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.

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 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.

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].

On the distribution of \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} \(\sqrt p\) \end{document} modulo one involving primes of special type

2013 ◽  
Vol 50 (4) ◽  
pp. 470-490 ◽  
Author(s):  
Yingchun Cai
Keyword(s):  
Infinitely Many Solutions ◽  
Prime Factors ◽  
Almost Prime

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper we show that the inequality \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\left\{ {\sqrt p } \right\} < p^{ - \tfrac{1} {{15.5}}}$$ \end{document} has infinitely many solutions in primes p such that p + 2 = P4.

On the distribution of αp modulo one for primes p of a special form

2010 ◽  
Vol 60 (6) ◽  
Author(s):  
T. Todorova ◽  
D. Tolev
Keyword(s):  
Number Theory ◽  
Prime Number ◽  
Special Form ◽  
Classical Problem ◽  
Analytic Number Theory ◽  
Analytic Number ◽  
Almost Prime

AbstractA classical problem in analytic number theory is to study the distribution of αp modulo 1, where α is irrational and p runs over the set of primes. We consider the subsequence generated by the primes p such that p+2 is an almost-prime (the existence of infinitely many such p is another topical result in prime number theory) and prove that its distribution has a similar property.

AN ADDITIVE PROBLEM INVOLVING PIATETSKI-SHAPIRO PRIMES

2011 ◽  
Vol 07 (05) ◽  
pp. 1359-1378 ◽  
Author(s):  
XINNA WANG ◽  
YINGCHUN CAI
Keyword(s):  
Positive Integer ◽  
Additive Problem ◽  
Prime Factors ◽  
Almost Prime

Let Pr denote an almost-prime with at most r prime factors, counted according to multiplicity. In this paper it is proved that there exist infinitely many primes of the form p = [nc] such that p + 2 = Pr, where r is the least positive integer satisfying certain inequalities. In particular for [Formula: see text] we have r = 5. This result constitutes an improvement upon that of T. P. Peneva.

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