Browsing Department of Mathematics (MU) by Author "Banks, William David, 1964"
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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 ... 
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). 
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 ... 
Character Sums over Integers with Restricted gary 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 ... 
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 ... 
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 ... 
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 baseb representation is the concatenation of the baseb representations of A1, ∙ ∙ ∙ ,At. In this paper, we prove that if ... 
Congruences and Exponential Sums with the Euler Function
Banks, William David, 1964; Shparlinski, Igor E. (2004)We give upper bounds for the number of solutions to congruences with the Euler function φ(n) and with the Carmichael function λ(n). We also give nontrivial bounds for certain exponential sums involving φ(n). Analogous ... 
A corollary to Bernstein's theorem and Whittaker functionals on the metaplectic group
Banks, William David, 1964 (1998)In this paper, we extend and apply a remarkable theorem due to Bernstein, which was proved in a letter to PiatetskiShapiro in the fall of 1985 [3]. The theorem is quite useful for establishing analytic continuation and ... 
Cryptographic applications of sparse polynomials over finite rings
Banks, William David, 1964; Lieman, Daniel, 1965; Shparlinski, Igor E.; To, Van Thuong (2001)This paper gives new examples that exploit the idea of using sparse polynomials with restricted coefficients over a finite ring for designing fast, reliable cryptosystems and identification schemes. 
Distribution of Inverses in Polynomial Rings
Banks, William David, 1964; Shparlinski, Igor E. (2001)Let IFp be the finite field with p elements, and let F(X) ∈ IFp[X] be a squarefree polynomial. We show that in the ring R = IFp[X]/F(X), the inverses of polynomials of small height are uniformly distributed. We also show ... 
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 ... 
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 ... 
An extremely small and efficient identification scheme
Banks, William David, 1964; Lieman, Daniel, 1965; Shparlinski, Igor E. (2000)We present a new identification scheme which is based on Legendre symbols modulo a certain hidden prime and which is naturally suited for low power, low memory applications. 
An identification scheme based on sparse polynomials
Banks, William David, 1964; Lieman, Daniel, 1965; Shparlinski, Igor E. (2000)This paper gives a new example of exploiting the idea of using polynomials with restricted coefficients over finite fields and rings to construct reliable cryptosystems and identification schemes. 
Incomplete exponential sums and DiffieHellman 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 DiffieHellman triples (Vx, Vy, Vxy) by considering the closely related ... 
Irrationality of Power Series for Various Number Theoretic Functions
Banks, William David, 1964; Luca, Florian; Shparlinski, Igor E. (200510)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 ... 
Matrix inequalities with applications to the theory of iterated kernels
Banks, William David, 1964; Harcharras, Asma; Neuwirth, Stefan; Ricard, Eric (2003)For an m × n matrix A with nonnegative real entries, Atkinson, Moran and Watterson proved the inequality s(A)3 ≤ mns(AAtA), where At is the transpose of A, and s(·) is the sum of the entries. We extend this result to finite ... 
Multiplicative Structure of Values of the Euler Function
Banks, William David, 1964; Friedlander, J. B. (John B.); Pomerance, Carl; Shparlinski, Igor E. (2004)We establish upper bounds for the number of smooth values of the Euler function. In particular, although the Euler function has a certain “smoothing” effect on its integer arguments, our results show that, in fact, most ... 
New examples of noncommutative Λ(p) sets
Banks, William David, 1964; Harcharras, Asma (2003)In this paper, we introduce a certain combinatorial property Z*(k), which is defined for every integer k ≥ 2, and show that every set E ⊂ Z with the property Z*(k) is necessarily a noncommutative Λ (2k) set. In particular, ...