Nonzero Integer Research Articles

On squares in Lucas sequences

Let P and Q be non-zero integers. The Lucas sequence { U n ( P , Q ) } is defined by U 0 = 0 , U 1 = 1 , U n = P U n − 1 − Q U n − 2 ( n ⩾ 2 ). The question of when U n ( P , Q ) can be a perfect square has generated interest in the literature. We show that for n = 2 , … , 7 , U n is a square for infinitely many pairs ( P , Q ) with gcd ( P , Q ) = 1 ; further, for n = 8 , … , 12 , the only non-degenerate sequences where gcd ( P , Q ) = 1 and U n ( P , Q ) = □ , are given by U 8 ( 1 , − 4 ) = 21 2 , U 8 ( 4 , − 17 ) = 620 2 , and U 12 ( 1 , − 1 ) = 12 2 .

On conjugate adjacency matrices of a graph

Let G be a simple graph of order n. Let c = a + b m and c ¯ = a - b m , where a and b are two nonzero integers and m is a positive integer such that m is not a perfect square. We say that A c = [ c ij ] is the conjugate adjacency matrix of the graph G if c ij = c for any two adjacent vertices i and j, c ij = c ¯ for any two nonadjacent vertices i and j, and c ij = 0 if i = j . Let P G ( λ ) = | λ I - A | and P G c ( λ ) = | λ I - A c | denote the characteristic polynomial and the conjugate characteristic polynomial of G, respectively. In this work we show that if P G c ( λ ) = P H c ( λ ) then P G ¯ c ( λ ) = P H ¯ c ( λ ) , where G ¯ denotes the complement of G. In particular, we prove that P G c ( λ ) = P H c ( λ ) if and only if P G ( λ ) = P H ( λ ) and P G ¯ ( λ ) = P H ¯ ( λ ) . Further, let P c ( G ) be the collection of conjugate characteristic polynomials P G i c ( λ ) of vertex-deleted subgraphs G i = G ⧹ i ( i = 1 , 2 , … , n ) . If P c ( G ) = P c ( H ) we prove that P G c ( λ ) = P H c ( λ ) , provided that the order of G is greater than 2.

On a distribution property of the residual order of a (mod p)- III

Let a be a positive integer which is not a perfect b -th power with b ≥ 2 , q be a prime number and Q a ( x ; q i , j ) be the set of primes p ≤ x such that the residual order of a ( m o d p ) in ( Z / p Z ) × is congruent to j modulo q i . In this paper, which is a sequel of our previous papers [1] and [6], under the assumption of the Generalized Riemann Hypothesis, we determine the natural densities of Q a ( x ; q i , j ) for i ≥ 3 if q = 2 , i ≥ 1 if q is an odd prime, and for an arbitrary nonzero integer j (the main results of this paper are announced without proof in [3], [7] and [2]).

On the set of integral solutions of the Pell equation in number fields

We investigate the set of integral solutions, over a given number field, of the equation X2 - dY2= 1, where d denotes some non-zero integer of this field. We define an operation on this set such that it is an abelian group and determine the structure of this abelian group in terms of the number of complex and real embeddings of the number field, and the number of positive embeddings of d.

A note on the size of the largest ball inside a convex polytope

Let $m>1$ be an integer, $B_m$ the set of all unit vectors of $\Bbb R^m$ pointing in the direction of a nonzero integer vector of the cube $[-1, 1]^m$. Denote by $s_m$ the radius of the largest ball contained in the convex hull of $B_m$. We determine the exact value of $s_m$ and obtain the asymptotic equality $s_m\sim\frac{2}{\sqrt{\log m}}$.

The abc–conjecture for Algebraic Numbers

The abc–conjecture for the ring of integers states that, for every e > 0 and every triple of relatively prime nonzero integers (a, b, c) satisfying a + b = c, we have max(|a|, |b|, |c|) ≤ rad(abc)1 + e with a finite number of exceptions. Here the radical rad(m) is the product of all distinct prime factors of m.

Two variations of a theorem of Kronecker

We present two variations of Kronecker's classical result that every nonzero algebraic integer that lies with its conjugates in the closed unit disc is a root of unity. The first is an analogue for algebraic nonintegers, while the second is a several variable version of the result, valid over any field.

On Chow and Brauer groups of a product of Mumford curves

Let C1,···,Cd be Mumford curves defined over a finite extension of Open image in new window and let X=C1×···×Cd. We shall show the following: (1) The cycle map CH0(X)/n → H2d(X, μn⊗d) is injective for any non-zero integer n. (2) The kernel of the canonical map CH0(X)→Hom(Br(X),Open image in new window) (defined by the Brauer-Manin pairing) coincides with the maximal divisible subgroup in CH0(X).

THE 3-PART OF CLASS NUMBERS OF QUADRATIC FIELDS

It is proved that the 3-part of the class number of a quadratic field is in general and if has a divisor of size . These bounds follow as results of nontrivial estimates for the number of solutions to the congruence modulo in the ranges and , where are nonzero integers and is a square-free positive integer. Furthermore, it is shown that the number of elliptic curves over with conductor is in general and if has a divisor of size . These results are the first improvements to the trivial bound and the resulting bound for the 3-part and the number of elliptic curves, respectively.

Test Ranks of Free Nilpotent Groups

ABSTRACT The test rank tr(G) of G is the minimum cardinality of a test set. In this paper we prove: I. Let N = N rc be a free nilpotent group of rank r ≥ 2 and class c ≥ 2. Then (i) tr(N) = 2 for r odd and c = 2; (ii) tr(N) = 1 in all other cases; (iii) an element g ∈ N 2q, 2 is a test element if and only if it can be written as g = ⋅ s , where t 1,…, t q are nonzero integers. II. Let F r be a free group of rank r ≥ 2 and A n be a free abelian group of rank n ≥ 1. Then the group G = G rn = F r ×A n has the test rank n. #Communicated by E. zelmanov.

The Constrained Shortest Path Problem: Algorithmic Approaches and an Algebraic Study with Generalization

The constrained shortest path (CSP) problem requires the determination of a minimum cost s-t path with delay at most a nonzero integer T. In this paper, we first point out the equivalence of certain algorithms, simply called the LARAC (Lagrangian Relaxation Based Aggregated Cost) algorithm presented independently in some earlier works. The LARAC algorithm solves the integer relaxation of the CSP problem (RELAX-CSP) and is based on a geometric approach. We then present an algebraic study of RELAX-CSP and establish several new properties of the optimal solution. These properties also hold for a class of combinatorial optimization problems involving two additive parameters. We follow this by establishing a characterization of optimal solutions for the general CSP problem involving more than two additive parameters. We present a new heuristic called LARAC-BIN based on binary search. This heuristic involves a parameter whose value can be specified in advance depending on the allowable deviation of the cost from the optimum. Using Megiddo's parametric search, we also present a strongly polynomial time algorithm for RELAX-CSP. This algorithm has the best complexity to date for RELAX-CSP. Finally, we present an integrated approach to the CSP problem and show how the LARAC algorithm can be used to achieve considerable speedup of ϵ-approximation algorithms for the CSP problem.

On sets which contain a qth power residue for almost all prime modules

A classical theorem of M. Fried \cite{fri} asserts that if non-zero integers $\beta_1,\ldots,\beta_l$ have the property that for each prime number $p$ there exists a quadratic residue $\beta_j$ mod $p$ then a certain product of an odd number of them is

Systems of quadratic diophantine inequalities

Let Q 1 ,⋯,Q r be quadratic forms with real coefficients. We prove that for any ϵ>0 the system of inequalities |Q 1 (x)|<ϵ,⋯,|Q r (x)|<ϵ has a nonzero integer solution, provided that the system Q 1 (x)=0,⋯,Q r (x)=0 has a nonsingular real solution and all forms in the real pencil generated by Q 1 ,⋯,Q r are irrational and have rank >8r.

Small prime solutions of quadratic equations II

Let b 1 , … , b 5 b_1, \ldots , b_5 be non-zero integers and n n any integer. Suppose that b 1 + ⋯ + b 5 ≡ n ( mod 24 ) b_1+\cdots +b_5 \equiv n \pmod {24} and ( b i , b j ) = 1 (b_i,b_j)=1 for 1 ≤ i &gt; j ≤ 5 1 \leq i &gt; j \leq 5 . In this paper we prove that (i) if the b j b_j are not all of the same sign, then the above quadratic equation has prime solutions satisfying p j ≪ | n | + max { | b j | } 25 / 2 + ε ; p_j\ll \sqrt {|n|}+ \max \{|b_j|\}^{25/2+\varepsilon }; and (ii) if all the b j b_j are positive and n ≫ max { | b j | } 26 + ε n \gg \max \{|b_j|\}^{26+\varepsilon } , then the quadratic equation b 1 p 1 2 + ⋯ + b 5 p 5 2 = n b_1p_1^2+\cdots +b_5p_5^2=n is soluble in primes p j . p_j. Our previous results are max { | b j | } 20 + ε \max \{|b_j|\}^{20+\varepsilon } and max { | b j | } 41 + ε \max \{|b_j|\}^{41+\varepsilon } in place of max { | b j | } 25 / 2 + ε \max \{|b_j|\}^{25/2+\varepsilon } and max { | b j | } 26 + ε \max \{|b_j|\}^{26+\varepsilon } above, respectively.

Reconstructing Integer Sets From Their Representation Functions

We give a simple common proof to recent results by Dombi and by Chen and Wang concerning the number of representations of an integer in the form $a_1+a_2$, where $a_1$ and $a_2$ are elements of a given infinite set of integers. Considering the similar problem for differences, we show that there exists a partition ${\Bbb N}=\cup_{k=1}^\infty A_k$ of the set of positive integers such that each $A_k$ is a perfect difference set (meaning that any non-zero integer has a unique representation as $a_1-a_2$ with $a_1,a_2\in A_k$). A number of open problems are presented.

Factor rings of the Gaussian integers

Whereas the homomorphic images of Z (the ring of integers) are well known, namely Z, {0} and Zn (the ring of integers modulo n), the same is not true for the homomorphic im-ages of Z[i] (the ring of Gaussian integers). More generally, let m be any nonzero square free integer (positive or negative), and consider the integral domain Z[ √m]={a + b √m | a, b ∈ Z}. Which rings can be homomorphic images of Z[ √m]? This ques-tion offers students an infinite number (one for each m) of investigations that require only undergraduate mathematics. It is the goal of this article to offer a guide to the in-vestigation of the possible homomorphic images of Z[ √m] using the Gaussian integers Z[i] as an example. We use the fact that Z[i] is a principal ideal domain to prove that if I =(a+bi) is a nonzero ideal of Z[i], then Z[i]/I ∼ = Zn for a positive integer n if and only if gcd{a, b} =1, in which case n = a2 + b2 . Our approach is novel in that it uses matrix techniques based on the row reduction of matrices with integer entries. By characterizing the integers n of the form n = a2 + b2 , with gcd{a, b} =1, we obtain the main result of the paper, which asserts that if n ≥ 2, then Zn is a homomorphic image of Z[i] if and only if the prime decomposition of n is 2α0 pα1 1 ··· pαk k , with α0 ∈{0, 1},pi ≡ 1(mod 4) and αi ≥ 0 for every i ≥ 1. All the fields which are homomorpic images of Z[i] are also determined.

On the ultimate complexity of factorials

It has long been observed that certain factorization algorithms provide a way to write the product of many different integers succinctly. In this paper, we study the problem of representing the product of all integers from 1 to n (i.e. n ! ) by straight-line programs. Formally, we say that a sequence of integers a n is ultimately f ( n ) -computable, if there exists a nonzero integer sequence m n such that for any n , a n m n can be computed by a straight-line program (using only additions, subtractions and multiplications) of length at most f ( n ) . Shub and Smale [12] showed that if n ! is ultimately hard to compute, then the algebraic version of NP ≠ P is true. Assuming a widely believed number theory conjecture concerning smooth numbers in a short interval, a subexponential upper bound ( exp ( c log n log log n ) ) for the ultimate complexity of n ! is proved in this paper, and a randomized subexponential algorithm constructing such a short straight-line program is presented as well.

Quadratic equations with five prime unknowns

Let b 1,⋯, b 5 be non-zero integers satisfying gcd( b i , b j , b k )=1, for 1⩽ i< j< k⩽5 and | b j |⩽| b 5| for 1⩽ j⩽5 and n an integer satisfying b 1+⋯+b 5≡n ( mod 24) . In this paper we improve earlier work by M.C. Liu and Tsang and by the first author and J.Y. Liu. In particular, we prove that if b j are not all of the same sign, then the quadratic equation b 1p 1 2+⋯+b 5p 5 2=n has prime solutions satisfying p j|| |n| +|b 1⋯b 4| 2+ε . If all b j are positive then the above equation is soluble provided that n≫ b 5( b 1⋯ b 4) 4+ ε .

A description of continued fraction expansions of quadratic surds represented by polynomials

The principal thrust of this investigation is to provide families of quadratic polynomials {D k(X)=f k 2X 2+2e kX+C} k∈ N , where e k 2− f k 2 C= n (for any given nonzero integer n) satisfying the property that for any X∈ N , the period length ℓ k=ℓ( D k(X) ) of the simple continued fraction expansion of D k(X) is constant for fixed k and lim k→∞ ℓ k =∞. This generalizes, and completes, numerous results in the literature, where the primary focus was upon | n|=1, including the work of this author, and coauthors, in Mollin (Far East J. Math. Sci. Special Vol. 1998, Part III, 257–293; Serdica Math. J. 27 (2001) 317) Mollin and Cheng (Math. Rep. Acad. Sci. Canada 24 (2002) 102; Internat Math J 2 (2002) 951) and Mollin et al. (JP J. Algebra Number Theory Appl. 2 (2002) 47).

Three-Descent and the Birch and Swinnerton-Dyer Conjecture

We give a 3-descent procedure to bound and, in some cases, compute the 3-part of the Selmer and Tate-Shafarevich group of the curves y2 = x3 + a, a a nonzero integer. This enables us to verify the whole Birch and Swinnerton-Dyer conjecture for some of such curves.