Small Prime Solutions of Diophantine Equations

We study the existence of small prime solutions to a Diophantine equation that combines quadratic and higher degree prime variables. Specifically, we consider the equation \(a_1p_1^2\)+\(a_2p_2^2\)+\(a_3p_3^2\)+\(a_4p_4^2\)+\(a_5p_5^k\)=n, where the ai are pairwise coprime nonzero integers, k is an integer with k \(\ge\) 3, and n is an integer. Using the Hardy–Littlewood circle method under a strong local solvability condition, we show that the potential influence of a Siegel zero can be completely avoided. This allows us to circumvent the Deuring–Heilbronn phenomenon—a common source of complication in such problems and obtain stronger bounds without relying on extensive numerical computation. The following results are proved for any \(\varepsilon\) > 0: （i）If all coefficients ai are all positive and \(n \gg \max \left\{\left|a_i\right|\right\}^{3 k \sigma_k^{-1}+1+\varepsilon}\), then the equation is solvable in primes. （ii）If the coefficients are not all of the same sign, then there exist prime solutions satisfying\(\left\{p_1^2, p_2^2, p_3^2, p_4^2, p_5^k\right\} \ll|n|+\max \left\{\left|a_i\right|\right\}^{3 k \sigma_k^{-1}+\varepsilon}\), where \(\sigma_k^{-1}=\min \left\{2^{k-1}, k(k-1)\right\}\) , and the implied constant depends only on \(\varepsilon\). These results extend earlier work on pure quadratic and linear prime equations, demonstrating how refined local hypotheses can lead to cleaner analytic conclusions in additive prime number theory.