Direct proof through an equilateral hyperbola to find the infinite Pythagorean triples and a direct proof to find the sum of all odd numbers through a telescopic series

• Odirley Willians Miranda Saraiva

  Universidade Regional do Noroeste do Estado do Rio Grande do Sul

  https://orcid.org/0000-0001-9090-3788

  Author

• Cássio Pinho dos Reis

  Federal University of Mato Grosso do Sul

  Author

• Antônio Thiago Madeira Beirão

  Federal Rural University of Amazonia

  Author

• Katiane Pereira Silva

  Federal Rural University of Amazonia

  https://orcid.org/0000-0001-7864-6467

  Author

• Herson Oliveira da Rocha

  Federal Rural University of Amazonia

  Author

• Fabrício da Silva Lobato

  State University of Pará

  https://orcid.org/0000-0002-4250-4763

  Author

• Daniele Cristina de Brito Soares

  Federal Rural University of Amazonia

  Author

• Adonai do Socorro da Cruz Gonçalves

  Federal University of Para

  Author

• Angélica Bittencourt da Cruz Galiza

  Federal University of Para

  Author

• Inocêncio Renato Gasparim

  SEASTER

  Author

In this article a prove new properties like a connection with Pythagorean triples and a hiperbolic equation. I prove a direct proof about a serie, a telescopic series about the even sum and odd sum. I had an article that use this properties but this results is mine. I discovered this theorem and prove in this article.

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