Related papers: Relations between e, $\pi$, golden ratios and $\sq…
In his notebooks, Gauss recorded various calculations with "infinite congruences". These infinite congruences are p-adic numbers; Gauss computes a square root of $5$ in the $11$-adic integers in order to find an $11$-adic approximation to a…
The notion of 'bifurcating continued fractions' is introduced. Two coupled sequences of non-negative integers are obtained from an ordered pair of positive real numbers in a manner that generalizes the notion of continued fractions. These…
We arrive at some new relations for the prime number $P_n$, based on the logarithmic and absolute-value properties of the function $\pi(x)$.
We consider the continued fraction expansion of real numbers under the action of a non-uniform lattice in PSL(2,R) and prove metric relations between the convergents and a natural geometric notion of good approximations.
We construct the sequences of Fibonacci and Lucas at any quadratic field $\mathbb{Q}(\sqrt{d}\ )$ with $d>0$ square free, noting in general that the properties remain valid as those given by the classical sequences of Fibonacci and Lucas…
The main objective of this paper is to introduced a new sequence space $l_{p}(\hat{F}(r,s)),$ $ 1\leq p \leq \infty$ by using the band matrix $\hat{F}(r,s).$ We also establish a few inclusion relations concerning this space and determine…
In this note, we present some basic properties of $q$-Fibonacci numbers and their relationship to the $q$-golden ratio and Catalan numbers. We then use this relationship to give a short proof of a combinatorial identity.
We introduce a geometric-arithmetic approach to the analysis of the Flint Hills series, linking its convergence behavior to the irrationality measure of pi. The framework highlights the interplay between the distribution of near-multiples…
Consider an algebraic function like $F(x) = \sqrt{x^3 - 1}$. If $p \in \mathbb{Q}$ is a rational number, how many iterates of $p$ under $F$ can also be rational? The dynamics of algebraic functions may be formalized in the language of…
We introduce a class of rings, namely the class of left or right $p$-nil rings, for which the adjoint groups behave regularly. Every $p$-ring is close to being left or right $p$-nil in the sense that it contains a large ideal belonging to…
In this paper we present a family of identities for recursive sequences arising from a second order recurrence relation, that gives instances of Zeckendorf representation. We prove these results using a special case of an universal property…
In this work, we prove the irrationality of $\pi$ based on the nested radicals with roots of $2$ of kind $c_k = \sqrt{2 + c_{k - 1}}$ and $c_0 = 0$. Sample computations showing how the rational approximation tends to $\pi$ with increasing…
This article is about Pi Formulas, infinite series of fractions which sum to multiples of Pi. Each such one can be associated with a unique set $S_k$ of rough numbers, where $k$ is a prime number. Given $S_k$ for any prime $k$, the set…
Thermal fits have consistently reproduced the experimental particles yields of heavy ion collisions, however, the proton to pion ratio from ALICE Pb+Pb $\sqrt{s_{NN}}=2.76$ TeV is over-predicted by thermal models- known at the $p/\pi$…
We study generalizations of the sequence of the n-anacci constants that consist of the ratio limits generated by linear recurrences of an arbitrary order n with equal positive weights p. We derive the analytic representation of these ratio…
Let $p$ be an odd prime. In the paper we collect the author's various conjectures on congruences modulo $p$ or $p^2$, which are concerned with sums of binomial coefficients, Lucas sequences, power residues and special binary quadratic…
Based on the structure of Fibonacci sequence, we give a new proof for the irrationality exponents of the Fibonacci real numbers. Moreover, we obtain all the irrationality exponents of the real numbers corresponding to the differences of…
We study Fibonacci compositions, which are compositions of natural numbers that only use Fibonacci numbers, in two different contexts. We first prove inequalities comparing the number of Fibonacci compositions to regular compositions where…
The generalised random Fibonacci chain is a stochastic extension of the classical Fibonacci substitution and is defined as the rule mapping $0\mapsto 1$ and $1 \mapsto 1^i01^{m-i}$ with probability $p_i$, where $p_i\geq 0$ with…
In this note, we extend the connection between the hydrogen atom and $\pi$ to the number $e$ via the Lerch's transcendent.