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Search: MSC category 11P32 ( Goldbach-type theorems; other additive questions involving primes )

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1. CMB 2012 (vol 56 pp. 785)

Liu, Zhixin
Small Prime Solutions to Cubic Diophantine Equations
Let $a_1, \cdots, a_9$ be non-zero integers and $n$ any integer. Suppose that $a_1+\cdots+a_9 \equiv n( \textrm{mod}\,2)$ and $(a_i, a_j)=1$ for $1 \leq i \lt j \leq 9$. In this paper we prove that (i) if $a_j$ are not all of the same sign, then the above cubic equation has prime solutions satisfying $p_j \ll |n|^{1/3}+\textrm{max}\{|a_j|\}^{14+\varepsilon};$ and (ii) if all $a_j$ are positive and $n \gg \textrm{max}\{|a_j|\}^{43+\varepsilon}$, then the cubic equation $a_1p_1^3+\cdots +a_9p_9^3=n$ is soluble in primes $p_j$. This result is the extension of the linear and quadratic relative problems.

Keywords:small prime, Waring-Goldbach problem, circle method
Categories:11P32, 11P05, 11P55

2. CMB 2008 (vol 51 pp. 399)

Meng, Xianmeng
Linear Equations with Small Prime and Almost Prime Solutions
Let $b_1, b_2$ be any integers such that $\gcd(b_1, b_2)=1$ and $c_1|b_1|<|b_2|\leq c_2|b_1|$, where $c_1, c_2$ are any given positive constants. Let $n$ be any integer satisfying $\{gcd(n, b_i)=1$, $i=1,2$. Let $P_k$ denote any integer with no more than $k$ prime factors, counted according to multiplicity. In this paper, for almost all $b_2$, we prove (i) a sharp lower bound for $n$ such that the equation $b_1p+b_2m=n$ is solvable in prime $p$ and almost prime $m=P_k$, $k\geq 3$ whenever both $b_i$ are positive, and (ii) a sharp upper bound for the least solutions $p, m$ of the above equation whenever $b_i$ are not of the same sign, where $p$ is a prime and $m=P_k, k\geq 3$.

Keywords:sieve method, additive problem
Categories:11P32, 11N36

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