This paper extends the concept of a $B$-happy number, for $B \geq 2$, from the rational integers, $\mathbb{Z}$, to the Gaussian integers, $\mathbb{Z}[i]$. We investigate the fixed points and cycles of the Gaussian $B$-happy functions, determining them for small values of $B$ and providing a method for computing them for any $B \geq 2$. We discuss heights of Gaussian $B$-happy numbers, proving results concerning the smallest Gaussian $B$-happy numbers of certain heights. Finally, we prove conditions for the existence and non-existence of arbitrarily long arithmetic sequences of Gaussian $B$-happy numbers.
This paper focuses on finding the smallest happy number for each height in any numerical base. Using the properties of the height, we deduce a recursive relationship between the smallest happy number and the height where the initial height is function of the numerical base. With the usage of the recursive relationship, we build an algorithm that exploits the properties of the height in order to find all of those smallest happy numbers with unknown height. However, with the modular arithmetic, we conclude on an equation that calculates the smallest happy numbers based on known heights for binar…
For $b\leq -2$ and $e \geq 2$, let $S_{e,b}:\mathbb{Z}\to\mathbb{Z}_{\geq 0}$ be the function taking an integer to the sum of the $e$-powers of the digits of its base $b$ expansion. An integer $a$ is a $b$-happy number if there exists $k\in\mathbb{Z}^+$ such that $S_{2,b}^k(a) = 1$. We prove that an integer is $-2$-happy if and only if it is congruent to 1 modulo 3 and that it is $-3$-happy if and only if it is odd. Defining a $d$-sequence to be an arithmetic sequence with constant difference $d$ and setting $d = \gcd(2,b - 1)$, we prove that if $b \leq -3$ odd or $b \in \{-4,-6,-8,-10\}$, the…
Let e>=1 and b>=2 be integers. For a positive integer n=\sum_{j=0}^ka_jb^j with 0<=a_j<b, define T_{e,b}(n)=\sum_{j=0}^ka_j^e. n is called (e,b)-happy if T_{e,b}^r(n)=1 for some r>=0, where T_{e,b}^r is the r-th iteration of T_{e,b}. In this paper, we prove that there exist arbitrarily long sequences of consecutive (e,b)-happy numbers provided that e-1 is not divisible by p-1 for any prime divisor p of b-1.
Given a positive integer $n$, the factorial base representation of $n$ is given by $n=\sum_{i=1}^ka_i\cdot i!$, where $a_k\neq 0$ and $0\leq a_i\leq i$ for all $1\leq i\leq k$. For $e\geq 1$, we define $S_{e,!}:\mathbb{Z}_{\geq0}\to\mathbb{Z}_{\geq0}$ by $S_{e,!}(0) = 0$ and $S_{e,!}(n)=\sum_{i=0}^{n}a_i^e$, for $n \neq 0$. For $\ell\geq 0$, we let $S_{e,!}^\ell(n)$ denote the $\ell$-th iteration of $S_{e,!}$, while $S_{e,!}^0(n)=n$. If $p\in\mathbb{Z}^+$ satisfies $S_{e,!}(p)=p$, then we say that $p$ is an $e$-power factoradic fixed point of $S_{e,!}$. Moreover, given $x\in \mathbb{Z}^+$, if …
The happy function $H: \mathbb{N} \rightarrow \mathbb{N}$ sends a positive integer to the sum of the squares of its digits. A number $x$ is said to be happy if the sequence $\{H^n(x)\}^\infty_{n=1}$ eventually reaches one. A basic open question regarding happy numbers is what bounds on the density can be proved. This paper uses probabilistic methods to reduce this problem to experimentally finding suitably large intervals containing a high (or low) density of happy numbers as a subset. Specifically we show that $\bar{d} > .18577$ and $\underline{d} < .1138$. We also prove that the asymptotic d…
An augmented generalized happy function $S_{[c,b]}$ maps a positive integer to the sum of the squares of its base $b$ digits plus $c$. In this paper, we study various properties of the fixed points of $S_{[c,b]}$; count the number of fixed points of $§_{[c,b]}$, for $b \geq 2$ and $0<c<3b-3$; and prove that, for each $b \geq 2$, there exist arbitrarily many consecutive values of $c$ for which $S_{[c,b]}$ has no fixed point.
A generalized happy function, $S_{e,b}$ maps a positive integer to the sum of its base $b$ digits raised to the $e^\text{th}$ power. We say that $x$ is a base $b$, $e$ power, height $h$, $u$ attracted number if $h$ is the smallest positive integer so that $S^{h}_{e,b}(x)=u$. Happy numbers are then base 10, 2 power, 1 attracted numbers of any height. Let $σ_{h,e,b}(u)$ denote the smallest height $h$, $u$ attracted number for a fixed base $b$ and exponent $e$ and let $g(e)$ denote the smallest number so that every integer can be written as $x_{1}^{e}+x_{2}^{e}+...+x_{g(e)}^{e}$ for some nonnegat…
In this paper, we define Gaussian generalized Tetranacci numbers and as special cases, we investigate Gaussian Tetranacci and Gaussian Tetranacci-Lucas numbers with their properties.
In the present article we introduce three new notions which are called Gaussian Mersenne Lucas numbers, Mersenne Lucas polynomials and Gaussian Mersenne Lucas polynomials. We present and prove our exciting properties and results of them such as: recurrence relations, Binet's formulas, explicit formulas, generating functions, symmetric functions and negative extensions.
The structure of the group $(\mathbb{Z}/n\mathbb{Z})^\star$ and Fermat's little theorem are the basis for some of best-known primality testing algorithms. Many related concepts arise: Euler's totient function and Carmichael's lambda function, Fermat pseudoprimes, Carmichael and cyclic numbers, Lehmer's totient problem, Giuga's conjecture, etc. In this paper, we present and study analogues to some of the previous concepts arising when we consider the underlying group $\mathcal{G}_n:=\{a+bi\in\mathbb{Z}[i]/n\mathbb{Z}[i] : a^2+b^2\equiv 1\ \textrm{$\pmod n$}\}$. In particular we characterize Gau…
The famous open problem of finding positive integer solutions to $a^5 + b^5 = c^5 + d^5$ is considered, and related solutions are found in two distinct settings: firstly, where $a$ and $b$ are both positive integers with $c$ and $d$ both Gaussian integers; secondly, where all of $a$, $b$, $c$, and $d$ are Gaussian integers.
{\em The Liouville number}, denoted $l$, is defined by $$l:=0.100101011101101111100...,$$ where the $n$th bit is given by ${1/2}(1+\gl(n))$; here $\gl$ is the Liouville function for the parity of prime divisors of $n$. Presumably the Liouville number is transcendental, though at present, a proof is unattainable. Similarly, define {\em the Gaussian Liouville number} by $$γ:=0.110110011100100111011...$$ where the $n$th bit reflects the parity of the number of rational Gaussian primes dividing $n$, 1 for even and 0 for odd. In this paper, we prove that the Gaussian Liouville number and its relati…
Abundancy index refers to the ratio of the sum of the divisors of a number to the number itself. It is a concept of great importance in defining friendly and perfect numbers. Here, we describe a suitable generalization of abundancy index to the ring of Gaussian integers ($\mathbb{Z}[i]$). We first show that this generalization possesses many of the useful properties of the traditional abundancy index in $\mathbb{Z}$. We then investigate $k$-powerful $τ$-perfect numbers and prove results regarding their existence in $\mathbb{Z}[i]$.
It has been observed many times, both in the Monthly and elsewhere, that the set of all quotients of prime numbers is dense in the positive real numbers. In this short note we answer the related question: "Is the set of all quotients of Gaussian primes dense in the complex plane?"
The Calkin-Wilf tree is an infinite binary tree whose vertices are the positive rational numbers. Each number occurs in the tree exactly once and in the form $a/b$, where are $a$ and $b$ are relatively prime positive integers. In this paper, certain subsemigroups of the modular group are used to construct similar trees in the set $\mathcal{D}_0$ of positive complex numbers. Associated to each semigroup is a forest of trees that partitions $\mathcal{D}_0$. The set of cusps and the fundamental domain of the semigroup are defined and computed.
Zeckendorf's Theorem states that any positive integer can be written uniquely as a sum of non-adjacent Fibonacci numbers. We consider higher-dimensional lattice analogues, where a legal decomposition of a number $n$ is a collection of lattice points such that each point is included at most once. Once a point is chosen, all future points must have strictly smaller coordinates, and the pairwise sum of the values of the points chosen equals $n$. We prove that the distribution of the number of summands in these lattice decompositions converges to a Gaussian distribution in any number of dimensions…
This paper studies the interplay between probability, number theory, and geometry in the context of relatively prime integers in the ring of integers of a number field. In particular, probabilistic ideas are coupled together with integer lattices and the theory of zeta functions over number fields in order to show that $$P(\gcd(z_{1},z_{2})=1) = \frac{1}{ζ_{\Q(i)}(2)}$$ where $z_{1},z_{2} \in \mathbb{Z}[i]$ are randomly chosen and $ζ_{\Q(i)}(s)$ is the Dedekind zeta function over the Gaussian integers. Our proof outlines a lattice-theoretic approach to proving the generalization of this theore…
This work is a sequel of a previous work of one of the authors (Y.Ô), which treated certain congruence relation between an elliptic Gauss sum and a coefficient of power series expansion at the origin of the lemniscate sine function. We extend the previous result (in \cite{O}) which concerned only for non-vanishing elliptic Gauss sums. We give new congruence relations between power series coefficients of the lemniscate cosine function, which hold if and only if the corresponding elliptic Gauss sum vanishes.
We extend the sum-of-divisors function to the complex plane via the Gaussian integers. Then we prove a modified form of Euler's classification of odd perfect numbers.
In this paper, we define new generalized k-Mersenne numbers and give a formula of generalized Mersenne polynomials and further we study their properties. Moreover, we define Gaussian Mersenne numbers and obtain some identities like Binet Formula, Cassini's identity, D'Ocagne's Identity, and generating functions. The generalized Gaussian Mersenne numbers are described and the relation with classical Mersenne numbers are explained. We also introduce a generalization of Gaussian Mersenne polynomials and establish some properties of these polynomials.
The purpose of this short communication is to make some observations on the connections between various existing formulas of counting the number of sublattices of a fixed index in an $n$-dimensional lattice and their connection with the Gaussian binomials.
We prove the Kloosterman-Spectral sum formula for PSL(2,Z[i])\PSL(2,C), and apply it to derive an explicit spectral expansion for the fourth power moment of the Dedekind zeta function of the Gaussian number field. This sum formula allows the extension of the spectral theory of Kloosterman sums to all algebraic number fields.
We introduce a new type of convergence in probability theory, which we call ``mod-Gaussian convergence''. It is directly inspired by theorems and conjectures, in random matrix theory and number theory, concerning moments of values of characteristic polynomials or zeta functions. We study this type of convergence in detail in the framework of infinitely divisible distributions, and exhibit some unconditional occurrences in number theory, in particular for families of $L$-functions over function fields in the Katz-Sarnak framework. A similar phenomenon of ``mod-Poisson convergence'' turns out to…
Manin's conjecture predicts the asymptotic behavior of the number of rational points of bounded height on algebraic varieties. For toric varieties, it was proved by Batyrev and Tschinkel via height zeta functions and an application of the Poisson formula. An alternative approach to Manin's conjecture via universal torsors was used so far mainly over the field Q of rational numbers. In this note, we give a proof of Manin's conjecture over the Gaussian rational numbers Q(i) and over other imaginary quadratic number fields with class number 1 for the singular toric cubic surface defined by t^3=xy…
This paper proves that a family of number field lattice codes simultaneously achieves a constant gap to capacity in Rayleigh fast fading and Gaussian channels. The key property in the proof is the existence of infinite towers of Hilbert class fields with bounded root discriminant. The gap to capacity of the proposed families is determined by the root discriminant. The comparison between the Gaussian and fading case reveals that in Rayleigh fading channels the normalized minimum product distance plays an analogous role to the Hermite invariant in Gaussian channels.
