Items in this collection are the scholarly output of the Department of Mathematics faculty, staff, and students, either alone or as co-authors, and which may or may not have been published in an alternate format. Items may contain more than one file type.

Recent Submissions

  • On integers with a special divisibility property 

    Banks, William David, 1964-; Luca, Florian (Masarykova Universita, 2006)
    In this note, we study those positive integers n which are divisible by the Carmichael function.
  • Non-residues and primitive roots in Beatty sequences 

    Banks, William David, 1964-; Shparlinski, Igor E. (Australian Mathematical Society, 2006)
    We study multiplicative character sums taken on the values of a non-homogeneous Beatty sequence Bα,β = {⌊αn + β⌋ : n = 1,2,3,…}, where α,β ∈ R, and α is irrational. In particular, our bounds imply that for every fixed ε > ...
  • Coincidences in the values of the Euler and Carmichael functions 

    Banks, William David, 1964-; Friedlander, J. B. (John B.); Luca, Florian; Pappalardi, Francesco; Shparlinski, Igor E. (Polish Academy of Sciences, Institute of Mathematics, 2006)
    The Euler function has long been regarded as one of the most basic of the arithmetic functions. More recently, partly driven by the rise in importance of computational number theory, the Carmichael function has drawn an ...
  • Incomplete exponential sums and Diffie-Hellman triples 

    Banks, William David, 1964-; Friedlander, J. B. (John B.); Koniagin, S. V. (Sergeĭ Vladimirovich); Shparlinski, Igor E. (Cambridge University Press, 2006)
    Let p be a prime and 79 an integer of order t in the multiplicative group modulo p. In this paper, we continue the study of the distribution of Diffie-Hellman triples (V-x, V-y, V-xy) by considering the closely related ...
  • Arithmetic properties of φ(n)/λ(n) and the structure of the multiplicative group modulo n 

    Banks, William David, 1964-; Luca, Florian; Shparlinski, Igor E. (European Mathematical Society, 2006)
    For a positive integer n, we let φ(n) and λ(n) denote the Euler function and the Carmichael function, respectively. We define ξ(n) as the ratio φ(n)/λ(n) and study various arithmetic properties of ξ(n).
  • Distributional Properties of the Largest Prime Factor 

    Banks, William David, 1964-; Harman, G. (Glyn), 1956-; Shparlinski, Igor E. (University of Michigan, 2005)
    Let P(n) denote the largest prime factor of an integer n ≥ 2, and put P(1) = 1. In this paper, we study the distribution of the sequence {P(n) : n ≥ 1} over the set of congruence classes modulo an integer q ≥ 2, and we ...
  • Some Divisibility Properties of the Euler Function 

    Banks, William David, 1964-; Luca, Florian; Shparlinski, Igor E. (Oxford University Press, 2005)
    Let '(・) denote the Euler function, and let a > 1 be a fixed integer. We study several divisibility conditions which exhibit typographical similarity with the standard formulation of the Euler theorem, such as a n ≡ 1 ...
  • Compositions with the Euler and Carmichael Functions 

    Banks, William David, 1964-; Luca, Florian; Saidak, Filip; Stănică, Pantelimon (Springer Verlag, 2005)
    Let ' and _ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that '(_(n)) = _('(n)). We also study the normal order of ...
  • Values of Arithmetical Functions Equal to a Sum of Two Squares 

    Banks, William David, 1964-; Luca, Florian; Saidak, Filip; Shparlinski, Igor E. (Oxford University Press, 2005)
    Let '(n) denote the Euler function. In this paper, we determine the order of growth for the number of positive integers n ≤ x for which '(n) is the sum of two square numbers. We also obtain similar results for the Dedekind ...
  • Nonaliquots and Robbins Numbers 

    Banks, William David, 1964-; Luca, Florian (Polish Academy of Sciences, Institute of Mathematics, 2005)
    Let '(•) and _(•) denote the Euler function and the sum of divisors function, respectively. In this paper, we give a lower bound for the number of m ≤ x for which the equation m = _(n)−n has no solution. We also show that ...
  • Prime divisors of palindromes 

    Banks, William David, 1964-; Shparlinski, Igor E. (Springer Verlag, 2005)
    In this paper, we study some divisibility properties of palindromic numbers in a fixed base g ≥ 2. In particular, if PL denotes the set of palindromes with precisely L digits, we show that for any sufficiently large value ...
  • Roughly squarefree values of the Euler and Carmichael functions 

    Banks, William David, 1964-; Luca, Florian (Polish Academy of Sciences, Institute of Mathematics, 2005)
    Let ' denote the Euler function. In this paper, we estimate the number of positive integers n ≤ x with the property that if a prime p > y divides '(n), then p2 ∤ '(n). We also give similar estimates for the Carmichael function _.
  • Towards Faster Cryptosystems, II 

    Banks, William David, 1964- (American Mathematical Society, 2005)
    We discuss three cryptosystems, NTRU, SPIFI , and ENROOT, that are based on the use of polynomials with restricted coefficients.
  • Concatenations with Binary Recurrent Sequences 

    Banks, William David, 1964-; Luca, Florian (University of Waterloo, 2005)
    Given positive integers A1,∙ ∙ ∙ ,At and b ≥ 2, we write A1 ∙ ∙ ∙ At(b) for the integer whose base-b representation is the concatenation of the base-b representations of A1, ∙ ∙ ∙ ,At. In this paper, we prove that if ...
  • Squares from products of integers 

    Banks, William David, 1964-; Van Der Poorten, A. J. (2006)
    Notice that 1_2_3_4+1 = 52 , 2_3_4_5+1 = 112 , 3_4_5_6+1 = 192 , . . . . Indeed, it is well known that the product of any four consecutive integers always differs by one from a perfect square. However, a little experimentation ...
  • Exponential Sums over Mersenne Numbers 

    Banks, William David, 1964-; Conflitti, Alessandro; Friedlander, J. B. (John B.); Shparlinski, Igor E. (2004)
    We give estimates for exponential sums of the form Σn≤N Λ(n) exp(2πiagn/m), where m is a positive integer, a and g are integers relatively prime to m, and Λ is the von Mangoldt function. In particular, our results yield ...
  • Number Theoretic Designs for Directed Regular Graphs of Small Diameter 

    Banks, William David, 1964-; Conflitti, Alessandro; Shparlinski, Igor E. (Society for Industrial and Applied Mathematics, 2004)
    In 1989, F. R. K. Chung gave a construction for certain directed h-regular graphs of small diameter. Her construction is based on finite fields, and the upper bound on the diameter of these graphs is derived from bounds ...
  • Short Kloosterman Sums for Polynomials over Finite Fields 

    Banks, William David, 1964-; Harcharras, Asma; Shparlinski, Igor E. (2003)
    We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ...
  • Character Sums over Integers with Restricted g-ary Digits 

    Banks, William David, 1964-; Conflitti, Alessandro; Shparlinski, Igor E. (2002)
    We establish upper bounds for multiplicative character sums and exponential sums over sets of integers that are described by various properties of their digits in a fixed base g ≥ 2. Our main tools are the Weil and ...
  • Average Normalisations of Elliptic Curves 

    Banks, William David, 1964-; Shparlinski, Igor E. (Australian Mathematical Society, 2002)
    Ciet, Quisquater, and Sica have recently shown that every elliptic curve E over a finite field Fp is isomorphic to a curve y2 = x3 +ax+b with a and b of size O(p3/4). In this paper, we show that almost all elliptic curves ...

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