Abstract: For a graph G = (V,E), a bijection g from V(G) ∪ E(G) into { 1,2, ..., ∣ V(G) ∣ + ∣ E(G) ∣ } is called (a,d)-edge-antimagic total labeling of G if the edge-weights w(xy) = g(x) + g(y) + g(xy), xy ∈ E(G), form an arithmetic progression with initial term a and common difference d. An (a,d)-edge-antimagic total labeling g is called super (a,d)-edge-antimagic total if g(V(G)) = { 1,2,..., ∣ V(G) ∣ } . We study super (a,d)-edge-antimagic total properties of stars Sn and caterpillar Sn1,n2,...,nr.

Abstract: Andrew Bremner (Experiment. Math. 8 (1999), 409{413) has described a tech- nique for producing inflnite families of elliptic curves containing length 7 and length 8 arithmetic progressions. This note describes another way to produce inflnite families of el- liptic curves containing length 7 and length 8 arithmetic progressions. We illustrate how the technique articulated here gives an easy way to produce an elliptic curve containing a length 12 progression and an inflnite family of elliptic curves containing a length 9 progression, with the caveat that these curves are not in Weierstrass form.

Abstract: In this paper we present a simple deterministic algorithm for testing whether a multivariate polynomial f(x1, ..., xn) is identically zero, in time polynomial in m, n, log(d + 1) and H. Here m is the number of monomials in f, d is the maximum degree of a variable in f and 2H is the least upper bound on the magnitude of the largest coefficient in f. We assume that f has integer coefficients.The main feature of our algorithm is its conceptual simplicity. The proof uses Linnik's Theorem which is a deep fact about distribution of primes in an arithmetic progression.

Abstract: For any non-empty subset I of the natural numbers, let {Lambda}I denote those numbers in the unit interval whose continued fraction digits all lie in I. Define the corresponding transfer operator Formula. for Formula, where Re (rs) = {theta}I is the abscissa of convergence of the series Formula. When acting on a certain Hilbert space HI, rs, we show that the operator LI, rs is conjugate to an integral operator KI, rs. If furthermore rs is real, then KI, rs is selfadjoint, so that LI, rs : HI, rs -> HI, rs has purely real spectrum. It is proved that LI, rs also has purely real spectrum when acting on various Hilbert or Banach spaces of holomorphic functions, on the nuclear space C{omega} [0, 1], and on the Frechet space C{infty} [0, 1]. The analytic properties of the map rs m LI, rs are investigated. For certain alphabets I of an arithmetic nature (for example, I = primes, I = squares, I an arithmetic progression, I the set of sums of two squares it is shown that rs m LI, rs admits an analytic continuation beyond the half-plane Re rs > {theta}I.

Abstract: Let K be a field and m 0 <ċ< m 3 be coprime positive integers, which form an arithmetic progression. Let 𝒞 be a monomial curve in the affine 4-space , defined parametrically by X 0 = T m 0 ,…, X 3 = T m 3 . Let A be the coordinate ring of 𝒞. In this article, we produce an explicit minimal free resolution for A.

Abstract: Let a1, a2, . . . , ak be relatively prime, positive integers arranged in increasing order. Let Γ denote the positive integers in the set { a1x1 + a2x2 + · · ·+ akxk : xj ≥ 0 }. Let S(a1, a2, . . . , ak) . = {n / ∈ Γ : n+ Γ ⊆ Γ }. We determine S(a1, a2, . . . , ak) in the case where the aj’s are in arithmetic progression. In particular, this determines g(a1, a2, . . . , ak) in this particular case.

Abstract: Let K be a field and let m(0),...,m(e-1) be a sequence of positive integers. Let W be a monomial curve in the affine e-space A(K)(e), defined parametrically by X-0 = T-m0,...,Xe-1 = Tme-1 and let p be the defining ideal of W. In this article, we assume that some e-1 terms of m(0), m(e-1) form an arithmetic sequence and produce a Grobner basis for p.

Abstract: Let A be a finite set of positive integers and let A∗ be the set of all subset sums of A. We show that if A is dense enough (say, A ⊆ [1, l] and |A| ≥ 5(l ln l)), then A∗ contains a long block of consecutive integers or at least a long homogeneous arithmetic progression. This refines earlier results due to Freiman and Sarkozy.

Abstract: The second moment ∑ q ≤ Q ∑ a = 1 q ( ψ ( x ; q , a ) − ρ ( x ; q , a ) ) 2 is investigated with the novel approximation ρ ( x ; q , a ) = ∑ n ≤ x n ≡ a ( mod q ) F R ( n ) , where F R ( n ) = ∑ r ≤ R μ ( r ) ϕ ( r ) ∑ b = 1 ( b , r ) = 1 r e ( b n / r ) , and it is shown that when R ≤ log A x , this leads to more precise conclusions than in the classical Montgomery-Hooley case.

Abstract: In this paper we prove that if ðr; 12Þp3; then the set of positive odd integers k such that k r � 2 n has at least two distinct prime factors for all positive integers n contains an infinite arithmetic progression. The same result corresponding to k r 2 n þ 1 is also true.

Abstract: In the paper the results concerning the linear approximation of n-bit arithmetic sum function are presented. In particular, the computationally effective algorithms are formulated, to compute values of the approximation tables and the distribution of values in these tables as well as the algorithm to generate the list of effective approximations, ordered decreasingly by the approximation effectiveness measure.

Abstract: In a recent article [2] in this journal, Charles Fleenor demonstrates the existence of Heron triangles having sides whose lengths are consecutive integers. He presents a list of examples of such triangles and observes a number of interesting relationships that appear to hold among their side lengths. The purpose of this article is to show how to derive all possible triangles satisfying these conditions (there are an infinite family of them) and to explain why the relationships he observes are indeed true. The more general problem of Heron triangles with sides having lengths in any arithmetic progression is also discussed and a complete solution is found, but using a different method. In addition, it is shown that there are no cyclic quadrilaterals having integer area and sides of consecutive integer lengths.

Abstract: We deal with the problem of labeling the vertices, edges and faces of a special class of plane graphs with 3-sided internal faces in such a way that the label of a face and the labels of the vertices and edges surrounding that face all together add up to the weight of that face. These face weights then form an arithmetic progression with common difference d.

Abstract: Let $\tau$ be the primitive Dirichlet character of conductor 4, let $\chi$ be the primitive even Dirichlet character of conductor 8 and let $k$ be an integer. Then the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $2k+1$ and character $\tau$ has slopes in the arithmetic progression ${2,4,...,2n,...}$, and the $U_2$ operator acting on cuspidal overconvergent modular forms of weight $k$ and character $\chi \cdot \tau^k$ has slopes in the arithmetic progression ${1,2,...,n,...}$. We then show that the characteristic polynomials of the Hecke operators $U_2$ and $T_p$ acting on the space of classical cusp forms of weight $k$ and character either $\tau$ or $\chi\cdot\tau^k$ split completely over $\qtwo$.

Abstract: The purpose of this paper is to provide a simple and self-contained exposition of the celebrated Roth's theorem on arithmetic progressions of length three. The original result is proved in [Roth53], while the proof given below is very similar to the exposition of Roth's original argument given in [GRS1990]. Definition. We say that a subset of positive integers A has positive upper density if (1) lim N sup |A ∩ [1, N ]| N > 0. Roth's Theorem. If A is a subset of positive integers of positive upper density, then A contains a three term arithmetic progression. Basic setup. Let S(n) denote the largest number of integers in [1, n] that can be chosen so that no three term arithmetic progression is formed. Let (2) c = lim n→∞ S(n) n. The existence of this limit exists follows from the fact that S : Z + → R is a sub-additive function. The (easy) details are left to the reader. Let (3) = c 2 10 6 , and let m be large enough so that (4) c ≤ S(n) n < c + for 2m + 1 ≤ n.

Abstract: Enumerating formulae are constructed which count the number of partitions of a positive integer into positive summands in arithmetic progression with common difference D. These enumerating formulae (denoted pD(n)) which are given in terms of elementary divisor functions together with auxiliary arithmetic functions (to be defined) are then used to establish a known characterisation for an integer to possess a partition of the form in question.

Abstract: We prove that if one has k non-intersecting arithmetic progressions of integers, with common differences 2 <= q_1,...,q_k <= x, then k < x exp((-1/6 + o(1)) sqrt(log x loglog x)). This improves a result of Szemeredi and Erdos.

Abstract: Certain sequences of numbers, as well as problems relating to them, appear throughout mathematics. Determining the general term of an arithmetic or geometric progression, as well as determining the sum of (finitely) many consecutive terms of these sequences, are common high school problems. A residue class for a given modulus also constitutes an arithmetic progression. In Chapter 2 various problems relating to both disjoint Systems of congruences as well as covering Systems of congruences for different moduli were discussed; many of these are still unsolved.

Abstract: In this note, an example of two homogeneous Cantor sets whose arithmetic sum is a homogeneous Cantor set with a positive Lebesgue measure is given.

Abstract: For any nonnegative integer k and natural numbers n and m, the following inequalities are obtained on the ratio for the geometric means of a positive arithmetic sequence with unit difference: where α ∈ [0, 1] is a constant. Moreover, some monotonicity results for the sequences involving are obtained, and the related inequalities are generalized.