Algebraic independence of the values of power series generated by linear recurrences
Algebraic Independence of Values of Elliptic Functions.
Algebraic independence over
Let be a power series , where is a strictly increasing linear recurrence sequence of non-negative integers, and a sequence of roots of unity in satisfying an appropriate technical condition. Then we are mainly interested in characterizing the algebraic independence over of the elements
Algebraic independence results for reciprocal sums of Fibonacci numbers
Algebraic integers as values of elliptic functions
Algebraic integers whose conjugates lie near the unit circle
Algebraic leaves of algebraic foliations over number fields
We prove an algebraicity criterion for leaves of algebraic foliations defined over number fields. Namely, consider a number field embedded in , a smooth algebraic variety over , equipped with a rational point , and an algebraic subbundle of the its tangent bundle , defined over . Assume moreover that the vector bundle is involutive, i.e., closed under Lie bracket. Then it defines an holomorphic foliation of the analytic manifold , and one may consider its leaf through . We prove...
Algebraic points of abelian functions in two variables
Algebraic relations between special values of multiple sine functions
Algebraic relations for reciprocal sums of Fibonacci numbers
Algebraic values of G-functions.
Algebraically unrelated sequences
Algebraische Abhängigkeit von Wurzeln
Algebraische Unabhängigkeit P-adischer Zahlen.
Algebraische Unabhängigkeit von Werten von Funktionen, die gewissen Differentialgleichungen genügen.
Algébricité de valeurs spéciales de fonctions L.
Algorithme de Jacobi-Perron dans un corps de séries formelles
All Liouville Numbers are Transcendental
In this Mizar article, we complete the formalization of one of the items from Abad and Abad’s challenge list of “Top 100 Theorems” about Liouville numbers and the existence of transcendental numbers. It is item #18 from the “Formalizing 100 Theorems” list maintained by Freek Wiedijk at http://www.cs.ru.nl/F.Wiedijk/100/. Liouville numbers were introduced by Joseph Liouville in 1844 [15] as an example of an object which can be approximated “quite closely” by a sequence of rational numbers. A real...
Almost powers in the Lucas sequence
The famous problem of determining all perfect powers in the Fibonacci sequence and in the Lucas sequence has recently been resolved [10]. The proofs of those results combine modular techniques from Wiles’ proof of Fermat’s Last Theorem with classical techniques from Baker’s theory and Diophantine approximation. In this paper, we solve the Diophantine equations , with and , for all primes and indeed for all but primes . Here the strategy of [10] is not sufficient due to the sizes of...
Almost-sure growth rate of generalized random Fibonacci sequences
We study the generalized random Fibonacci sequences defined by their first non-negative terms and for n≥1, Fn+2=λFn+1±Fn (linear case) and ̃Fn+2=|λ̃Fn+1±̃Fn| (non-linear case), where each ± sign is independent and either + with probability p or − with probability 1−p (0<p≤1). Our main result is that, when λ is of the form λk=2cos(π/k) for some integer k≥3, the exponential growth of Fn for 0<p≤1, and of ̃Fn for 1/k<p≤1, is almost surely positive and given by ∫0∞log x dνk, ρ(x),...