It is widely conjectured that every sufficiently large integer satisfying certain necessary congruence conditions is the sum of 4 cubes of prime numbers. As an approximation to this conjecture, we shall establish two results in this paper. Firstly, we show that every large odd integer is the sum of a prime, 4 cubes of primes and 15 powers of 2. Secondly, we show that the conjecture is true for at least $$8.25\%$$ 8.25 % of the positive integers.

中文翻译：

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涉及素数立方体的 Waring-Goldbach 问题

人们普遍推测，每个满足某些必要同余条件的足够大的整数都是 4 个素数的立方之和。作为对这一猜想的近似，我们将在本文中建立两个结果。首先，我们证明每个大奇数都是一个素数、4 个素数的立方和 2 的 15 次幂之和。其次，我们证明该猜想对至少 $$8.25\%$$ 8.25% 的正整数成立.