Solutions Of Congruences Research Articles

The Hardy-Littlewood Prime K-Tuple Conjecture Is False

Using Jiang function we prove Jiang prime k -tuple theorem. We prove that the Hardy-Littlewood prime k -tuple conjecture is false. Jiang prime k -tuple theorem can replace the Hardy-Littlewood prime k -tuple conjecture. (A) Jiang prime k -tuple theorem [1, 2]. We define the prime k -tuple equation , i p p n + , (1) where 2 , 1, 1 i n i k = − L . we have Jiang function [1, 2] 2 ( ) ( 1 ( )) P J P P ω χ = Π − − , (2) where P P ω = Π , ( ) P χ is the number of solutions of congruence

The number of steps for construction of a Boolean solution to polynomial congruences and systems of polynomial congruences

Fixing the parameter, we decompose some NP-complete problems connected with the solution of congruences (and incongruences) of arithmetic terms modulo prime numbers into an infinite collection of problems for which polynomial-time algorithms are developed.

Functoriality and number of solutions of congruences

Functoriality and number of solutions of congruences

Small solutions of congruences. II

Small solutions of congruences. II

Small Solutions of Congruences in a Large Number of Variables

AbstractIt is shown that a system of congruences 1(x) ≡ . . . ≡ (x) = 0 (mod m)where each i(x) = i,(x1, .. . ,x2,) is a form of degree at most k has a nontrivial solution x satisfying |xi|≦cm(½)+∊ (i=1,...,S)with c = c(k,r,∊), provided that ∊ > 0 and that S > S1(k,r,∊).

Numbers of Solutions of Congruences: Poincare Series for Strongly Nondegenerate Forms

Numbers of Solutions of Congruences: Poincare Series for Strongly Nondegenerate Forms

The distribution of solutions of congruences

The distribution of solutions of congruences

The Numbers of Solutions of Congruences Involving Only kTH Powers

The Numbers of Solutions of Congruences Involving Only kTH Powers

The numbers of solutions of congruences involving only 𝑘th powers

The numbers of solutions of congruences involving only 𝑘th powers

Solutions of Congruences by Means of Divergent Series

It is shown in this note that if the solution, in the form of an infinite power series, of the algebraic equation f(x) = 0 with integral co-efficients, is known, we can derive the solutions (if any) of the congruence f(x)= 0 (mod. K n ) when we have solved it for the special case n = 1.