Proth Numbers

Schwarzweller, Christoph

doi:10.2478/forma-2014-0013

Abstract

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.

References

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.
Search in Google Scholar
[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.
Search in Google Scholar
[3] J. Buchmann and V. Müller. Primality testing. 1992.
Search in Google Scholar
[4] Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.
Search in Google Scholar
[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.
Search in Google Scholar
[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/v10037-012-0012-2.10.2478/v10037-012-0012-2
Search in Google Scholar
[7] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.
Search in Google Scholar
[8] Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.
Search in Google Scholar
[9] Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.
Search in Google Scholar
[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.
Search in Google Scholar
[11] Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.
Search in Google Scholar
[12] Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.
Search in Google Scholar
[13] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.
Search in Google Scholar
[14] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.
Search in Google Scholar
[15] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.
Search in Google Scholar
[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/v10037-008-0004-4.10.2478/v10037-008-0004-4
Search in Google Scholar

Proth Numbers

Abstract

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