On the Waring-Goldbach problem for one cube and nine biquadrates

Abstract

Let N be a sufficiently large integer. In this paper, it is proved that with at most O\left(N^{\frac{17}{18}+\epsilon }\right)exceptions, all positive integers satisfying some necessary congruence conditions up to N can be represented in the form p_1^3+p_2^4+p_3^4+p_4^4+p_5^4+p_6^4+p_7^4+p_8^4+p_9^4+p_{10}^4, where p_1,p_2,\dots ,p_{10} are prime numbers.

Keywords

Waring-Goldbach problem / Hardy-Littlewood method / exceptional set

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