Proth Numbers

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.WSB Schools of Banking Gdańsk, PolandGrzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41–46, 1990.Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91–96, 1990.J. Buchmann and V. Müller. Primality testing. 1992.Czesław Byliński. Some basic properties of sets. Formalized Mathematics, 1(1):47–53, 1990.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.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.Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5):841–845, 1990.Rafał Kwiatek. Factorial and Newton coefficients. Formalized Mathematics, 1(5):887–890, 1990.Rafał Kwiatek and Grzegorz Zwara. The divisibility of integers and integer relatively primes. Formalized Mathematics, 1(5):829–832, 1990.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.Piotr Rudnicki and Andrzej Trybulec. Abian’s fixed point theorem. Formalized Mathematics, 6(3):335–338, 1997.Andrzej Trybulec and Czesław Byliński. Some properties of real numbers. Formalized Mathematics, 1(3):445–449, 1990.Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501–505, 1990.Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821–827, 1990.Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67–71, 1990.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

On Quasi-Periodicity in Proth-Gilbreath Triangles

Let PG be the Proth-Gilbreath operator that transforms a sequence of integers into the sequence of the absolute values of the differences between all pairs of neighbor terms. Consider the infinite tables obtained by successive iterations of PG applied to different initial sequences of integers. We study these tables of higher order differences and characterize those that have near-periodic features. As a biproduct, we also obtain two results on a class of formal power series over the field with two elements F2 that can be expressed as rational functions in several ways