For integers a and n>0, let a(n) denote the residue class {x\in Z: x=a (mod n)}. Let A be a collection {a_s(n_s)}_{s=1}^k of finitely many residue classes such that A covers all the integers at least m times but {a_s(n_s)}_{s=1}^{k-1} does not. We show that if n_k is a period of the covering function w_A(x)=|{1\le s\le k: x\in a_s(n_s)}| then for any r=0,...,n_k-1 there are at least m integers in the form $\sum_{s\in I}1/n_s-r/n_k$ with I contained in {1,...,k-1}.
In this paper we construct a cover {a_s(mod n_s)}_{s=1}^k of Z with odd moduli such that there are distinct primes p_1,...,p_k dividing 2^{n_1}-1,...,2^{n_k}-1 respectively. Using this cover we show that for any positive integer m divisible by none of 3, 5, 7, 11, 13 there exists an infinite arithmetic progression of positive odd integers the m-th powers of whose terms are never of the form $2^n\pm p^a$ with p a prime and a,n in {0,1,2,...}. We also construct another cover of Z with odd moduli and use it to prove that $x^2-F_{3n}/2$ has at least two distinct prime factors whenever n is a nonne…
Zero-sum problems for abelian groups and covers of the integers by residue classes, are two different active topics initiated by P. Erdos more than 40 years ago and investigated by many researchers separately since then. In an earlier announcement [Electron. Res. Announc. Amer. Math. Soc. 9(2003), 51-60], the author claimed some surprising connections among these seemingly unrelated fascinating areas. In this paper we establish further connections between zero-sum problems for abelian p-groups and covers of the integers. For example, we extend the famous Erdos-Ginzburg-Ziv theorem in the follo…
In this paper we make a Gaussian integer version of the Erdős-Straus conjecture and we solve the Erdős-Straus diophantine equation over the rings of integers of norm-Euclidean quadratic fields.
The Euler phi function on a given integer $n$ yields the number of positive integers less than $n$ that are relatively prime to $n$. Equivalently, it gives the order of the group of units in the quotient ring $\mathbb{Z}/(n)$. We generalize the Euler phi function to the Eisenstein integer ring $\mathbb{Z}[ρ]$ where $ρ$ is the primitive third root of unity $e^{2πi/3}$ by finding the order of the group of units in the ring $\mathbb{Z}[ρ]/(θ)$ for any given Eisenstein integer $θ$. As one application we investigate a sufficiency criterion for when certain unit groups $\left(\mathbb{Z}[ρ]/(γ^n)\rig…
In the paper, I consider appearance of unit's digits in minor totals of a few integer sequences. The sequences include the sequence of even integers, sequence of odd integers and Faulhaber polynomial at $p = 2$. Application of difference tables allows predicting of which digits can appear as unit's digits in minor totals of the sequences. Absence of some digits ("gaps" in frequency distributions) depends often on numbering system applied. However, in case of odd numbers' integers the gaps are found under all numbering systems with bases from 3 to 10.
It is proved that if the least modulus of a distinct covering system is 4, its largest modulus is at least 60; also if the least modulus is 3, the LCM of the moduli is at least 120; finally, if the least modulus is 4, the LCM of the moduli is at least 360. The constants 60, 120, and 360 can not be replaced by larger constants.
Let A={a_s(mod n_s)}_{s=0}^k be a system of residue classes. With the help of cyclotomic fields we obtain a theorem which unifies several previously known results concerning system A. In particular, we show that if every integer lies in more than m=[sum_{s=1}^k 1/n_s] members of A, then for any a=0,1,2,... there are at least binom{m}{[a/n_0]} subsets I of {1,...,k} with sum_{s in I}1/n_s=a/n_0. We also characterize when any integer lies in at most m members of A, where m is a fixed positive integer.
We give an affirmative answer to the following question by Jarden and Narkiewicz: Is it true that every number field has a finite extension L such that the ring of integers of L is generated by its units (as a ring)? As a part of the proof, we generalize a theorem by Hinz on power-free values of polynomials in number fields.
We generalise Uspensky's theorem characterising eventual exact (e.e.) covers of the positive integers by homogeneous Beatty sequences, to e.e. m-covers, for any m \in \N, by homogeneous sequences with irrational moduli. We also consider inhomogeneous sequences, again with irrational moduli, and obtain a purely arithmetical characterisation of e.e. m-covers. This generalises a result of Graham for m = 1, but when m > 1 the arithmetical description is more complicated. Finally we speculate on how one might make sense of the notion of an exact m-cover when m is not an integer, and present a "frac…
Let $N \geq2$ and let $1 < a_1 < ... < a_N$ be relatively prime integers. Frobenius number of this $N$-tuple is defined to be the largest positive integer that cannot be expressed as $\sum_{i=1}^N a_i x_i$ where $x_1,...,x_N$ are non-negative integers. The condition that $gcd(a_1,...,a_N)=1$ implies that such number exists. The general problem of determining the Frobenius number given $N$ and $a_1,...,a_N$ is NP-hard, but there has been a number of different bounds on the Frobenius number produced by various authors. We use techniques from the geometry of numbers to produce a new bound, relati…
An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such a covering of the integers, then the least member of S is also bounded, thus confirming a conjecture of Erdos and Selfridge. We also prove a conjecture of Erdos and Graham, that, for each fixed number K>1, the complement in the integers of any union of residue classes r(n) mod n, for distinct n in (N,KN], has density at least d_K for N sufficiently large. He…
Let $φ(τ)=η((τ+1)/2)^2/\sqrt{2π}e^\frac{πi}{4}η(τ+1)$ where $η(τ)$ is the Dedekind eta-function. We show that if $τ_0$ is an imaginary quadratic number with $\mathrm{Im}(τ_0)>0$ and $m$ is an odd integer, then $\sqrt{m}φ(mτ_0)/φ(τ_0)$ is an algebraic integer dividing $\sqrt{m}$. This is a generalization of Theorem 4.4 given in [B. C. Berndt, H. H. Chan and L. C. Zhang, Ramanujan's remarkable product of theta-functions, Proc. Edinburgh Math. Soc. (2) 40 (1997), no. 3, 583-612]. On the other hand, let $K$ be an imaginary quadratic field and $θ_K$ be an element of $K$ with $\mathrm{Im}(θ_K)>0$ wh…
Let O be the ring of S-integers in a number field k. We prove that if the group of units O^* is infinite then every matrix in $Γ$ = SL_2(O) is a product of at most 9 elementary matrices. This completes a long line of research in this direction. As a consequence, we obtain that $Γ$ is boundedly generated as an abstract group.
Let A={a_s+n_sZ}_{s=1}^k be a finite system of arithmetic sequences which forms an m-cover of Z (i.e., every integer belongs at least to m members of A). In this paper we show the following sharp result: For any positive integers m_1,...,m_k and theta in [0,1), if there is a subset I of {1,...,k} such that the fractional part of sum_{s in I}m_s/n_s is theta, then there are at least 2^m such subsets of {1,...,k}. This extends an earlier result of M. Z. Zhang and an extension by Z. W. Sun. Also, we generalize the above result to m-covers of the integral ring of any algebraic number field with a …
Masser and Vaaler have given an asymptotic formula for the number of algebraic numbers of given degree $d$ and increasing height. This problem was solved by counting lattice points (which correspond to minimal polynomials over $\mathbb{Z}$) in a homogeneously expanding star body in $\mathbb{R}^{d+1}$. The volume of this star body was computed by Chern and Vaaler, who also computed the volume of the codimension-one "slice" corresponding to monic polynomials -- this led to results of Barroero on counting algebraic integers. We show how to estimate the volume of higher-codimension slices, which a…
A sequence of integers $ \{ s_n \}_{n \in \mathbb{N}} $ is called a T-sequence if there exists a Hausdorff group topology on $ \mathbb{Z} $ such that $ \{ s_n \}_{n \in \mathbb{N}} $ converges to zero. For every finite set of primes $ S $ we build a Hausdorff group topology on $ \mathbb{Z} $ such that every growing sequence of $ S $-integers converges to zero. As a corollary, we solve in the affirmative an open problem by I.V. Protasov and E.G. Zelenuk asking if $ \{ 2^n + 3^n \}_{n \in \mathbb{N}} $ is a T-sequence. Our results rely on a nontrivial number-theoretic fact about $ S $-unit equat…
We give a Schinzel-Zassenhaus-type lower bound for the maximum modulus of roots of a monic integer polynomial with all roots symmetric with respect to the unit circle. Our results extend a recent work of Dimitrov, who proved the general Schinzel-Zassenhaus conjecture by using the Pólya rationality theorem for a power series with integer coefficients, and some estimates for logarithmic capacity (transfinite diameter) of sets. We use an enhancement of Pólya's result obtained by Robinson, which involves Laurent-type rational functions with small supremum norms, thereby replacing the logarithmic c…
Let R be the ring of S-integers of an algebraic function field (in one variable) over a perfect field, where S is finite and not empty. It is shown that for every positive integer N there exist elements of R that can not be written as a sum of at most N units.
For any fixed coprime positive integers $a,b$ and $c$ with $\min\{a,b,c\}>1$, we prove that the equation $a^x+b^y=c^z$ has at most two solutions in positive integers $x,y$ and $z$, except for one specific case which exactly gives three solutions. Our result is essentially sharp in the sense that there are infinitely many examples allowing the equation to have two solutions in positive integers. From the viewpoint of a well-known generalization of Fermat's equation, it is also regarded as a 3-variable generalization of the celebrated theorem of Bennett [M.A.Bennett, On some exponential equation…
For an element $a$ of an integral domain D under an equivalence relation τ, the τ-factorization of a is defined as λa_1 a_2... a_k, where λis a unit in D and a_i τa_j for all i, j. An irreducible element has no proper τ-factorization; that is, a τ-factorization in which there is more than one distinct non-unit factor. In this paper, the irreducible integers under the congruence modulo n relation for some values of n are found, and these findings are generalized in the first step toward a general characterization of the irreducible integers under this relation for any prime n.
In this paper, we investigate generalizations of the Mahler-Popkens complexity of integers. Specifically, we generalize to $k$-th roots of unity, polynomials over the naturals, and the integers mod $m$. In cyclotomic rings, we establish upper and lower bounds for integer complexity, investigate the complexity of roots of unity using cyclotomic polynomials, and introduce a concept of "minimality.'' In polynomials over the naturals, we establish bounds on the sizes of complexity classes and establish a trivial but useful upper bound. In the integers mod $m$, we introduce the concepts of "ineffic…
Let $R$ be a ring with identity, $\mathcal{U}(R)$ the group of units of $R$ and $k$ a positive integer. We say that $a\in \mathcal{U}(R)$ is $k$-unit if $a^k=1$. Particularly, if the ring $R$ is $\mathbb{Z}_n$, for a positive integer $n$, we will say that $a$ is a $k$-unit modulo $n$. We denote with $\mathcal{U}_k(n)$ the set of $k$-units modulo $n$. By $\text{du}_k(n)$ we represent the number of $k$-units modulo $n$ and with $\text{rdu}_k(n)=\frac{ϕ(n)}{\text{du}_k(n)}$ the ratio of $k$-units modulo $n$, where $ϕ$ is the Euler phi function. Recently, S. K. Chebolu proved that the solutions of…
For a positive integer $N$, let $\mathscr C(N)$ be the subgroup of $J_0(N)$ generated by the equivalence classes of cuspidal divisors of degree $0$ and $\mathscr C(N)(\mathbb Q):=\mathscr C(N)\cap J_0(N)(\mathbb Q)$ be its $\mathbb Q$-rational subgroup. Let also $\mathscr C_{\mathbb Q}(N)$ be the subgroup of $\mathscr C(N)(\mathbb Q)$ generated by $\mathbb Q$-rational cuspidal divisors. We prove that when $N=n^2M$ for some integer $n$ dividing $24$ and some squarefree integer $M$, the two groups $\mathscr C(N)(\mathbb Q)$ and $\mathscr C_{\mathbb Q}(N)$ are equal. To achieve this, we show that…
In 2007, Jarden and Narkiewicz raised the following question: Is it true that each algebraic number field has a finite extension L such that the ring of integers of L is generated by its units (as a ring)? In this article, we answer the analogous question in the function field case. More precisely, it is shown that for every finite non-empty set S of places of an algebraic function field F | K over a perfect field K, there exists a finite extension F' | F, such that the integral closure of the ring of S-integers of F in F' is generated by its units (as a ring).
For a fixed integer $n$, we study the question whether at least one of the numbers $\Re Xω^k$, $1\leq k\leq n$, is $\varepsilon$-close to an integer, for any possible value of $X\in\mathbb{C}$, where $ω$ is a primitive $n$th root of unity. It turns out that there is always a $X$ for which the above numbers are concentrated around $1/2\bmod1$. The distance from $1/2$ depends only on the local properties of $n$, rather than its magnitude. This is directly related the so-called "pyjama" problem which was solved recently.
