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.
  • Values of the Euler Function in Various Sequences 

    Banks, William David, 1964-; Ford, Kevin; Luca, Florian; Pappalardi, Francesco; Shparlinski, Igor E. (2005)
    Let φ (n) and λ(n) denote the Euler and Carmichael functions, respectively. In this paper, we investigate the equation φ (n)r = λ(n)s, where r ≥ s ≥ 1 are fixed positive integers. We also study those positive integers n, ...
  • On the Value Set of n! Modulo a Prime 

    Banks, William David, 1964-; Luca, Florian; Shparlinski, Igor E.; Stichtenoth, H. (Henning), 1944- (2005)
    We show that for infinitely many prime numbers p there are at least log log p/ log log log p distinct residue classes modulo p that are not congruent to n! for any integer n.
  • Irrationality of Power Series for Various Number Theoretic Functions 

    Banks, William David, 1964-; Luca, Florian; Shparlinski, Igor E. (2005-10)
    We study formal power series whose coefficients are taken to be a variety of number theoretic functions, such as the Euler, Möbius and divisor functions. We show that these power series are irrational over ℤ [X], and we ...
  • 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 ...
  • On the average value of divisor sums in arithmetic progressions 

    Banks, William David, 1964-; Heath-Brown, D. R.; Shparlinski, Igor E. (2005)
    We consider very short sums of the divisor function in arithmetic progressions prime to a fixed modulus and show that "on average" these sums are close to the expected value. We also give applications of our result to sums ...
  • Almost All Palindromes Are Composite 

    Banks, William David, 1964-; Hart, Derrick, 1980-; Sakata, Mayumi (2004)
    We study the distribution of palindromic numbers (with respect to a fixed base g ≥ 2) over certain congruence classes, and we derive a nontrivial upper bound for the number of prime palindromes n ≤ x as x → ∞. Our results ...
  • On the norm of an idempotent Schur multiplier on the Schatten class 

    Banks, William David, 1964-; Harcharras, Asma (American Mathematical Society, 2004-02)
    We show that if the norm of an idempotent Schur multiplier on the Schatten class Sp lies sufficiently close to 1, then it is necessarily equal to 1. We also give a simple characterization of those idempotent Schur multipliers ...

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