Topic
Sylvester's sequence
About: Sylvester's sequence is a research topic. Over the lifetime, 3 publications have been published within this topic receiving 31 citations.
Papers
TL;DR: In this article, the authors considered the cyclic system of n ≥ 2 simultaneous congruences for fixed nonzero integers with r > 0 and (r, s) = 1, and showed that there are only finitely many solutions in positive integers qi ≥ 2.
Abstract: This paper considers the cyclic system of n ≥ 2 simultaneous congruences for fixed nonzero integers (r, s) with r > 0 and (r, s) = 1. It shows there are only finitely many solutions in positive integers qi ≥ 2, with gcd(q1q2 ⋯ qn, s) = 1 and obtains sharp bounds on the maximal size of solutions for almost all (r, s). The extremal solutions for r = s = 1 are related to Sylvester's sequence 2, 3, 7, 43, 1807,…. If the positivity condition on the integers qi is dropped, then for r = 1 these systems of congruences, taken (mod|qi|), have infinitely many solutions, while for r ≥ 2 they have finitely many solutions. The problem is reduced to studying integer solutions of the family of Diophantine equations depending on three parameters (r, s, m).
4 citations
1 Jan 2019
TL;DR: In this article, the rationality of functions satisfying certain functional equations was studied and generalized by a variant of Mahler's method, and the result of Becker was generalized by a Mahler method.
Abstract: Cahen’s constant is defined by the alternating sum of reciprocals of terms of Sylvester’s sequence minus 1. Davison and Shallit proved the transcendence of the constant and Becker improved it. In this paper, we study rationality of functions satisfying certain functional equations and generalize the result of Becker by a variant of Mahler’s method.
3 citations
TL;DR: In this paper, it was shown that Cahen's constant continued fraction is transcendental, i.e., its partial quotients grow doubly exponentially and they are all squares.
Abstract: We discuss certain simple continued fractions that exhibit a type of “self-similar” structure: their partial quotients are formed by perturbing and shifting the denominators of their convergents. We prove that all such continued fractions represent transcendental numbers. As an application, we prove that Cahen's constant
$$C = \sum\limits_{i \geqslant 0} {\frac{{( - 1)^i }}{{S_i - 1}}}$$
is transcendental. Here (S
n
) isSylvester's sequence defined byS
0=2 andS
n+1
=S
2
−S
n
+1 forn≥0. We also explicitly compute the continued fraction for the numberC; its partial quotients grow doubly exponentially and they are all squares.
Performance Metrics
| Year | Papers |
|---|---|
| 2019 | 1 |
| 2010 | 1 |
| 1991 | 1 |