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Ilker Inam

Publications and source records attributed to Ilker Inam.

11 recordsLinked to original sources
arXiv2026

Inverses of Fibonacci and Lucas Numbers via Rational Indices

Fibonacci and Lucas numbers have been extensively studied in various algebraic and number-theoretic contexts, including their modular inverses and generalizations. Motivated by these developments, we study the inverses of Fibonacci and Lucas numbers with rational indices through the codenominator function $F$. Uluda\u{g} and G\"okmen (2022) showed that the rational-indexed Fibonacci number $F_X$, where $X\in\mathbb{Q}_{>0}$, can be expressed using $F$, and that infinitely many such representations exist. In this paper, we extend their work by deriving a general explicit formula for $F_X$ through the codenominator function. We establish precise conditions under which $F_X$ coincides with a Lucas number or deviates from the classical Fibonacci sequence. Moreover, by means of these generalized formulas, we compute the units of Fibonacci and Lucas numbers, thereby proving the existence of multiplicative inverses for all Fibonacci and Lucas numbers within this framework. These results reveal new structural properties of Fibonacci- and Lucas-related sequences and suggest further directions for number-theoretic exploration.

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arXiv2026

On a Diophantine Equation with Jacobsthal and Fibonacci Numbers

In the present paper, we identify all Jacobsthal numbers that may be expressed as a product of three Fibonacci numbers. More precisely, our main result shows that the only solution to the Diophantine equation \[ F_kF_lF_m=J_n \] for $2<k<l<m$ is \[ (k,l,m,n)=(5,7,8,12). \] The proof relies on techniques involving linear forms in logarithms.

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arXiv2023

On The Hecke Eigenforms of Half-Integral Weight and Dedekind-Eta Products

Systematic choice of the Hecke eigenforms of half-integral weight is an interesting problem in the theory of modular forms. In this paper, we find all Dedekind-eta products of half-integral weight which are Hecke eigenforms up to weight 15/2 with varying levels. Proof is based on the Shimura lift.

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arXiv2022

A Short Note on the Pell-Lucas-Eisenstein Series

In this work, we define a new type of Eisenstein-like series by using Pell-Lucas numbers and call them the Pell-Lucas-Eisenstein Series. Firstly, we show that the Pell-Lucas-Eisenstein series are convergent on their domain. Afterwards we prove that they satisfy some certain functional equations. Proofs follows from some on calculations on Pell-Lucas numbers.

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arXiv2022

Fast computation of half-integral weight modular forms

To study statistical properties of modular forms, including for instance Sato-Tate like problems, it is essential to have a large number of Fourier coefficients. In this article, we exhibit three bases for the space of modular forms of any half-integral weight and level 4, which have the property that many coefficients can be computed (relatively) quickly on a computer.

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arXiv2021

On the distribution of coefficients of half-integral weight modular forms and the Bruinier-Kohnen Conjecture

This work represents a systematic computational study of the distribution of the Fourier coefficients of cuspidal Hecke eigenforms of level $Γ_0(4)$ and half-integral weights. Based on substantial calculations, the question is raised whether the distribution of normalised Fourier coefficients with bounded indices can be approximated by a generalised Gaussian distribution. Moreover, it is argued that the apparent symmetry around zero of the data lends strong evidence to the Bruinier-Kohnen Conjecture on the equidistribution of signs and even suggests the strengthening that signs and absolute values are distributed independently.

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arXiv2013

On conjectures of Sato-Tate and Bruinier-Kohnen

This article covers three topics. (1) It establishes links between the density of certain subsets of the set of primes and related subsets of the set of natural numbers. (2) It extends previous results on a conjecture of Bruinier and Kohnen in three ways: the CM-case is included; under the assumption of the same error term as in previous work one obtains the result in terms of natural density instead of Dedekind-Dirichlet density; the latter type of density can already be achieved by an error term like in the prime number theorem. (3) It also provides a complete proof of Sato-Tate equidistribution for CM modular forms with an error term similar to that in the prime number theorem.

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arXiv2013

Equidistribution of Signs for Modular Eigenforms of Half Integral Weight

Let f be a cusp form of weight k+1/2 and at most quadratic nebentype character whose Fourier coefficients a(n) are all real. We study an equidistribution conjecture of Bruinier and Kohnen for the signs of a(n). We prove this conjecture for certain subfamilies of coefficients that are accessible via the Shimura lift by using the Sato-Tate equidistribution theorem for integral weight modular forms. Firstly, an unconditional proof is given for the family {a(tp^2)}_p where t is a squarefree number and p runs through the primes. In this case, the result is in terms of natural density. To prove it for the family {a(tn^2)}_n where t is a squarefree number and n runs through all natural numbers, we assume the existence of a suitable error term for the convergence of the Sato-Tate distribution, which is weaker than one conjectured by Akiyama and Tanigawa. In this case, the results are in terms of Dedekind-Dirichlet density.

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arXiv2012

Selmer Groups in Twist Families of Elliptic Curves

The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute $% L_{E}(1)$. Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type $α\frac{(\log \log (X))^{1+\varepsilon}}{\log (X)}$ with $\varepsilon $ small. We discuss how the "best choice" of $α$ is depending on the conductor of the chosen elliptic curves and the congruence classes of twist factors.

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