scholarly journals Sign Changes of Kloosterman Sums with Almost Prime Moduli. II: Corrigendum

2020 ◽  
Author(s):  
Ping Xi
Keyword(s):  
Kloosterman Sums ◽  
Kloosterman Sum ◽  
Sign Changes ◽  
Prime Factors ◽  
Quantitative Statement ◽  
Almost Prime

Abstract We give a corrigendum to the previous paper [ 8] and recover the same quantitative statement: the Kloosterman sum changes sign infinitely often as the modulus runs over squarefree numbers with at most seven prime factors.

scholarly journals Sign changes of Kloosterman sums and exceptional characters

2018 ◽  
Vol 147 (1) ◽  
pp. 61-75 ◽  
Author(s):  
Sary Drappeau ◽  
James Maynard
Keyword(s):  
Kloosterman Sums ◽  
Sign Changes

On the hyper-Kloosterman sum and its fourth power mean

2009 ◽  
Vol 46 (4) ◽  
pp. 479-492
Author(s):  
Zhefeng Xu
Keyword(s):  
Kloosterman Sums ◽  
Fourth Power ◽  
Kloosterman Sum ◽  
Power Mean ◽  
Fourth Power Mean

The main purpose of this paper is to study the calculating problem of the fourth power mean of the hyper-Kloosterman sums, and give an exact calculating formula for them.

scholarly journals On the Equation x21 + x22 + x23 + x24 = N with Variables such that x1x2x3x4 + 1 is an Almost-Prime

2014 ◽  
Vol 59 (1) ◽  
pp. 1-26
Author(s):  
T. L. Todorova ◽  
D. I. Tolev
Keyword(s):  
Natural Numbers ◽  
Lagrange’S Equation ◽  
Prime Factors ◽  
Almost Prime

Abstract We consider Lagrange’s equation x21 +x22 +x23 +x24 = N, where N is a sufficiently large and odd integer, and prove that it has a solution in natural numbers x1, . . . , x4 such that x1x2x3x4 + 1 has no more than 48 prime factors.

scholarly journals Distribution of α p 2 Modulo One with Prime Variable p of a Special Form

2021 ◽  
Vol 2021 ◽  
pp. 1-22
Author(s):  
Fei Xue ◽  
Jinjiang Li ◽  
Min Zhang
Keyword(s):  
Special Form ◽  
Prime Factors ◽  
Prime Variable ◽  
Almost Prime

Let P r denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, it is proved that, for α ∈ ℝ / ℚ ,   β ∈ ℝ , and 0 < θ < 10 / 1561 , there exist infinitely many primes p , such that α p 2 + β < p − θ and p + 2 = P 4 , which constitutes an improvement upon the previous result.

Linear Equations with Small Prime and Almost Prime Solutions

2008 ◽  
Vol 51 (3) ◽  
pp. 399-405
Author(s):  
Xianmeng Meng
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Linear Equations ◽  
Prime Factors ◽  
Almost All ◽  
Sharp Lower Bound ◽  
Almost Prime

AbstractLet b1, b2 be any integers such that gcd(b1, b2) = 1 and c1|b1| < |b2| ≤ c2|b1|, where c1, c2 are any given positive constants. Let n be any integer satisfying gcd(n, bi) = 1, i = 1, 2. Let Pk denote any integer with no more than k prime factors, counted according to multiplicity. In this paper, for almost all b2, we prove (i) a sharp lower bound for n such that the equation b1p + b2m = n is solvable in prime p and almost prime m = Pk, k ≥ 3 whenever both bi are positive, and (ii) a sharp upper bound for the least solutions p, m of the above equation whenever bi are not of the same sign, where p is a prime and m = Pk, k ≥ 3.

On Waring–Goldbach problem for unlike powers

2018 ◽  
Vol 14 (06) ◽  
pp. 1669-1687
Author(s):  
Jinjiang Li ◽  
Min Zhang
Keyword(s):  
The Other ◽  
Goldbach Problem ◽  
Prime Factors ◽  
Almost Prime

Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In this paper, it is proved that, for [Formula: see text] and for every sufficiently large even integer [Formula: see text], the equation [Formula: see text] is solvable with [Formula: see text] being an almost-prime [Formula: see text] and the other variables primes, which constitutes an extension upon that of Lü.

LAGRANGE'S FOUR SQUARES THEOREM WITH VARIABLES OF SPECIAL TYPE

2010 ◽  
Vol 06 (08) ◽  
pp. 1801-1817 ◽  
Author(s):  
YINGCHUN CAI
Keyword(s):  
Large Integer ◽  
Prime Factors ◽  
Almost Prime

Let N denote a sufficiently large integer satisfying N ≡ 4 (mod 24), and Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper, we proved that the equation [Formula: see text] is solvable in one prime and three P42, or in four P13. These results constitute improvements upon that of Heath-Brown and Tolev.

scholarly journals On the Waring–Goldbach problem for one square and five cubes

2018 ◽  
Vol 14 (09) ◽  
pp. 2425-2440
Author(s):  
Jinjiang Li ◽  
Min Zhang
Keyword(s):  
The Other ◽  
Goldbach Problem ◽  
Prime Factors ◽  
Almost Prime

Let [Formula: see text] denote an almost-prime with at most [Formula: see text] prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer [Formula: see text], the equation [Formula: see text] is solvable with [Formula: see text] being an almost-prime [Formula: see text] and the other variables primes. This result constitutes an improvement upon that of Cai, who obtained the same conclusion, but with [Formula: see text] in place of [Formula: see text].

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