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On a certain class of arithmetic functions

Antonio M. Oller-Marcén (2017)

Mathematica Bohemica

A homothetic arithmetic function of ratio K is a function f : ℕ → R such that f ( K n ) = f ( n ) for every n ∈ ℕ . Periodic arithmetic funtions are always homothetic, while the converse is not true in general. In this paper we study homothetic and periodic arithmetic functions. In particular we give an upper bound for the number of elements of f ( ℕ ) in terms of the period and the ratio of f .

On pseudoprimes having special forms and a solution of K. Szymiczek’s problem

Andrzej Rotkiewicz (2005)

Acta Mathematica Universitatis Ostraviensis

We use the properties of p -adic integrals and measures to obtain general congruences for Genocchi numbers and polynomials and tangent coefficients. These congruences are analogues of the usual Kummer congruences for Bernoulli numbers, generalize known congruences for Genocchi numbers, and provide new congruences systems for Genocchi polynomials and tangent coefficients.

On the Diophantine equation x 2 - k x y + y 2 - 2 n = 0

Refik Keskin, Zafer Şiar, Olcay Karaatlı (2013)

Czechoslovak Mathematical Journal

In this study, we determine when the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 has an infinite number of positive integer solutions x and y for 0 ≤ n ≤ 10 . Moreover, we give all positive integer solutions of the same equation for 0 ≤ n ≤ 10 in terms of generalized Fibonacci sequence. Lastly, we formulate a conjecture related to the Diophantine equation x 2 - k x y + y 2 - 2 n = 0 .

On the number of places of convergence for Newton’s method over number fields

Xander Faber, José Felipe Voloch (2011)

Journal de Théorie des Nombres de Bordeaux

Let f be a polynomial of degree at least 2 with coefficients in a number field K , let x 0 be a sufficiently general element of K , and let α be a root of f . We give precise conditions under which Newton iteration, started at the point x 0 , converges v -adically to the root α for infinitely many places v of K . As a corollary we show that if f is irreducible over K of degree at least 3, then Newton iteration converges v -adically to any given root of f for infinitely many places v . We also conjecture that...

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