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Small prime solutions of equations with powers 2 and 3. (Chinese. English summary) Zbl 1499.11300

Summary: This paper concerns the solvability of the prime variables equations
\[ a_1 p_1^2 +a_2 p_2^2 +a_3 p_3^2 +a_4 p_4^3 +a_5 p_5^3 +a_6 p_6^3 = b, \]
where \(a_1, a_2, \ldots, a_6\) are non-zero integers and \(b\) is any integer. Suppose that \(a_1, a_2, \ldots, a_6\) satisfy some related conditions. Using the circle method we prove that
(i) if all \(a_1, a_2, \ldots, a_6\) are positive, and \(b\gg \max\{|a_j|\}^{33+\varepsilon}\), then the equation is soluble in primes \(p_j, j = 1,\cdots, 6\), and
(ii) if \(a_1, a_2, \ldots, a_6\) are not all of the same sign, then the equation has prime solutions satisfying
\[ \max\{p_1^2, p_2^2, p_3^2, p_4^3, p_5^3, p_6^3\}\ll |b| + \max\{|a_j|\}^{32+\varepsilon}. \]

MSC:

11P32 Goldbach-type theorems; other additive questions involving primes
11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
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