Proth Numbers

• Autorzy: Christoph Schwarzweller
• Języki publikacji: EN

Abstrakty

EN

In this article we introduce Proth numbers and prove two theorems on such numbers being prime [3]. We also give revised versions of Pocklington’s theorem and of the Legendre symbol. Finally, we prove Pepin’s theorem and that the fifth Fermat number is not prime.

Słowa kluczowe

EN

prime numbers; Pocklington’s theorem; Proth’s theorem; Pepin’s theorem

Kategorie tematyczne

• 03B35: Mechanization of proofs and logical operations
• 11A41: Primes

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2014
• Tom: 22
• Numer: 2
• Strony: 111-118

Daty

otrzymano

2014-06-09

online

2015-02-05

Twórcy

autor

Christoph Schwarzweller

• WSB Schools of Banking Gdańsk, Poland

Bibliografia

• [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
• [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
• [3] J. Buchmann and V. Müller. Primality testing. 1992.
• [4] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
• [5] 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.
• [6] 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]
• [7] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.
• [8] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.
• [9] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.
• [10] Hiroyuki Okazaki and Yasunari Shidama. Uniqueness of factoring an integer and multiplicative group Z/pZ∗. Formalized Mathematics, 16(2):103–107, 2008. doi:10.2478/v10037- 008-0015-1.[Crossref]
• [11] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.
• [12] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.
• [13] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.
• [14] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.
• [15] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
• [16] Li Yan, Xiquan Liang, and Junjie Zhao. Gauss lemma and law of quadratic reciprocity. Formalized Mathematics, 16(1):23–28, 2008. doi:10.2478/v10037-008-0004-4.[Crossref]

Identyfikatory

DOI

10.2478/forma-2014-0013