Waring's problem involving D.H. Lehmer numbers

Abstract

For every positive integer $a$ which is coprime with $p$, $p$ is an odd prime, we denote by $\overline{a}$ the unique integer satisfying $1\le \overline{a}\le p$ and $a\overline{a}\equiv 1(\mathrm{mod}p)$. Put $$L(p)=\{a\in Z^{+}:(a,p)=1,2\nmid a+\overline{a}\}.$$ The elements of $L(p)$ are called D.H. Lehmer numbers. The main purpose of this paper is to prove that for any fixed odd prime $p$, every sufficiently large number unless it is congruent to 15 or 16$(\mathrm{mod}16)$ is representable as the sum of 14 fourth powers of D.H. Lehmer numbers. Furthermore, every sufficiently large number is representable as the sum of 16 fourth powers of D.H. Lehmer numbers.