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The aim of this note is to offer a summary of the definitions and properties of arithmetic symbols on the linear group Gl(n, F) -F being an arbitrary discrete valuation field- and to show that the natural generalizations of the Parshin symbol on an algebraic surface S to the linear group Gl(n, ΣS) do not allow us to define new 2-dimensional symbols on S.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

A note on $(a, b)$ -Fibonacci sequences and specially multiplicative arithmetic functions

Emil Daniel Schwab, Gabriela Schwab (2024)

Mathematica Bohemica

A specially multiplicative arithmetic function is the Dirichlet convolution of two completely multiplicative arithmetic functions. The aim of this paper is to prove explicitly that two mathematical objects, namely $(a, b)$ -Fibonacci sequences and specially multiplicative prime-independent arithmetic functions, are equivalent in the sense that each can be reconstructed from the other. Replacing one with another, the exploration space of both mathematical objects expands significantly.

A note on a conjecture of Erdős-Turán.

Sándor, Csaba (2008)

Integers

A note on a conjecture of Jeśmanowicz

Moujie Deng, G. Cohen (2000)

Colloquium Mathematicae

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.