Proth Numbers

Christoph Schwarzweller

Displaying similar documents to “Proth Numbers”

A note on a conjecture of Jeśmanowicz

Moujie Deng, G. Cohen (2000)

Colloquium Mathematicae

Similarity:

Let a, b, c be relatively prime positive integers such that $a^{2} + b^{2} = c^{2}$ . Jeśmanowicz conjectured in 1956 that for any given positive integer n the only solution of ${(a n)}^{x} + {(b n)}^{y} = {(c n)}^{z}$ in positive integers is x=y=z=2. If n=1, then, equivalently, the equation ${(u^{2} - v^{2})}^{x} + {(2 u v)}^{y} = {(u^{2} + v^{2})}^{z}$ , for integers u>v>0, has only the solution x=y=z=2. We prove that this is the case when one of u, v has no prime factor of the form 4l+1 and certain congruence and inequality conditions on u, v are satisfied.

The Diophantine equation $x^{4} - y^{4} = z^{2}$ in three quadratic fields.

Szabó, Sándor (2004)

Acta Mathematica Academiae Paedagogicae Nyí regyháziensis. New Series [electronic only]

Similarity:

Squares in Lucas sequenceshaving an even first parameter

Paulo Ribenboim, Wayne McDaniel (1998)

Colloquium Mathematicae

Similarity:

When the group-counting function assumes a prescribed integer value at squarefree integers frequently, but not extremely frequently

Claudia A. Spiro-Silverman (1992)

Acta Arithmetica

Similarity:

Basic Properties of Primitive Root and Order Function

Na Ma, Xiquan Liang (2012)

Formalized Mathematics

Similarity:

In this paper we defined the reduced residue system and proved its fundamental properties. Then we proved the basic properties of the order function. Finally, we defined the primitive root and proved its fundamental properties. Our work is based on [12], [8], and [11].

Lagrange’s Four-Square Theorem

Yasushige Watase (2014)

Formalized Mathematics

Similarity:

This article provides a formalized proof of the so-called “the four-square theorem”, namely any natural number can be expressed by a sum of four squares, which was proved by Lagrange in 1770. An informal proof of the theorem can be found in the number theory literature, e.g. in [14], [1] or [23]. This theorem is item #19 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/.

A note on primes p with $σ (p^{m}) = z^{n}$

Maohua Le (1991)

Colloquium Mathematicae

Similarity:

Set of Points on Elliptic Curve in Projective Coordinates

Yuichi Futa, Hiroyuki Okazaki, Yasunari Shidama (2011)

Formalized Mathematics

Similarity:

In this article, we formalize a set of points on an elliptic curve over GF(p). Elliptic curve cryptography [10], whose security is based on a difficulty of discrete logarithm problem of elliptic curves, is important for information security.

Decomposition of an integer as a sum of two cubes to a fixed modulus

David Tsirekidze, Ala Avoyan (2013)

Matematički Vesnik

Similarity:

On integers not of the form n - φ (n)

J. Browkin, A. Schinzel (1995)

Colloquium Mathematicae

Similarity:

W. Sierpiński asked in 1959 (see [4], pp. 200-201, cf. [2]) whether there exist infinitely many positive integers not of the form n - φ(n), where φ is the Euler function. We answer this question in the affirmative by proving Theorem. None of the numbers $2^{k} \cdot 509203$ (k = 1, 2,...) is of the form n - φ(n).