2019). A formal proof of Hensel’s lemma over the p-adic integers. CPP’19, January p. 14–15, Lisbon, PT. Disponível em: http://robertylewis.com/padics/padics.pdf
NATARAJAN, P N; RANGANATHAN, K. N. (2000). A Geometry in which all Triangles are Isosceles An Introduction to Non-Archimedean Analysis. Resonance, October, 32 – 42. Disponível em: https://www.ias.ac.in/article/fulltext/reso/005/10/0032-0042
NEUKIRCH, J. (1990). The p-Adic numbers. EBBINGHAUS, H.-D. Numbers. New York: Springer. 155 – 179.
OLIVEIRA, Graciano. (2009). O corpo dos p-ádicos. Gazeta Matemática, Dezembro, 7 – 18. Disponível em: http://gazeta.spm.pt/getArtigo?gid=258
ORBEGOSO, Jorge Luis Rojas. Completitud y clausura algebraica de campos P-ádicos. Universidad nacional mayor de san marcos. Perú: Lima, 2016. Disponível em: http://cybertesis.unmsm.edu.pe/bitstream/handle/cybertesis/6315/Rojas_oj.pdf?sequence=1
PITANEN, Matti. (2015). How Imagination Could Be Realized p-Adically?. Journal of Consciousness Exploration & Research, v 6, nº 6, 354 – 356. Disponível em: https://jcer.com/index.php/jcj/article/viewFile/468/488
REDDY, B. Surender; SHANKARAIAH, D. (2013). On i-cauchy sequences in p-adic linear 2-normed spaces. International Journal of Pure and Applied Mathematics. v. 89, nº 4, 483 – 496. Disponivel em: https://ijpam.eu/contents/2013-89-4/4/4.pdf
RIBENBOIM, Paulo. (1999). The Theory of Classical Valuations. New York: Springer.
ROBERT, Alain. (1996). Qu´est-qe que le nombres p- ádique? Societé des enseignants neuchâtelois des Sciences. Bulletin, setembre, 5 – 12. Disponível em: http://www.sens-neuchatel.ch/bulletin/anciens-no-pdf/BULL18.PDF
ROZIKOV, U. A. (2013). What are p-Adic Numbers? What are They Used for?. Asia Pacific Mathematics Newsletter. v. 3, nº 4, 1 – 6, October. Disponível em: http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf
SCHIKHOF W. H. (1984). Ultrametric Calculus: An introduction to p-adic Analysis. Cambridge: University Press.
SCHLICHENMAIER, Martin. (2007). An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Springer: New York.
SILVERMANM, Joseph H. (2013). What is the p-adic Mandelbrot Set?. Notices of the AMS, setember, v. 60, nº 8, 1048 – 1050. Disponível em: http://www.ams.org/notices/201308/201308-full-issue.pdf
STEUDING, Jorn. (2002). The world of p-adic numbers and p-adic functions. Faculty of Physics and Mathematics. nº 5, 90 – 107. Disponível em: http://siauliaims.su.lt/pdfai/2002/STEUD-02.pdf
TEITELBAUM. Jeremy. (1995). The Geometry of p-adic Symmetric Spaces. Notice in American Mathematical Monthly. v. 42, nº 10, 1120 – 1126. Disponível em: https://www.ams.org/notices/199510/teitelbaum.pdf
TELLER, Jacek. (2012). Newton polygons on p-adic number fields. (dissertation in Arts and Mathematics). East Caroline University. Disponível em: http://thescholarship.ecu.edu/bitstream/handle/10342/3848/Teller_ecu_0600M_10676.pdf?sequence=1
TORRES, Sergio Carrillo; AMAYA, Carlos Hurtado. (2001). Una introduccion a los numeros p-adicos. Memorias XVIII encuentro de geometrıa y VI de aritmetica, 359 – 369. Disponível em: http://funes.uniandes.edu.co/5615/1/CarrilloUnaintroduccionGeometr%C3%ADa2008.PDF
VACCON. Tristan. (2015). Précision p-adique. (thésis de doctorat). Rennes: Université de Rennes. Disponível em: https://tel.archives-ouvertes.fr/tel-01205269v2/document
ALVES, Francisco, R. V. (2017). Fórmula de de moivre, ou de binet ou de lamé: demonstrações e generalidades sobre a sequência generalizada de fibonacci – SGF. Revista Brasileira de História da Matemática, v, 17, nº 33, 1 – 16.
AMICE, Ivete. (1975). Le nombres p-adique. Paris: Presses Universitaire. Disponível em: https://www-fourier.ujf-grenoble.fr/~panchish/GDL16/amice_les-nombres-p-adiques%20%5B2755599%5D.pdf
AÇIKGÖZ, Mehmet; ASLAN,Nurgül; Köşkeroğlu, Nurten e Aracı, Serkan. (2009). p-Adic Approach to Linear 2-Normed Spaces. Mathematica Moravica, v. 13, nº 2, 7 – 21. Disponível em: http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2009/1450-59320902007A.pdf
ADERBURG, Drew. (2002). A Mathematical Seduction. Math Horizons, 9:3, 12-15. Disponível em: https://web.williams.edu/Mathematics/eburger/BurgerMathHorizons.pdf
BELAIR, Luc. (2012). Panorama of p-adic model theory, Annales des Sciences Mathematiques du Quebec, nº 36, p. 43-75, 2012. Disponível em: http://www.logique.jussieu.fr/modnet/Publications/Introductory%20Notes%20and%20surveys/Belair.pdf
BOREVICH, Z. I.; SHAFAREVICH, I. R. (1966). Number Theory. New York: Academic Press.
BURGER, Edward, B. (2000). Exploring the number jungle: a journey into diophantine analysis. New York: Americal Mathematical Society.
CĂLIN, Mureşan Alexe. (2006). Non-Archimedian Fields. Topological Properties of Zp, Qp (p-adics Numbers). Matematică - Informatică – Fizică, v. 63, nº 2, 43 – 48. Disponível em: http://bmif.unde.ro/docs/20062/7%20MuresanA.pdf
CARVALHO, Maria Pires de; LOURENÇO, João Nuno P. (2015). Convergência de séries p-ádicas. Boletim da Sociedade Portuguesa de Matemática, 1 – 30, Disponível em: https://repositorio-aberto.up.pt/bitstream/10216/90725/2/104694.pdf
CHINEA, C. (2000). Matemática. ¿qué son los números p-adicos?. carlos s. chinea, octubr. Divulgación de la matemática en la red. 1 – 5. Disponível em: https://casanchi.com/casanchi_2000/19_padicos01.pdf
CORNELISSEN, Gunther; KATO, Fumiharu. (2005). The p-adic icosaedron. Notes in American Mathematical Association - AMS. v. 52, nº 7, 720 – 727. Disponível em: https://www.ams.org/notices/200507/fea-cornelissen.pdf
COHEN, Henri. (2007). Number Theory Volume I: Tools and Diophantine Equations. New York: Springer.
CORRY, Leo. (2004). Modern Algebra and the Rise of Mathematical Structures, Basel and Boston, Birkhäuser.
CUOCO, Albert. A. Visualizing the p-adic integers. American Mathematical Monthly. v. 98, nº 4, 355 – 364. Disponível em: https://www.jstor.org/stable/pdf/2323809.pdf?refreqid=excelsior%3Ada3401d345ac7e62ccac8f6378fb2cfc
CROMPTON, Catherine (2007) Some Geometry of the p-adic rationals. Rose-Hulman Undergraduate Mathematics Journal: v. 8 : Iss. 1 , Article 2. 1 – 13. Disponível em: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?referer=https://www.google.com.br/&httpsredir=1&article=1183&context=rhumj
DELAHAYE. Jean-Paul. (2001). Les nombres infinit vers la gauche. Pour la Science, nº 279, 100 – 104. Disponível em: http://www.lifl.fr/~jdelahay/pls/081.pdf
EHRLICH. Philip. (2006). The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes. The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Archive History in Exact Sciences. V. 60, 1 – 121. Disponível em: http://prima.lnu.edu.ua/faculty/mechmat/Departments/MFAUKR/attachments/erlich.pdf
FERREIRA, Jamil. (2010). A construção dos números. Textos Universitários. Rio de Janeiro: SBM.
GOLDBLATT, Robert. (1998). Lectures on the Hyprreeals numbers: an introduction to non standard analisys. New York: Springer.
GOUVÊA, F. Q. (1997). p-adic numbers. New York: Springer.
GUSMÃO, Italo, B. (2016). Números p-ádicos. Dissertação de Mestrado Profissional PROFMAT, João Pessoa: UFPB.. Disponível em: https://repositorio.ufpb.br/jspui/bitstream/tede/9337/2/arquivototal.pdf
GAUTSCHI, Walter. Alessandro M. (2002). Ostrowski (1893–1986): la sua vita e le opere. Boll. Docenti Matem. v. 45, 9–19. Disponível em: https://www.cs.purdue.edu/homes/wxg/AMOital.pdf
HASSE, Helmut. (1980). Number Theory. Springer: New York.
HEFEZ, Abramo. (2013). Um curso de Àlgebra, v. 1, Rio de Janeiro: SBM.
HOLLY, Jan E. (2001). Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields. The mathematical association of America. October, 721 – 728. Disponível em: https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf
HUAMAN, Ronald M. (2015). Raıces p-adicas de la unidad. (Tesis de Maestrıa). Pontificia Universidad del Perú: Lima. Disponível em: http://tesis.pucp.edu.pe/repositorio/bitstream/handle/123456789/6415/MAS_HUAMAN_RONALD_PADICAS.pdf;sequence=1
LAGES, Elon. (2010). Curso de Análise, v. 1. Rio de Janeiro: SBM.
LAPIDUS, Michel L.; HUNG. Lu. (2011). The Geometry of p-Adic Fractal Strings: A Comparative Survey. Contemporary Mathematics, v. 551, 163 – 206. Disponível em: http://www.math.ucr.edu/~lapidus/papers/ContMath/GeometrypAdicStringsSurvey10893.pdf
LAPIDUS, Michel L.; HUNG. Lu. (2008). Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and Applications. v. 4, p. 1 – 10. Disponível em: https://pdfs.semanticscholar.org/6dee/45208234f48fe156d174d4ad56b17363d2ef.pdf
LEGUAY, Mathieu; HENRI, Joseph. (1992). Distance p-adique : une distance qui n'est pas habituelle. MATh.en.JEANS” au Palais de la Découverte. 33 – 36. Disponível em: http://mathenjeans.free.fr/amej/edition/actes/actespdf/92033036.pdf
MARCOS, José. E. (2006). The algebraic closure of a p-adic number field is a complete topological field. Mathematica Slovaca,v. 56, No. 3, p. 317—331. Disponível em: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1010.7259&rep=rep1&type=pdf
MASIAS, Henry Zorrilla. (2011). Complecion no arquimedeana. Pro Mathematica, v. 25, 49-50. Disponível em: http://revistas.pucp.edu.pe/index.php/promathematica/article/viewFile/2666/2610
NARICI, Lawrence; BECKENSTEIN, Edward. (1981). Strange Terrain--Nonarchimedean Spaces. The American Mathematical Monthly, Vol. 88, No. 9 (Nov., 1981), pp. 667-676. Disponível em: https://www.jstor.org/stable/pdf/2320670.pdf
OSTROWSKI, A. (1916). Uber einige L¨osungen der Funktionalgleichung ¨ φ(x)φ(y) = φ(xy)”, Acta Mathematica (2nd ed.), nº 41, no. 1, 271–284. Disponível em: https://projecteuclid.org/euclid.acta/1485887472
KATO, Kazuya; KUROKAWA, Nobushige; SAITO, Takeshi, (2000). Number theory 1: Fermat’s dream, Translations of Mathematical Monographs, vol. 186, American Mathematical Society, Providence, RI.
KATOK, S. (2007). p-adic analysis compared with real. Student mathematical library, American Mathematical Soc.
KOBLITZ, Neal. (1984). p-adic Numbers, p-adic Analysis, and Zeta- Functions. New York: Springer.
KHRENNIKOV, Andrei; KOTOVICHM, Nikolay. (2017). Image Segmentation with the Aid of the p-Adic Metrics. In: TONI, Bourama. New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. 143 – 155.
KUMAR, Ashish; RANI, Mamta; CHUGH, Renu. (2013). New 5-adic Cantor sets and fractal string. SpringerPlus, 1 – 7. Disponível em: https://link.springer.com/content/pdf/10.1186%2F2193-1801-2-654.pdf
LEWIS, Robert Y. (2019). A formal proof of Hensel’s lemma over the p-adic integers. CPP’19, January p. 14–15, Lisbon, PT. Disponível em: http://robertylewis.com/padics/padics.pdf
NATARAJAN, P N; RANGANATHAN, K. N. (2000). A Geometry in which all Triangles are Isosceles An Introduction to Non-Archimedean Analysis. Resonance, October, 32 – 42. Disponível em: https://www.ias.ac.in/article/fulltext/reso/005/10/0032-0042
NEUKIRCH, J. (1990). The p-Adic numbers. EBBINGHAUS, H.-D. Numbers. New York: Springer. 155 – 179.
OLIVEIRA, Graciano. (2009). O corpo dos p-ádicos. Gazeta Matemática, Dezembro, 7 – 18. Disponível em: http://gazeta.spm.pt/getArtigo?gid=258
ORBEGOSO, Jorge Luis Rojas. Completitud y clausura algebraica de campos P-ádicos. Universidad nacional mayor de san marcos. Perú: Lima, 2016. Disponível em: http://cybertesis.unmsm.edu.pe/bitstream/handle/cybertesis/6315/Rojas_oj.pdf?sequence=1
PITANEN, Matti. (2015). How Imagination Could Be Realized p-Adically?. Journal of Consciousness Exploration & Research, v 6, nº 6, 354 – 356. Disponível em: https://jcer.com/index.php/jcj/article/viewFile/468/488
REDDY, B. Surender; SHANKARAIAH, D. (2013). On i-cauchy sequences in p-adic linear 2-normed spaces. International Journal of Pure and Applied Mathematics. v. 89, nº 4, 483 – 496. Disponivel em: https://ijpam.eu/contents/2013-89-4/4/4.pdf
RIBENBOIM, Paulo. (1999). The Theory of Classical Valuations. New York: Springer.
ROBERT, Alain. (1996). Qu´est-qe que le nombres p- ádique? Societé des enseignants neuchâtelois des Sciences. Bulletin, setembre, 5 – 12. Disponível em: http://www.sens-neuchatel.ch/bulletin/anciens-no-pdf/BULL18.PDF
ROZIKOV, U. A. (2013). What are p-Adic Numbers? What are They Used for?. Asia Pacific Mathematics Newsletter. v. 3, nº 4, 1 – 6, October. Disponível em: http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf
SCHIKHOF W. H. (1984). Ultrametric Calculus: An introduction to p-adic Analysis. Cambridge: University Press.
SCHLICHENMAIER, Martin. (2007). An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Springer: New York.
SILVERMANM, Joseph H. (2013). What is the p-adic Mandelbrot Set?. Notices of the AMS, setember, v. 60, nº 8, 1048 – 1050. Disponível em: http://www.ams.org/notices/201308/201308-full-issue.pdf
STEUDING, Jorn. (2002). The world of p-adic numbers and p-adic functions. Faculty of Physics and Mathematics. nº 5, 90 – 107. Disponível em: http://siauliaims.su.lt/pdfai/2002/STEUD-02.pdf
TEITELBAUM. Jeremy. (1995). The Geometry of p-adic Symmetric Spaces. Notice in American Mathematical Monthly. v. 42, nº 10, 1120 – 1126. Disponível em: https://www.ams.org/notices/199510/teitelbaum.pdf
TELLER, Jacek. (2012). Newton polygons on p-adic number fields. (dissertation in Arts and Mathematics). East Caroline University. Disponível em: http://thescholarship.ecu.edu/bitstream/handle/10342/3848/Teller_ecu_0600M_10676.pdf?sequence=1
TORRES, Sergio Carrillo; AMAYA, Carlos Hurtado. (2001). Una introduccion a los numeros p-adicos. Memorias XVIII encuentro de geometrıa y VI de aritmetica, 359 – 369. Disponível em: http://funes.uniandes.edu.co/5615/1/CarrilloUnaintroduccionGeometr%C3%ADa2008.PDF
VACCON. Tristan. (2015). Précision p-adique. (thésis de doctorat). Rennes: Université de Rennes. Disponível em: https://tel.archives-ouvertes.fr/tel-01205269v2/document
AMICE, Ivete. (1975). Le nombres p-adique. Paris: Presses Universitaire. Disponível em: https://www-fourier.ujf-grenoble.fr/~panchish/GDL16/amice_les-nombres-p-adiques%20%5B2755599%5D.pdf
AÇIKGÖZ, Mehmet; ASLAN,Nurgül; Köşkeroğlu, Nurten e Aracı, Serkan. (2009). p-Adic Approach to Linear 2-Normed Spaces. Mathematica Moravica, v. 13, nº 2, 7 – 21. Disponível em: http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2009/1450-59320902007A.pdf
ADERBURG, Drew. (2002). A Mathematical Seduction. Math Horizons, 9:3, 12-15. Disponível em: https://web.williams.edu/Mathematics/eburger/BurgerMathHorizons.pdf
BELAIR, Luc. (2012). Panorama of p-adic model theory, Annales des Sciences Mathematiques du Quebec, nº 36, p. 43-75, 2012. Disponível em: http://www.logique.jussieu.fr/modnet/Publications/Introductory%20Notes%20and%20surveys/Belair.pdf
BOREVICH, Z. I.; SHAFAREVICH, I. R. (1966). Number Theory. New York: Academic Press.
BURGER, Edward, B. (2000). Exploring the number jungle: a journey into diophantine analysis. New York: Americal Mathematical Society.
CĂLIN, Mureşan Alexe. (2006). Non-Archimedian Fields. Topological Properties of Zp, Qp (p-adics Numbers). Matematică - Informatică – Fizică, v. 63, nº 2, 43 – 48. Disponível em: http://bmif.unde.ro/docs/20062/7%20MuresanA.pdf
CARVALHO, Maria Pires de; LOURENÇO, João Nuno P. (2015). Convergência de séries p-ádicas. Boletim da Sociedade Portuguesa de Matemática, 1 – 30, Disponível em: https://repositorio-aberto.up.pt/bitstream/10216/90725/2/104694.pdf
CHINEA, C. (2000). Matemática. ¿qué son los números p-adicos?. carlos s. chinea, octubr. Divulgación de la matemática en la red. 1 – 5. Disponível em: https://casanchi.com/casanchi_2000/19_padicos01.pdf
CORNELISSEN, Gunther; KATO, Fumiharu. (2005). The p-adic icosaedron. Notes in American Mathematical Association - AMS. v. 52, nº 7, 720 – 727. Disponível em: https://www.ams.org/notices/200507/fea-cornelissen.pdf
COHEN, Henri. (2007). Number Theory Volume I: Tools and Diophantine Equations. New York: Springer.
CORRY, Leo. (2004). Modern Algebra and the Rise of Mathematical Structures, Basel and Boston, Birkhäuser.
CUOCO, Albert. A. Visualizing the p-adic integers. American Mathematical Monthly. v. 98, nº 4, 355 – 364. Disponível em: https://www.jstor.org/stable/pdf/2323809.pdf?refreqid=excelsior%3Ada3401d345ac7e62ccac8f6378fb2cfc
CROMPTON, Catherine (2007) Some Geometry of the p-adic rationals. Rose-Hulman Undergraduate Mathematics Journal: v. 8 : Iss. 1 , Article 2. 1 – 13. Disponível em: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?referer=https://www.google.com.br/&httpsredir=1&article=1183&context=rhumj
DELAHAYE. Jean-Paul. (2001). Les nombres infinit vers la gauche. Pour la Science, nº 279, 100 – 104. Disponível em: http://www.lifl.fr/~jdelahay/pls/081.pdf
EHRLICH. Philip. (2006). The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes. The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Archive History in Exact Sciences. V. 60, 1 – 121. Disponível em: http://prima.lnu.edu.ua/faculty/mechmat/Departments/MFAUKR/attachments/erlich.pdf
FERREIRA, Jamil. (2010). A construção dos números. Textos Universitários. Rio de Janeiro: SBM.
GOLDBLATT, Robert. (1998). Lectures on the Hyprreeals numbers: an introduction to non standard analisys. New York: Springer.
GOUVÊA, F. Q. (1997). p-adic numbers. New York: Springer.
GUSMÃO, Italo, B. (2016). Números p-ádicos. Dissertação de Mestrado Profissional PROFMAT, João Pessoa: UFPB.. Disponível em: https://repositorio.ufpb.br/jspui/bitstream/tede/9337/2/arquivototal.pdf
GAUTSCHI, Walter. Alessandro M. (2002). Ostrowski (1893–1986): la sua vita e le opere. Boll. Docenti Matem. v. 45, 9–19. Disponível em: https://www.cs.purdue.edu/homes/wxg/AMOital.pdf
HASSE, Helmut. (1980). Number Theory. Springer: New York.
HEFEZ, Abramo. (2013). Um curso de Àlgebra, v. 1, Rio de Janeiro: SBM.
HOLLY, Jan E. (2001). Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields. The mathematical association of America. October, 721 – 728. Disponível em: https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf
HUAMAN, Ronald M. (2015). Raıces p-adicas de la unidad. (Tesis de Maestrıa). Pontificia Universidad del Perú: Lima. Disponível em: http://tesis.pucp.edu.pe/repositorio/bitstream/handle/123456789/6415/MAS_HUAMAN_RONALD_PADICAS.pdf;sequence=1
LAGES, Elon. (2010). Curso de Análise, v. 1. Rio de Janeiro: SBM.
LAPIDUS, Michel L.; HUNG. Lu. (2011). The Geometry of p-Adic Fractal Strings: A Comparative Survey. Contemporary Mathematics, v. 551, 163 – 206. Disponível em: http://www.math.ucr.edu/~lapidus/papers/ContMath/GeometrypAdicStringsSurvey10893.pdf
LAPIDUS, Michel L.; HUNG. Lu. (2008). Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and Applications. v. 4, p. 1 – 10. Disponível em: https://pdfs.semanticscholar.org/6dee/45208234f48fe156d174d4ad56b17363d2ef.pdf
LEGUAY, Mathieu; HENRI, Joseph. (1992). Distance p-adique : une distance qui n'est pas habituelle. MATh.en.JEANS” au Palais de la Découverte. 33 – 36. Disponível em: http://mathenjeans.free.fr/amej/edition/actes/actespdf/92033036.pdf
MARCOS, José. E. (2006). The algebraic closure of a p-adic number field is a complete topological field. Mathematica Slovaca,v. 56, No. 3, p. 317—331. Disponível em: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1010.7259&rep=rep1&type=pdf
MASIAS, Henry Zorrilla. (2011). Complecion no arquimedeana. Pro Mathematica, v. 25, 49-50. Disponível em: http://revistas.pucp.edu.pe/index.php/promathematica/article/viewFile/2666/2610
NARICI, Lawrence; BECKENSTEIN, Edward. (1981). Strange Terrain--Nonarchimedean Spaces. The American Mathematical Monthly, Vol. 88, No. 9 (Nov., 1981), pp. 667-676. Disponível em: https://www.jstor.org/stable/pdf/2320670.pdf
OSTROWSKI, A. (1916). Uber einige L¨osungen der Funktionalgleichung ¨ φ(x)φ(y) = φ(xy)”, Acta Mathematica (2nd ed.), nº 41, no. 1, 271–284. Disponível em: https://projecteuclid.org/euclid.acta/1485887472
KATO, Kazuya; KUROKAWA, Nobushige; SAITO, Takeshi, (2000). Number theory 1: Fermat’s dream, Translations of Mathematical Monographs, vol. 186, American Mathematical Society, Providence, RI.
KATOK, S. (2007). p-adic analysis compared with real. Student mathematical library, American Mathematical Soc.
KOBLITZ, Neal. (1984). p-adic Numbers, p-adic Analysis, and Zeta- Functions. New York: Springer.
KHRENNIKOV, Andrei; KOTOVICHM, Nikolay. (2017). Image Segmentation with the Aid of the p-Adic Metrics. In: TONI, Bourama. New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. 143 – 155.
KUMAR, Ashish; RANI, Mamta; CHUGH, Renu. (2013). New 5-adic Cantor sets and fractal string. SpringerPlus, 1 – 7. Disponível em: https://link.springer.com/content/pdf/10.1186%2F2193-1801-2-654.pdf
LEWIS, Robert Y. (2019). A formal proof of Hensel’s lemma over the p-adic integers. CPP’19, January p. 14–15, Lisbon, PT. Disponível em: http://robertylewis.com/padics/padics.pdf
NATARAJAN, P N; RANGANATHAN, K. N. (2000). A Geometry in which all Triangles are Isosceles An Introduction to Non-Archimedean Analysis. Resonance, October, 32 – 42. Disponível em: https://www.ias.ac.in/article/fulltext/reso/005/10/0032-0042
NEUKIRCH, J. (1990). The p-Adic numbers. EBBINGHAUS, H.-D. Numbers. New York: Springer. 155 – 179.
OLIVEIRA, Graciano. (2009). O corpo dos p-ádicos. Gazeta Matemática, Dezembro, 7 – 18. Disponível em: http://gazeta.spm.pt/getArtigo?gid=258
ORBEGOSO, Jorge Luis Rojas. Completitud y clausura algebraica de campos P-ádicos. Universidad nacional mayor de san marcos. Perú: Lima, 2016. Disponível em: http://cybertesis.unmsm.edu.pe/bitstream/handle/cybertesis/6315/Rojas_oj.pdf?sequence=1
PITANEN, Matti. (2015). How Imagination Could Be Realized p-Adically?. Journal of Consciousness Exploration & Research, v 6, nº 6, 354 – 356. Disponível em: https://jcer.com/index.php/jcj/article/viewFile/468/488
REDDY, B. Surender; SHANKARAIAH, D. (2013). On i-cauchy sequences in p-adic linear 2-normed spaces. International Journal of Pure and Applied Mathematics. v. 89, nº 4, 483 – 496. Disponivel em: https://ijpam.eu/contents/2013-89-4/4/4.pdf
RIBENBOIM, Paulo. (1999). The Theory of Classical Valuations. New York: Springer.
ROBERT, Alain. (1996). Qu´est-qe que le nombres p- ádique? Societé des enseignants neuchâtelois des Sciences. Bulletin, setembre, 5 – 12. Disponível em: http://www.sens-neuchatel.ch/bulletin/anciens-no-pdf/BULL18.PDF
ROZIKOV, U. A. (2013). What are p-Adic Numbers? What are They Used for?. Asia Pacific Mathematics Newsletter. v. 3, nº 4, 1 – 6, October. Disponível em: http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf
SCHIKHOF W. H. (1984). Ultrametric Calculus: An introduction to p-adic Analysis. Cambridge: University Press.
SCHLICHENMAIER, Martin. (2007). An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Springer: New York.
SILVERMANM, Joseph H. (2013). What is the p-adic Mandelbrot Set?. Notices of the AMS, setember, v. 60, nº 8, 1048 – 1050. Disponível em: http://www.ams.org/notices/201308/201308-full-issue.pdf
STEUDING, Jorn. (2002). The world of p-adic numbers and p-adic functions. Faculty of Physics and Mathematics. nº 5, 90 – 107. Disponível em: http://siauliaims.su.lt/pdfai/2002/STEUD-02.pdf
TEITELBAUM. Jeremy. (1995). The Geometry of p-adic Symmetric Spaces. Notice in American Mathematical Monthly. v. 42, nº 10, 1120 – 1126. Disponível em: https://www.ams.org/notices/199510/teitelbaum.pdf
TELLER, Jacek. (2012). Newton polygons on p-adic number fields. (dissertation in Arts and Mathematics). East Caroline University. Disponível em: http://thescholarship.ecu.edu/bitstream/handle/10342/3848/Teller_ecu_0600M_10676.pdf?sequence=1
TORRES, Sergio Carrillo; AMAYA, Carlos Hurtado. (2001). Una introduccion a los numeros p-adicos. Memorias XVIII encuentro de geometrıa y VI de aritmetica, 359 – 369. Disponível em: http://funes.uniandes.edu.co/5615/1/CarrilloUnaintroduccionGeometr%C3%ADa2008.PDF
VACCON. Tristan. (2015). Précision p-adique. (thésis de doctorat). Rennes: Université de Rennes. Disponível em: https://tel.archives-ouvertes.fr/tel-01205269v2/document
ALVES, Francisco, R. V. (2017). Fórmula de de moivre, ou de binet ou de lamé: demonstrações e generalidades sobre a sequência generalizada de fibonacci – SGF. Revista Brasileira de História da Matemática, v, 17, nº 33, 1 – 16.
AMICE, Ivete. (1975). Le nombres p-adique. Paris: Presses Universitaire. Disponível em: https://www-fourier.ujf-grenoble.fr/~panchish/GDL16/amice_les-nombres-p-adiques%20%5B2755599%5D.pdf
AÇIKGÖZ, Mehmet; ASLAN,Nurgül; Köşkeroğlu, Nurten e Aracı, Serkan. (2009). p-Adic Approach to Linear 2-Normed Spaces. Mathematica Moravica, v. 13, nº 2, 7 – 21. Disponível em: http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2009/1450-59320902007A.pdf
ADERBURG, Drew. (2002). A Mathematical Seduction. Math Horizons, 9:3, 12-15. Disponível em: https://web.williams.edu/Mathematics/eburger/BurgerMathHorizons.pdf
BELAIR, Luc. (2012). Panorama of p-adic model theory, Annales des Sciences Mathematiques du Quebec, nº 36, p. 43-75, 2012. Disponível em: http://www.logique.jussieu.fr/modnet/Publications/Introductory%20Notes%20and%20surveys/Belair.pdf
BOREVICH, Z. I.; SHAFAREVICH, I. R. (1966). Number Theory. New York: Academic Press.
BURGER, Edward, B. (2000). Exploring the number jungle: a journey into diophantine analysis. New York: Americal Mathematical Society.
CĂLIN, Mureşan Alexe. (2006). Non-Archimedian Fields. Topological Properties of Zp, Qp (p-adics Numbers). Matematică - Informatică – Fizică, v. 63, nº 2, 43 – 48. Disponível em: http://bmif.unde.ro/docs/20062/7%20MuresanA.pdf
CARVALHO, Maria Pires de; LOURENÇO, João Nuno P. (2015). Convergência de séries p-ádicas. Boletim da Sociedade Portuguesa de Matemática, 1 – 30, Disponível em: https://repositorio-aberto.up.pt/bitstream/10216/90725/2/104694.pdf
CHINEA, C. (2000). Matemática. ¿qué son los números p-adicos?. carlos s. chinea, octubr. Divulgación de la matemática en la red. 1 – 5. Disponível em: https://casanchi.com/casanchi_2000/19_padicos01.pdf
CORNELISSEN, Gunther; KATO, Fumiharu. (2005). The p-adic icosaedron. Notes in American Mathematical Association - AMS. v. 52, nº 7, 720 – 727. Disponível em: https://www.ams.org/notices/200507/fea-cornelissen.pdf
COHEN, Henri. (2007). Number Theory Volume I: Tools and Diophantine Equations. New York: Springer.
CORRY, Leo. (2004). Modern Algebra and the Rise of Mathematical Structures, Basel and Boston, Birkhäuser.
CUOCO, Albert. A. Visualizing the p-adic integers. American Mathematical Monthly. v. 98, nº 4, 355 – 364. Disponível em: https://www.jstor.org/stable/pdf/2323809.pdf?refreqid=excelsior%3Ada3401d345ac7e62ccac8f6378fb2cfc
CROMPTON, Catherine (2007) Some Geometry of the p-adic rationals. Rose-Hulman Undergraduate Mathematics Journal: v. 8 : Iss. 1 , Article 2. 1 – 13. Disponível em: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?referer=https://www.google.com.br/&httpsredir=1&article=1183&context=rhumj
DELAHAYE. Jean-Paul. (2001). Les nombres infinit vers la gauche. Pour la Science, nº 279, 100 – 104. Disponível em: http://www.lifl.fr/~jdelahay/pls/081.pdf
EHRLICH. Philip. (2006). The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes. The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Archive History in Exact Sciences. V. 60, 1 – 121. Disponível em: http://prima.lnu.edu.ua/faculty/mechmat/Departments/MFAUKR/attachments/erlich.pdf
FERREIRA, Jamil. (2010). A construção dos números. Textos Universitários. Rio de Janeiro: SBM.
GOLDBLATT, Robert. (1998). Lectures on the Hyprreeals numbers: an introduction to non standard analisys. New York: Springer.
GOUVÊA, F. Q. (1997). p-adic numbers. New York: Springer.
GUSMÃO, Italo, B. (2016). Números p-ádicos. Dissertação de Mestrado Profissional PROFMAT, João Pessoa: UFPB.. Disponível em: https://repositorio.ufpb.br/jspui/bitstream/tede/9337/2/arquivototal.pdf
GAUTSCHI, Walter. Alessandro M. (2002). Ostrowski (1893–1986): la sua vita e le opere. Boll. Docenti Matem. v. 45, 9–19. Disponível em: https://www.cs.purdue.edu/homes/wxg/AMOital.pdf
HASSE, Helmut. (1980). Number Theory. Springer: New York.
HEFEZ, Abramo. (2013). Um curso de Àlgebra, v. 1, Rio de Janeiro: SBM.
HOLLY, Jan E. (2001). Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields. The mathematical association of America. October, 721 – 728. Disponível em: https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf
HUAMAN, Ronald M. (2015). Raıces p-adicas de la unidad. (Tesis de Maestrıa). Pontificia Universidad del Perú: Lima. Disponível em: http://tesis.pucp.edu.pe/repositorio/bitstream/handle/123456789/6415/MAS_HUAMAN_RONALD_PADICAS.pdf;sequence=1
LAGES, Elon. (2010). Curso de Análise, v. 1. Rio de Janeiro: SBM.
LAPIDUS, Michel L.; HUNG. Lu. (2011). The Geometry of p-Adic Fractal Strings: A Comparative Survey. Contemporary Mathematics, v. 551, 163 – 206. Disponível em: http://www.math.ucr.edu/~lapidus/papers/ContMath/GeometrypAdicStringsSurvey10893.pdf
LAPIDUS, Michel L.; HUNG. Lu. (2008). Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and Applications. v. 4, p. 1 – 10. Disponível em: https://pdfs.semanticscholar.org/6dee/45208234f48fe156d174d4ad56b17363d2ef.pdf
LEGUAY, Mathieu; HENRI, Joseph. (1992). Distance p-adique : une distance qui n'est pas habituelle. MATh.en.JEANS” au Palais de la Découverte. 33 – 36. Disponível em: http://mathenjeans.free.fr/amej/edition/actes/actespdf/92033036.pdf
MARCOS, José. E. (2006). The algebraic closure of a p-adic number field is a complete topological field. Mathematica Slovaca,v. 56, No. 3, p. 317—331. Disponível em: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1010.7259&rep=rep1&type=pdf
MASIAS, Henry Zorrilla. (2011). Complecion no arquimedeana. Pro Mathematica, v. 25, 49-50. Disponível em: http://revistas.pucp.edu.pe/index.php/promathematica/article/viewFile/2666/2610
NARICI, Lawrence; BECKENSTEIN, Edward. (1981). Strange Terrain--Nonarchimedean Spaces. The American Mathematical Monthly, Vol. 88, No. 9 (Nov., 1981), pp. 667-676. Disponível em: https://www.jstor.org/stable/pdf/2320670.pdf
OSTROWSKI, A. (1916). Uber einige L¨osungen der Funktionalgleichung ¨ φ(x)φ(y) = φ(xy)”, Acta Mathematica (2nd ed.), nº 41, no. 1, 271–284. Disponível em: https://projecteuclid.org/euclid.acta/1485887472
KATO, Kazuya; KUROKAWA, Nobushige; SAITO, Takeshi, (2000). Number theory 1: Fermat’s dream, Translations of Mathematical Monographs, vol. 186, American Mathematical Society, Providence, RI.
KATOK, S. (2007). p-adic analysis compared with real. Student mathematical library, American Mathematical Soc.
KOBLITZ, Neal. (1984). p-adic Numbers, p-adic Analysis, and Zeta- Functions. New York: Springer.
KHRENNIKOV, Andrei; KOTOVICHM, Nikolay. (2017). Image Segmentation with the Aid of the p-Adic Metrics. In: TONI, Bourama. New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. 143 – 155.
KUMAR, Ashish; RANI, Mamta; CHUGH, Renu. (2013). New 5-adic Cantor sets and fractal string. SpringerPlus, 1 – 7. Disponível em: https://link.springer.com/content/pdf/10.1186%2F2193-1801-2-654.pdf
LEWIS, Robert Y. (2019). A formal proof of Hensel’s lemma over the p-adic integers. CPP’19, January p. 14–15, Lisbon, PT. Disponível em: http://robertylewis.com/padics/padics.pdf
NATARAJAN, P N; RANGANATHAN, K. N. (2000). A Geometry in which all Triangles are Isosceles An Introduction to Non-Archimedean Analysis. Resonance, October, 32 – 42. Disponível em: https://www.ias.ac.in/article/fulltext/reso/005/10/0032-0042
NEUKIRCH, J. (1990). The p-Adic numbers. EBBINGHAUS, H.-D. Numbers. New York: Springer. 155 – 179.
OLIVEIRA, Graciano. (2009). O corpo dos p-ádicos. Gazeta Matemática, Dezembro, 7 – 18. Disponível em: http://gazeta.spm.pt/getArtigo?gid=258
ORBEGOSO, Jorge Luis Rojas. Completitud y clausura algebraica de campos P-ádicos. Universidad nacional mayor de san marcos. Perú: Lima, 2016. Disponível em: http://cybertesis.unmsm.edu.pe/bitstream/handle/cybertesis/6315/Rojas_oj.pdf?sequence=1
PITANEN, Matti. (2015). How Imagination Could Be Realized p-Adically?. Journal of Consciousness Exploration & Research, v 6, nº 6, 354 – 356. Disponível em: https://jcer.com/index.php/jcj/article/viewFile/468/488
REDDY, B. Surender; SHANKARAIAH, D. (2013). On i-cauchy sequences in p-adic linear 2-normed spaces. International Journal of Pure and Applied Mathematics. v. 89, nº 4, 483 – 496. Disponivel em: https://ijpam.eu/contents/2013-89-4/4/4.pdf
RIBENBOIM, Paulo. (1999). The Theory of Classical Valuations. New York: Springer.
ROBERT, Alain. (1996). Qu´est-qe que le nombres p- ádique? Societé des enseignants neuchâtelois des Sciences. Bulletin, setembre, 5 – 12. Disponível em: http://www.sens-neuchatel.ch/bulletin/anciens-no-pdf/BULL18.PDF
ROZIKOV, U. A. (2013). What are p-Adic Numbers? What are They Used for?. Asia Pacific Mathematics Newsletter. v. 3, nº 4, 1 – 6, October. Disponível em: http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf
SCHIKHOF W. H. (1984). Ultrametric Calculus: An introduction to p-adic Analysis. Cambridge: University Press.
SCHLICHENMAIER, Martin. (2007). An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Springer: New York.
SILVERMANM, Joseph H. (2013). What is the p-adic Mandelbrot Set?. Notices of the AMS, setember, v. 60, nº 8, 1048 – 1050. Disponível em: http://www.ams.org/notices/201308/201308-full-issue.pdf
STEUDING, Jorn. (2002). The world of p-adic numbers and p-adic functions. Faculty of Physics and Mathematics. nº 5, 90 – 107. Disponível em: http://siauliaims.su.lt/pdfai/2002/STEUD-02.pdf
TEITELBAUM. Jeremy. (1995). The Geometry of p-adic Symmetric Spaces. Notice in American Mathematical Monthly. v. 42, nº 10, 1120 – 1126. Disponível em: https://www.ams.org/notices/199510/teitelbaum.pdf
TELLER, Jacek. (2012). Newton polygons on p-adic number fields. (dissertation in Arts and Mathematics). East Caroline University. Disponível em: http://thescholarship.ecu.edu/bitstream/handle/10342/3848/Teller_ecu_0600M_10676.pdf?sequence=1
TORRES, Sergio Carrillo; AMAYA, Carlos Hurtado. (2001). Una introduccion a los numeros p-adicos. Memorias XVIII encuentro de geometrıa y VI de aritmetica, 359 – 369. Disponível em: http://funes.uniandes.edu.co/5615/1/CarrilloUnaintroduccionGeometr%C3%ADa2008.PDF
VACCON. Tristan. (2015). Précision p-adique. (thésis de doctorat). Rennes: Université de Rennes. Disponível em: https://tel.archives-ouvertes.fr/tel-01205269v2/document
ALVES, Francisco, R. V. (2017). Fórmula de de moivre, ou de binet ou de lamé: demonstrações e generalidades sobre a sequência generalizada de fibonacci – SGF. Revista Brasileira de História da Matemática, v, 17, nº 33, 1 – 16.
AMICE, Ivete. (1975). Le nombres p-adique. Paris: Presses Universitaire. Disponível em: https://www-fourier.ujf-grenoble.fr/~panchish/GDL16/amice_les-nombres-p-adiques%20%5B2755599%5D.pdf
AÇIKGÖZ, Mehmet; ASLAN,Nurgül; Köşkeroğlu, Nurten e Aracı, Serkan. (2009). p-Adic Approach to Linear 2-Normed Spaces. Mathematica Moravica, v. 13, nº 2, 7 – 21. Disponível em: http://scindeks-clanci.ceon.rs/data/pdf/1450-5932/2009/1450-59320902007A.pdf
ADERBURG, Drew. (2002). A Mathematical Seduction. Math Horizons, 9:3, 12-15. Disponível em: https://web.williams.edu/Mathematics/eburger/BurgerMathHorizons.pdf
BELAIR, Luc. (2012). Panorama of p-adic model theory, Annales des Sciences Mathematiques du Quebec, nº 36, p. 43-75, 2012. Disponível em: http://www.logique.jussieu.fr/modnet/Publications/Introductory%20Notes%20and%20surveys/Belair.pdf
BOREVICH, Z. I.; SHAFAREVICH, I. R. (1966). Number Theory. New York: Academic Press.
BURGER, Edward, B. (2000). Exploring the number jungle: a journey into diophantine analysis. New York: Americal Mathematical Society.
CĂLIN, Mureşan Alexe. (2006). Non-Archimedian Fields. Topological Properties of Zp, Qp (p-adics Numbers). Matematică - Informatică – Fizică, v. 63, nº 2, 43 – 48. Disponível em: http://bmif.unde.ro/docs/20062/7%20MuresanA.pdf
CARVALHO, Maria Pires de; LOURENÇO, João Nuno P. (2015). Convergência de séries p-ádicas. Boletim da Sociedade Portuguesa de Matemática, 1 – 30, Disponível em: https://repositorio-aberto.up.pt/bitstream/10216/90725/2/104694.pdf
CHINEA, C. (2000). Matemática. ¿qué son los números p-adicos?. carlos s. chinea, octubr. Divulgación de la matemática en la red. 1 – 5. Disponível em: https://casanchi.com/casanchi_2000/19_padicos01.pdf
CORNELISSEN, Gunther; KATO, Fumiharu. (2005). The p-adic icosaedron. Notes in American Mathematical Association - AMS. v. 52, nº 7, 720 – 727. Disponível em: https://www.ams.org/notices/200507/fea-cornelissen.pdf
COHEN, Henri. (2007). Number Theory Volume I: Tools and Diophantine Equations. New York: Springer.
CORRY, Leo. (2004). Modern Algebra and the Rise of Mathematical Structures, Basel and Boston, Birkhäuser.
CUOCO, Albert. A. Visualizing the p-adic integers. American Mathematical Monthly. v. 98, nº 4, 355 – 364. Disponível em: https://www.jstor.org/stable/pdf/2323809.pdf?refreqid=excelsior%3Ada3401d345ac7e62ccac8f6378fb2cfc
CROMPTON, Catherine (2007) Some Geometry of the p-adic rationals. Rose-Hulman Undergraduate Mathematics Journal: v. 8 : Iss. 1 , Article 2. 1 – 13. Disponível em: https://scholar.rose-hulman.edu/cgi/viewcontent.cgi?referer=https://www.google.com.br/&httpsredir=1&article=1183&context=rhumj
DELAHAYE. Jean-Paul. (2001). Les nombres infinit vers la gauche. Pour la Science, nº 279, 100 – 104. Disponível em: http://www.lifl.fr/~jdelahay/pls/081.pdf
EHRLICH. Philip. (2006). The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Emergence of non-Archimedean Systems of Magnitudes. The Rise of non-Archimedean Mathematics and the Roots of a Misconception I: The Archive History in Exact Sciences. V. 60, 1 – 121. Disponível em: http://prima.lnu.edu.ua/faculty/mechmat/Departments/MFAUKR/attachments/erlich.pdf
FERREIRA, Jamil. (2010). A construção dos números. Textos Universitários. Rio de Janeiro: SBM.
GOLDBLATT, Robert. (1998). Lectures on the Hyprreeals numbers: an introduction to non standard analisys. New York: Springer.
GOUVÊA, F. Q. (1997). p-adic numbers. New York: Springer.
GUSMÃO, Italo, B. (2016). Números p-ádicos. Dissertação de Mestrado Profissional PROFMAT, João Pessoa: UFPB.. Disponível em: https://repositorio.ufpb.br/jspui/bitstream/tede/9337/2/arquivototal.pdf
GAUTSCHI, Walter. Alessandro M. (2002). Ostrowski (1893–1986): la sua vita e le opere. Boll. Docenti Matem. v. 45, 9–19. Disponível em: https://www.cs.purdue.edu/homes/wxg/AMOital.pdf
HASSE, Helmut. (1980). Number Theory. Springer: New York.
HEFEZ, Abramo. (2013). Um curso de Àlgebra, v. 1, Rio de Janeiro: SBM.
HOLLY, Jan E. (2001). Pictures of Ultrametric Spaces, the p-adic Numbers, and Valued Fields. The mathematical association of America. October, 721 – 728. Disponível em: https://www.colby.edu/math/faculty/Faculty_files/hollydir/Holly01.pdf
HUAMAN, Ronald M. (2015). Raıces p-adicas de la unidad. (Tesis de Maestrıa). Pontificia Universidad del Perú: Lima. Disponível em: http://tesis.pucp.edu.pe/repositorio/bitstream/handle/123456789/6415/MAS_HUAMAN_RONALD_PADICAS.pdf;sequence=1
LAGES, Elon. (2010). Curso de Análise, v. 1. Rio de Janeiro: SBM.
LAPIDUS, Michel L.; HUNG. Lu. (2011). The Geometry of p-Adic Fractal Strings: A Comparative Survey. Contemporary Mathematics, v. 551, 163 – 206. Disponível em: http://www.math.ucr.edu/~lapidus/papers/ContMath/GeometrypAdicStringsSurvey10893.pdf
LAPIDUS, Michel L.; HUNG. Lu. (2008). Nonarchimedean Cantor set and string. Journal of Fixed Point Theory and Applications. v. 4, p. 1 – 10. Disponível em: https://pdfs.semanticscholar.org/6dee/45208234f48fe156d174d4ad56b17363d2ef.pdf
LEGUAY, Mathieu; HENRI, Joseph. (1992). Distance p-adique : une distance qui n'est pas habituelle. MATh.en.JEANS” au Palais de la Découverte. 33 – 36. Disponível em: http://mathenjeans.free.fr/amej/edition/actes/actespdf/92033036.pdf
MARCOS, José. E. (2006). The algebraic closure of a p-adic number field is a complete topological field. Mathematica Slovaca,v. 56, No. 3, p. 317—331. Disponível em: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.1010.7259&rep=rep1&type=pdf
MASIAS, Henry Zorrilla. (2011). Complecion no arquimedeana. Pro Mathematica, v. 25, 49-50. Disponível em: http://revistas.pucp.edu.pe/index.php/promathematica/article/viewFile/2666/2610
NARICI, Lawrence; BECKENSTEIN, Edward. (1981). Strange Terrain--Nonarchimedean Spaces. The American Mathematical Monthly, Vol. 88, No. 9 (Nov., 1981), pp. 667-676. Disponível em: https://www.jstor.org/stable/pdf/2320670.pdf
OSTROWSKI, A. (1916). Uber einige L¨osungen der Funktionalgleichung ¨ φ(x)φ(y) = φ(xy)”, Acta Mathematica (2nd ed.), nº 41, no. 1, 271–284. Disponível em: https://projecteuclid.org/euclid.acta/1485887472
KATO, Kazuya; KUROKAWA, Nobushige; SAITO, Takeshi, (2000). Number theory 1: Fermat’s dream, Translations of Mathematical Monographs, vol. 186, American Mathematical Society, Providence, RI.
KATOK, S. (2007). p-adic analysis compared with real. Student mathematical library, American Mathematical Soc.
KOBLITZ, Neal. (1984). p-adic Numbers, p-adic Analysis, and Zeta- Functions. New York: Springer.
KHRENNIKOV, Andrei; KOTOVICHM, Nikolay. (2017). Image Segmentation with the Aid of the p-Adic Metrics. In: TONI, Bourama. New Trends and Advanced Methods in Interdisciplinary Mathematical Sciences. 143 – 155.
KUMAR, Ashish; RANI, Mamta; CHUGH, Renu. (2013). New 5-adic Cantor sets and fractal string. SpringerPlus, 1 – 7. Disponível em: https://link.springer.com/content/pdf/10.1186%2F2193-1801-2-654.pdf
LEWIS, Robert Y. (2019). A formal proof of Hensel’s lemma over the p-adic integers. CPP’19, January p. 14–15, Lisbon, PT. Disponível em: http://robertylewis.com/padics/padics.pdf
NATARAJAN, P N; RANGANATHAN, K. N. (2000). A Geometry in which all Triangles are Isosceles An Introduction to Non-Archimedean Analysis. Resonance, October, 32 – 42. Disponível em: https://www.ias.ac.in/article/fulltext/reso/005/10/0032-0042
NEUKIRCH, J. (1990). The p-Adic numbers. EBBINGHAUS, H.-D. Numbers. New York: Springer. 155 – 179.
OLIVEIRA, Graciano. (2009). O corpo dos p-ádicos. Gazeta Matemática, Dezembro, 7 – 18. Disponível em: http://gazeta.spm.pt/getArtigo?gid=258
ORBEGOSO, Jorge Luis Rojas. Completitud y clausura algebraica de campos P-ádicos. Universidad nacional mayor de san marcos. Perú: Lima, 2016. Disponível em: http://cybertesis.unmsm.edu.pe/bitstream/handle/cybertesis/6315/Rojas_oj.pdf?sequence=1
PITANEN, Matti. (2015). How Imagination Could Be Realized p-Adically?. Journal of Consciousness Exploration & Research, v 6, nº 6, 354 – 356. Disponível em: https://jcer.com/index.php/jcj/article/viewFile/468/488
REDDY, B. Surender; SHANKARAIAH, D. (2013). On i-cauchy sequences in p-adic linear 2-normed spaces. International Journal of Pure and Applied Mathematics. v. 89, nº 4, 483 – 496. Disponivel em: https://ijpam.eu/contents/2013-89-4/4/4.pdf
RIBENBOIM, Paulo. (1999). The Theory of Classical Valuations. New York: Springer.
ROBERT, Alain. (1996). Qu´est-qe que le nombres p- ádique? Societé des enseignants neuchâtelois des Sciences. Bulletin, setembre, 5 – 12. Disponível em: http://www.sens-neuchatel.ch/bulletin/anciens-no-pdf/BULL18.PDF
ROZIKOV, U. A. (2013). What are p-Adic Numbers? What are They Used for?. Asia Pacific Mathematics Newsletter. v. 3, nº 4, 1 – 6, October. Disponível em: http://www.asiapacific-mathnews.com/03/0304/0001_0006.pdf
SCHIKHOF W. H. (1984). Ultrametric Calculus: An introduction to p-adic Analysis. Cambridge: University Press.
SCHLICHENMAIER, Martin. (2007). An Introduction to Riemann Surfaces, Algebraic Curves and Moduli Spaces. Springer: New York.
SILVERMANM, Joseph H. (2013). What is the p-adic Mandelbrot Set?. Notices of the AMS, setember, v. 60, nº 8, 1048 – 1050. Disponível em: http://www.ams.org/notices/201308/201308-full-issue.pdf
STEUDING, Jorn. (2002). The world of p-adic numbers and p-adic functions. Faculty of Physics and Mathematics. nº 5, 90 – 107. Disponível em: http://siauliaims.su.lt/pdfai/2002/STEUD-02.pdf
TEITELBAUM. Jeremy. (1995). The Geometry of p-adic Symmetric Spaces. Notice in American Mathematical Monthly. v. 42, nº 10, 1120 – 1126. Disponível em: https://www.ams.org/notices/199510/teitelbaum.pdf
TELLER, Jacek. (2012). Newton polygons on p-adic number fields. (dissertation in Arts and Mathematics). East Caroline University. Disponível em: http://thescholarship.ecu.edu/bitstream/handle/10342/3848/Teller_ecu_0600M_10676.pdf?sequence=1
TORRES, Sergio Carrillo; AMAYA, Carlos Hurtado. (2001). Una introduccion a los numeros p-adicos. Memorias XVIII encuentro de geometrıa y VI de aritmetica, 359 – 369. Disponível em: http://funes.uniandes.edu.co/5615/1/CarrilloUnaintroduccionGeometr%C3%ADa2008.PDF
VACCON. Tristan. (2015). Précision p-adique. (thésis de doctorat). Rennes: Université de Rennes. Disponível em: https://tel.archives-ouvertes.fr/tel-01205269v2/document
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ALVES, Francisco Regis Vieira; OLIVEIRA, Paulo César Cavalcante de. Sobre os números P-ádicos: aspectos históricos, matemáticos e epistemológicos. Revista Brasileira de História da Matemática, São Paulo, v. 24, n. 48, p. 1–32, 2024. DOI: 10.47976/RBHM2024v24n481-32. Disponível em: https://rbhm.org.br/index.php/RBHM/article/view/389. Acesso em: 27 dez. 2024.
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