Expand all Collapse all | Results 1 - 2 of 2 |
1. CMB 2012 (vol 56 pp. 785)
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)
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 |