An asymptotic formula for the number of prime solutions for multivariate linear equations

In this paper, we study the multivariate linear equations with arbitrary positive integral coefficients. Under the Generalized Riemann Hypothesis, we obtained the asymptotic formula for the linear equations with more than five prime variables. This asymptotic formula is composed of three parts, that is, the first main term, the explicit second main term and the error term. Among them, the first main term is similar with the former one, the explicit second main term is relative to the non-trivial zeros of Dirichlet L-functions, and our error term improves the former one.     

On the least prime primitive root modulo a prime

We derive a conditional formula for the natural density of prime numbers having its least prime primitive root equal to , and compare theoretical results with the numerical evidence.

     

Small prime solutions of a pair of linear equations in five prime variables

Assume an additional congruent condition on the coefficients. We prove that the pair 5 of linear equations ∑j=1^5 αλjpj = bλ （λ= 1, 2） has solutions in primes pj satisfying pj 〈〈 （｜b1｜＋｜b2｜＋1） maxλ,j ｜αλj｜^2318＋ε. This improves the exponent 79680 without assuming the additional condition of the second author＇s.     

On the distinctness of modular reductions of maximal length sequences modulo odd prime powers

We discuss the distinctness problem of the reductions modulo of maximal length sequences modulo powers of an odd prime , where the integer has a prime factor different from . For any two different maximal length sequences generated by the same polynomial, we prove that their reductions modulo are distinct. In other words, the reduction modulo of a maximal length sequence is proved to contain all the information of the original sequence.

     

On short Kloosterman sums modulo a prime

Using the Karatsuba method, we obtain new estimates for Kloosterman sums modulo a prime, which, under certain constraints on the number of summands, are sharper than similar estimates found earlier.

     

Conditional bounds for small prime solutions of linear equations

Leta ₁,a ₂,a ₃ be non-zero integers with gcd(a ₁ a ₂,a ₃)=1 and letb be an arbitrary integer satisfying gcd (b, a _i,a _j) =1 fori≠j andb≡a ₁+a ₂+a ₃ (mod 2). In a previous paper [3] which completely settled a problem of A. Baker, the 2nd and 3rd authors proved that ifa ₁,a ₂,a ₃ are not all of the same sign, then the equationa ₁ p ₁+a ₂ p ₂+a ₃ p ₃=b has a solution in primesp _j satisfying $$\mathop {\max }\limits_{1 \leqslant j \leqslant 3} p_j \leqslant 3\left| b \right| + (3\mathop {\max }\limits_{1 \leqslant j \leqslant 3} \left| {a_j } \right|)^A $$ whereA>0 is an absolute constant. In this paper, under the Generalized Riemann Hypothesis, the authors obtain a more precise bound for the solutionsp _j . In particular they obtainA<4+∈ for some ∈>0. An immediate consquence of the main result is that the Linnik's courtant is less than or equal to 2.     

A linear function associated to asymptotic prime divisors

Let be a Noetherian standard -graded ring and finitely generated, -graded -modules. Let be finitely many homogeneous ideals of . We show that there exist linear functions such that the associated primes over of and are stable whenever satisfies and , respectively.

     

A distribution property of the solutions of a congruence modulo a large prime

A regularity in the distribution of the solutions of the congruence f(X₁ ,…, X_n)  0 (modp) is shown.     

The maximal nonuniqueness classes of solutions to the Cauchy problem for linear equations

We study linear partial differential equations with increasing coefficients in a half-plane. We establish maximal nonuniqueness classes of solutions to the Cauchy problem for these equations. The proof is based on a new estimation method for a solution to the dual differential equation with a parameter.     

Small prime solutions to cubic equations

Let a_1,..., a_9 be nonzero integers not of the same sign, and let b be an integer. Suppose that a_1,..., a_9 are pairwise coprime and a_1 + + a_9 ≡ b(mod 2). We apply the p-adic method of Davenport to find an explicit P = P(a_1,..., a_9, n) such that the cubic equation a_1p_1~3+ + a9p_9~3= b is solvable with p_j 《 P for all 1 ≤ j ≤ 9. It is proved that one can take P = max{|a_1|,..., |a_9|}~c+ |b|~(1/3) with c = 2. This improves upon the earlier result with c = 14 due to Liu(2013).     

Transformation of approximate solutions of linear parabolic equations into asymptotic solutions of quasilinear parabolic equations

Moscow Institute of Electronics and Mathematics. Translated from Matematicheskie Zametki, Vol. 56, No. 6, pp. 122–126, December, 1994.     

On the cycle structure of repeated exponentiation modulo a prime

In a recent work, Shallit and Vasiga have obtained several results about tails and cycles in orbits of repeated squaring. Some of these results have been based on the Extended Riemann Hypothesis. Here, we extend their result to repeated exponentiation with any fixed exponent e and also show that in fact classical unconditional results about the distribution of primes in arithmetic progressions, combined with very elementary arguments, are quite sufficient to generalise and give an unconditional proof of their asymptotic formulas.     

An asymptotic property of the solutions to second order linear nonautonomous delay differential equations

Second order linear nonautonomous delay differential equations are considered, and a fundamental asymptotic criterion for the solutions is established, by the use of the concept of generalized characteristic equation.     

On asymptotic formulas for solutions of systems of linear differential equations with a degerate matrix with derivatives

In this paper, we suggest a method for the construction of asymptotic formulas for solutions of systems of differential equations in the case where the roots of the characteristic equation are simple Ukrainian Pedagogical University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1278–1285, September, 1996.