Formulation of Solutions of a Class of Solvable Standard Quadratic Congruence of Composite Modulus- an Odd Prime Positive Integer Multiple of Eight

2018 ◽  
Vol 61 (3) ◽  
pp. 227-229
Author(s):  
B M Roy ◽  
Keyword(s):  
Positive Integer ◽  
Integer Multiple ◽  
Composite Modulus ◽  
Quadratic Congruence

scholarly journals RP-179: Formulation of Solutions of Standard Bi-Quadratic Congruence Modulo a POSITIVE INTEGER MULTIPLE to nth Power of Four

2021 ◽  
Vol 9 (VII) ◽  
pp. 2598-2601
Author(s):  
Prof. B. M. Roy
Keyword(s):  
Positive Integer ◽  
Integer Multiple ◽  
Composite Modulus ◽  
First Case ◽  
Congruence Modulo ◽  
Quadratic Congruence ◽  
First Time

In this paper, the author has formulated the solutions of the standard bi-quadratic congruence of an even composite modulus modulo a positive integer multiple to nth power of four. First time a formula is established for the solutions. No literature is available for the current congruence. The author analysed the formulation of solutions in two different cases. In the first case of analysis, the congruence has the formulation which gives exactly eight incongruence solutions while in the second case of the analysis, the congruence has a different formulation of solutions and gives thirty-two incongruent solutions. A very simple and easy formulation to find all the solutions is presented here. Formulation is the merit of the paper.

scholarly journals RP-181: Formulation of Standard Quadratic Congruence of Even Composite Modulus modulo an Even Prime Raised to the Power n

2021 ◽  
Vol 9 (8) ◽  
pp. 3064-3069
Author(s):  
Prof B M Roy
Keyword(s):  
Number Theory ◽  
Positive Integer ◽  
Complete Formulation ◽  
Composite Modulus ◽  
Quadratic Congruence ◽  
Keywords Composite

Abstract: The paper presented here, is a standard quadratic congruence of composite modulus, studied rigorously and found the formulation incomplete. It was partially formulated by the earlier mathematicians. The present authors realised that the earlier formulation need a completion and a reformulation of the solutions is done along with two more results. The author considered the problem for reformulation, studied and reformulated the solutions completely. A partial formulation is found in a books of Number Theory by Zuckerman at el. There the formulation is only for an odd positive integer but nothing is said about even positive integer. The authors have provided a complete formulation of the said quadratic congruence and presented here. Keywords: Composite Modulus, Quadratic Congruence, Reformulation.

On the addition of two weighted squares of units mod n

2016 ◽  
Vol 12 (07) ◽  
pp. 1783-1790 ◽  
Author(s):  
Cui-Fang Sun ◽  
Zhi Cheng
Keyword(s):  
Positive Integer ◽  
Number Of Solutions ◽  
Residue Classes ◽  
Quadratic Congruence

For any positive integer [Formula: see text], let [Formula: see text] be the ring of residue classes modulo [Formula: see text] and [Formula: see text] be the group of its units. Recently, for any [Formula: see text], Yang and Tang obtained a formula for the number of solutions of the quadratic congruence [Formula: see text] with [Formula: see text] units, nonunits and mixed pairs, respectively. In this paper, for any [Formula: see text], we give a formula for the number of representations of [Formula: see text] as the sum of two weighted squares of units modulo [Formula: see text]. We resolve a problem recently posed by Yang and Tang.

scholarly journals Li-Yorke chaotic eigen set of the backward shift operator on ℓ2(𝕅)

2020 ◽  
Vol 7 (1) ◽  
pp. 180-182
Author(s):  
B. Sanooj ◽  
P.B. Vinodkumar
Keyword(s):  
Positive Integer ◽  
Complex Plane ◽  
Shift Operator ◽  
Integer Multiple ◽  
Backward Shift

AbstractIn this paper, we prove our main result that the Li-Yorke chaotic eigen set of a positive integer multiple of the backward shift operator on ℓ2 (𝕅) is a disk in the complex plane 𝔺 and the union of such Li-Yorke chaotic eigen set’s is the whole complex plane 𝔺.

scholarly journals A RESTRICTION OF EUCLID

2012 ◽  
Vol 86 (3) ◽  
pp. 506-509 ◽  
Author(s):  
GRANT CAIRNS ◽  
NHAN BAO HO
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Combinatorial Game ◽  
Integer Multiple ◽  
Positive Integers

AbstractEuclid is a well-known two-player impartial combinatorial game. A position in Euclid is a pair of positive integers and the players move alternately by subtracting a positive integer multiple of one of the integers from the other integer without making the result negative. The player who makes the last move wins. There is a variation of Euclid due to Grossman in which the game stops when the two entries are equal. We examine a further variation which we called M-Euclid where the game stops when one of the entries is a positive integer multiple of the other. We solve the Sprague–Grundy function for M-Euclid and compare the Sprague–Grundy functions of the three games.

k-barrier coverage reliability evaluation for wireless sensor networks using two-dimensional k-within-consecutive-r × s-out-of-m × n:F system

2019 ◽  
Vol 233 (6) ◽  
pp. 1029-1039
Author(s):  
Qiang Liu
Keyword(s):  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Positive Integer ◽  
Recursive Algorithm ◽  
Wireless Sensor ◽  
Barrier Coverage ◽  
Horizontal Distance ◽  
Two Dimensional ◽  
Coverage Problem ◽  
Integer Multiple

The reliability of k-barrier coverage for wireless sensor networks is evaluated in this article. We construct a two-dimensional k-within-consecutive- r ×  s-out-of- m ×  n:F system to describe the k-barrier coverage problem and propose the definition of the k-barrier coverage reliability. If the sensing radius r of each sensor is equal to half of the horizontal distance d between two adjacent sensors, then the formula for calculating the k-barrier coverage reliability can be derived directly. If r is a positive integer multiple of d, then a recursive algorithm is applied to calculate the coverage reliability. The effects of the probability that sensors are active, the number of rows of deployed sensors and the number of sensors in each row on the coverage reliability are also analyzed using some illustrative examples. The results show that, if r =  d/2, the reliability of a barrier decreases with an increase in the length of the barrier; if r =  t ×  d ( t is a positive integer), the reliability of a barrier hardly decreases with an increase in the length of the barrier.

scholarly journals Variations of the Game 3-Euclid

2012 ◽  
Vol 2012 ◽  
pp. 1-11 ◽  
Author(s):  
Nhan Bao Ho
Keyword(s):  
Positive Integer ◽  
The Other ◽  
Integer Multiple ◽  
Positive Integers

We present two variations of the game 3-Euclid. The games involve a triplet of positive integers. Two players move alternately. In the first game, each move is to subtract a positive integer multiple of the smallest integer from one of the other integers as long as the result remains positive. In the second game, each move is to subtract a positive integer multiple of the smallest integer from the largest integer as long as the result remains positive. The player who makes the last move wins. We show that the two games have the same -positions and positions of Sprague-Grundy value 1. We present three theorems on the periodicity of -positions and positions of Sprague-Grundy value 1. We also obtain a theorem on the partition of Sprague-Grundy values for each game. In addition, we examine the misère versions of the two games and show that the Sprague-Grundy functions of each game and its misère version differ slightly.

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