For positive integers q, Dirichlet’s theorem states that there are infinitely many primes in each reduced residue class modulo q. Extending a proof of Dirichlet’s theorem shows that the primes are equidistributed among the φ(q) reduced residue classes modulo q. This project considers patterns of sequences of consecutive primes (pn, pn+1, . . . , pn+k) modulo q. Numerical evidence suggests a pre...

Let p be a prime number such that p ≡ 1 (mod r) for some integer r > 1. Let g > 1 be an integer such that g has order r in (Z/pZ)∗. Let 1 p = ∞ ∑ k=1 xk gk be the g-adic expansion. Our result implies that the “average” digit in the g-adic expansion of 1/p is (g − 1)/2 with error term involving the generalized Bernoulli numbers B1,χ (where χ is a character modulo p of order r with χ(−1) = −1). A...

Let α be a real number with convergents pm/qm from the continued fraction expansion of α. In this paper we investigate the functions E(α) := � m≥0 |αqm − pm| and E∗(α) := � m≥0(αqm − pm) depending only on α and prove that they take every value in [0, (1 + √ 5)/2] and [0, 1], respectively. For any sequence (αμ)μ≥1, which is uniformly distributed modulo 1, we show that both sequences (E(αμ))μ≥1 a...

Given two words u and v, the binomial coefficient ( u v ) is the number of ways v appears as a subword (or subsequence) of u. The Thue-Morse sequence is the infinite word t = abbabaab · · · obtained by iteration of the morphism τ (a) = ab and τ (b) = ba. We show that, for every prime p, and every positive integer n, there exists an integer m = f(p, n), such that, for every non-empty word v of l...

Here {y\ = y — [y] = the fractional part of y. The number N is an integer >1. The number x0 is a given initial value such that 0 ^ Xo < 1. The number 0 is fixed. Some early references to numerical work with sequences of the type (1) are given by 0. Taussky and J. Todd in [1]. Regarding the sequence x„ as a function of Xo, I proved in [2] that for almost all x0 the sequence x„ is equidistributed...

ABSTRACT. A Diophantine m-tuple is a set A of m positive integers such that ab + 1 is a perfect square for every pair a, b of distinct elements of A. We derive an asymptotic formula for the number of Diophantine quadruples whose elements are bounded by x. In doing so, we extend two existing tools in ways that might be of independent interest. The Erdős-Turán inequality bounds the discrepancy be...

Take XN to be a symmetric matrix with real independent (modulo the symmetry constraint) equidistributed entries with law P and denote (λ1, · · · , λN) its eigenvalues. Then, Wigner [14] has shown that, if ∫ xdP (x) is finite, N−1 ∑N i=1 δλi/ √ N converges in expectation towards the semi-circle distribution. In this paper, we consider the case where P has a heavy tail and belong to the domain of...

A 4-wheel is a simple graph on 5 vertices with 8 edges, consisting of a 4-cycle, with a fifth vertex joined to each vertex in the 4-cycle. A A-fold 4-wheel system of order n is an edge-disjoint decomposition of AKn into 4-wheels. If two non-adjacent edges of the 4-cycle are removed, the result is a bowtie (that is, two triangles with a common vertex). In this paper necessary and sufficient cond...

In stream cipher the terms of an infinite binary seOluel1ce are added modulo two to binary plaintext to form One method Groth for this is to sum second products of the stages of maximal length sequence the products are defined by a Langford It was claimed that could be produced in this fashion with maximum provided are used. In this paper these sequences are by using recent results on stream

Some variable Krasnonsel’skiı̆-Mann iteration algorithms generate some sequences {xn}, {yn}, and {zn}, respectively, via the formula xn 1 1 − αn xn αnTN · · · T2T1xn, yn 1 1 − βn yn βn ∑N i 1 λiTiyn, zn 1 1 − γn 1 zn γn 1T n 1 zn, where T n TnmodN and the mod function takes values in {1, 2, . . . ,N}, {αn}, {βn}, and {γn} are sequences in 0, 1 , and {T1, T2, . . . , TN} are sequences of nonexpan...