scholarly journals QUADRATIC NONRESIDUES AND NONPRIMITIVE ROOTS SATISFYING A COPRIMALITY CONDITION

2018 ◽  
Vol 99 (2) ◽  
pp. 177-183
Author(s):  
JAITRA CHATTOPADHYAY ◽  
BIDISHA ROY ◽  
SUBHA SARKAR ◽  
R. THANGADURAI
Keyword(s):  
Real Number ◽  
Primitive Root ◽  
Log P ◽  
Primitive Root Modulo ◽  
Image Position ◽  
Quadratic Nonresidue

Let $q\geq 1$ be any integer and let $\unicode[STIX]{x1D716}\in [\frac{1}{11},\frac{1}{2})$ be a given real number. We prove that for all primes $p$ satisfying $$\begin{eqnarray}p\equiv 1\!\!\!\!\hspace{0.6em}({\rm mod}\hspace{0.2em}q),\quad \log \log p>\frac{2\log 6.83}{1-2\unicode[STIX]{x1D716}}\quad \text{and}\quad \frac{\unicode[STIX]{x1D719}(p-1)}{p-1}\leq \frac{1}{2}-\unicode[STIX]{x1D716},\end{eqnarray}$$ there exists a quadratic nonresidue $g$ which is not a primitive root modulo $p$ such that $\text{gcd}(g,(p-1)/q)=1$.

scholarly journals Congruences involving product of intervals and sets with small multiplicative doubling modulo a prime and applications

2016 ◽  
Vol 160 (3) ◽  
pp. 477-494 ◽  
Author(s):  
J. CILLERUELO ◽  
M. Z. GARAEV
Keyword(s):  
Upper Bound ◽  
Absolute Constant ◽  
Primitive Root ◽  
Multiple Roots ◽  
Number Of Solutions ◽  
Residue Classes ◽  
Primitive Root Modulo ◽  
Image Position ◽  
Almost All ◽  
Positive Integers

AbstractIn this paper we obtain new upper bound estimates for the number of solutions of the congruence $$\begin{equation} x\equiv y r\pmod p;\quad x,y\in \mathbb{N},\quad x,y\le H,\quad r\in \mathcal{U}, \end{equation}$$ for certain ranges of H and |${\mathcal U}$|, where ${\mathcal U}$ is a subset of the field of residue classes modulo p having small multiplicative doubling. We then use these estimates to show that the number of solutions of the congruence $$\begin{equation} x^n\equiv \lambda\pmod p; \quad x\in \mathbb{N}, \quad L<x<L+p/n, \end{equation}$$ is at most $p^{\frac{1}{3}-c}$ uniformly over positive integers n, λ and L, for some absolute constant c > 0. This implies, in particular, that if f(x) ∈ $\mathbb{Z}$[x] is a fixed polynomial without multiple roots in $\mathbb{C}$, then the congruence xf(x) ≡ 1 (mod p), x ∈ $\mathbb{N}$, x ⩽ p, has at most $p^{\frac{1}{3}-c}$ solutions as p → ∞, improving some recent results of Kurlberg, Luca and Shparlinski and of Balog, Broughan and Shparlinski. We use our results to show that almost all the residue classes modulo p can be represented in the form xgy (mod p) with positive integers x < p5/8+ϵ and y < p3/8. Here g denotes a primitive root modulo p. We also prove that almost all the residue classes modulo p can be represented in the form xyzgt (mod p) with positive integers x, y, z, t < p1/4+ϵ.

scholarly journals Homogeneous orbit closures and applications

2011 ◽  
Vol 32 (2) ◽  
pp. 785-807 ◽  
Author(s):  
ELON LINDENSTRAUSS ◽  
URI SHAPIRA
Keyword(s):  
Real Number ◽  
Unit Volume ◽  
Orbit Closures ◽  
Image Position

AbstractWe give new classes of examples of orbits of the diagonal group in the space of unit volume lattices in ℝd for d≥3 with nice (homogeneous) orbit closures, as well as examples of orbits with explicitly computable but irregular orbit closures. We give Diophantine applications to the former; for instance, we show that, for all γ,δ∈ℝ, where 〈c〉 denotes the distance of a real number c to the integers.

Explicit upper bound on the least primitive root modulo p2

2021 ◽  
pp. 1-7
Author(s):  
Bo Chen
Keyword(s):  
Recent Result ◽  
Upper Bound ◽  
Primitive Root ◽  
Character Sums ◽  
Primitive Root Modulo ◽  
Primitive Roots

In this paper, we give an explicit upper bound on [Formula: see text], the least primitive root modulo [Formula: see text]. Since a primitive root modulo [Formula: see text] is not primitive modulo [Formula: see text] if and only if it belongs to the set of integers less than [Formula: see text] which are [Formula: see text]th power residues modulo [Formula: see text], we seek the bounds for [Formula: see text] and [Formula: see text] to find [Formula: see text] which satisfies [Formula: see text], where, [Formula: see text] denotes the number of primitive roots modulo [Formula: see text] not exceeding [Formula: see text], and [Formula: see text] denotes the number of [Formula: see text]th powers modulo [Formula: see text] not exceeding [Formula: see text]. The method we mainly use is to estimate the character sums contained in the expressions of the [Formula: see text] and [Formula: see text] above. Finally, we show that [Formula: see text] for all primes [Formula: see text]. This improves the recent result of Kerr et al.

Small Solutions of the Congruence ax2 + by2 ≡ c(mod k)

1969 ◽  
Vol 12 (3) ◽  
pp. 311-320 ◽  
Author(s):  
Kenneth S. Williams
Keyword(s):  
Asymptotic Formula ◽  
Absolute Constant ◽  
Log P ◽  
Positive Absolute Constant ◽  
Image Position

In 1957, Mordell [3] provedTheorem. If p is an odd prime there exist non-negative integers x, y ≤ A p3/4 log p, where A is a positive absolute constant, such that(1.1)provided (abc, p) = 1.Recently Smith [5] has obtained a sharp asymptotic formula for the sum where r(n) denotes the number of representations of n as the sum of two squares.

On Cyclotomic Numbers of Order Sixteen

1954 ◽  
Vol 6 ◽  
pp. 449-454 ◽  
Author(s):  
Emma Lehmer
Keyword(s):  
Linear Combination ◽  
Primitive Root ◽  
Number Of Solutions ◽  
Integer Coefficients ◽  
Cyclotomic Numbers ◽  
Image Position ◽  
Mod P

It has been shown by Dickson (1) that if (i, j)8 is the number of solutions of (mod p),then 64(i,j)8 is expressible for each i,j, as a linear combination with integer coefficients of p, x, y, a, and b where,anda ≡ b ≡ 1 (mod 4),while the sign of y and b depends on the choice of the primitive root g. There are actually four sets of such formulas depending on whether p is of the form 16n + 1 or 16n + 9 and whether 2 is a quartic residue or not.

Discounted branching random walks

1985 ◽  
Vol 17 (01) ◽  
pp. 53-66
Author(s):  
K. B. Athreya
Keyword(s):  
Functional Equation ◽  
Random Walks ◽  
Unique Solution ◽  
Log P ◽  
Branching Random Walks ◽  
Probabilistic Construction ◽  
Image Position

Let F(·) be a c.d.f. on [0,∞), f(s) = ∑∞ 0 pjsi a p.g.f. with p 0 = 0, &lt; 1 &lt; m = Σj p j &lt; ∞ and 1 &lt; ρ &lt;∞. For the functional equation for a c.d.f. H(·) on [0,∞] we establish that if 1 – F(x) = O(x –θ ) for some θ &gt; α =(log m)/(log p) there exists a unique solution H(·) to (∗) in the class C of c.d.f.’s satisfying 1 – H(x) = o(x –α ). We give a probabilistic construction of this solution via branching random walks with discounting. We also show non-uniqueness if the condition 1 – H(x) = o(x –α ) is relaxed.

scholarly journals An estimate for trigonometrical sums over square-free integers with a constant number of prime factors

1967 ◽  
Vol 15 (4) ◽  
pp. 249-255
Author(s):  
Sean Mc Donagh
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Prime Number ◽  
Rational Number ◽  
Large Integer ◽  
Constant Number ◽  
The Real ◽  
Prime Factors ◽  
Squarefree Number ◽  
Image Position

1. In deriving an expression for the number of representations of a sufficiently large integer N in the formwhere k: is a positive integer, s(k) a suitably large function of k and pi is a prime number, i = 1, 2, …, s(k), by Vinogradov's method it is necessary to obtain estimates for trigonometrical sums of the typewhere ω = l/k and the real number a satisfies 0 ≦ α ≦ 1 and is “near” a rational number a/q, (a, q) = 1, with “large” denominator q. See Estermann (1), Chapter 3, for the case k = 1 or Hua (2), for the general case. The meaning of “near” and “arge” is made clear below—Lemma 4—as it is necessary for us to quote Hua's estimate. In this paper, in Theorem 1, an estimate is obtained for the trigonometrical sumwhere α satisfies the same conditions as above and where π denotes a squarefree number with r prime factors. This estimate enables one to derive expressions for the number of representations of a sufficiently large integer N in the formwhere s(k) has the same meaning as above and where πri, i = 1, 2, …, s(k), denotes a square-free integer with ri prime factors.

AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

Eigenvalues of Finite Band-Width Hilbert Space Operators and Their Application to Orthogonal Polynomials

1989 ◽  
Vol 41 (1) ◽  
pp. 106-122 ◽  
Author(s):  
Attila Máté ◽  
Paul Nevai
Keyword(s):  
Hilbert Space ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Main Diagonal ◽  
Band Width ◽  
Positive Part ◽  
Hilbert Space Operators ◽  
Image Position ◽  
Finite Band

The main result of this paper concerns the eigenvalues of an operator in the Hilbert space l2that is represented by a matrix having zeros everywhere except in a neighborhood of the main diagonal. Write (c)+ for the positive part of a real number c, i.e., put (c+ = cif c≧ 0 and (c)+=0 otherwise. Then this result can be formulated as follows. Theorem 1.1. Let k ≧ 1 be an integer, and consider the operator S on l2 such that

An integral which occurs in statistics

1933 ◽  
Vol 29 (2) ◽  
pp. 271-276 ◽  
Author(s):  
A. E. Ingham
Keyword(s):  
Real Number ◽  
Positive Definite ◽  
Direct Evaluation ◽  
Independent Variables ◽  
Symmetrical Matrices ◽  
Image Position

1. In this note we give a direct evaluation of the integralwhose value has been inferred from the theory of statistics. Here A = Ap = (αμν) and C = Cp = (Cμν) are real symmetrical matrices, of which A is positive definite; there are ½ p (p + 1) independent variables of integration tμν (1 ≤ μ ≤ ν ≤ p), and tμν is written also as tνμ for symmetry of notation; in the summation ∑ the variables μ, ν run independently from 1 to p; k is a real number. A word of explanation is necessary with regard to the determination of the power |A − iT|−k. Since A is positive definite and T real and symmetric, the roots of the equation

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