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Linear Equations with Small Prime and Almost Prime Solutions

 Printed: Sep 2008
  • Xianmeng Meng
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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 sieve method, additive problem
MSC Classifications: 11P32, 11N36 show english descriptions Goldbach-type theorems; other additive questions involving primes
Applications of sieve methods
11P32 - Goldbach-type theorems; other additive questions involving primes
11N36 - Applications of sieve methods

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