On a Waring-Goldbach problem involving squares, cubes and biquadrates
Bull. Korean Math. Soc. 2018 Vol. 55, No. 6, 1659-1666 https://doi.org/10.4134/BKMS.b170937 Published online November 7, 2018 Printed November 30, 2018
Yuhui Liu Tongji University
Abstract : Let $P_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, it is proved that for every sufficiently large even integer $N$, the equation \begin{gather*} N=x^2+p_1^2+p_2^3+p_3^3+p_4^4+p_5^4 \end{gather*} is solvable with $x$ being an almost-prime $P_{4}$ and the other variables primes. This result constitutes an improvement upon that of L\"{u} \cite{7}.
Keywords : Waring-Goldbach problem, Hardy-Littlewood method, almost-prime, sieve theory
