Robert Tichy,Ingrid Vukusic,Daodao Yang et al.
Robert Tichy et al.
Let { U n } n ≥ 0 and { V m } m ≥ 0 be two linear recurrence sequences. We establish an asymptotic formula for the number of integers c in the range [ - x , x ] which can be represented as differences U n - V m . In pa...
Paul Surer
Paul Surer
We present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently publish...
An explicit form of the polynomial part of a restricted partition function [0.03%]
限制性分区函数的多项式部分的显式形式
Karl Dilcher,Christophe Vignat
Karl Dilcher
We prove an explicit formula for the polynomial part of a restricted partition function, also known as the first Sylvester wave. This is achieved by way of some identities for higher-order Bernoulli polynomials, one of which is analogous to...
Vandita Patel,Samir Siksek
Vandita Patel
Let k ≥ 2 be even, and let r be a non-zero integer. We show that for almost all d ≥ 2 (in the sense of natural density), the equation x k + ( x + r ) k + ⋯ + ( x + ( d - 1 ) r ) k = y n , x , y , n ∈ Z , n ...
Decomposition types in minimally tamely ramified extensions of [Formula: see text] [0.03%]
最小温和奇点扩张中的分解类型公式大小正确公式大小正确公式大小正确
David S Dummit,Hershy Kisilevsky
David S Dummit
We examine whether it is possible to realize finite groups G as Galois groups of minimally tamely ramified extensions of Q and also specify both the inertia groups and the further decomposition of the ramified primes. ...
Robin de Jong
Robin de Jong
We show that there exists a sequence of genus three curves defined over the rationals in which the height of a canonical Gross-Schoen cycle tends to infinity. ...
George C Ţurcaş
George C Ţurcaş
Assuming a deep but standard conjecture in the Langlands programme, we prove Fermat's Last Theorem over Q ( i ) . Under the same assumption, we also prove that, for all prime exponents p ≥ 5 , Fermat's equation a p + b p + c p = 0 ...
Beth Romano,Jack A Thorne
Beth Romano
An ADE Dynkin diagram gives rise to a family of algebraic curves. In this paper, we use arithmetic invariant theory to study the integral points of the curves associated to the exceptional diagrams E 6 , E 7 , E 8 . These curves are non...
Clemens Fuchs,Christoph Hutle,Florian Luca
Clemens Fuchs
The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence...