The integral part of a nonlinear form with five cubes of primes*

This paper shows that if μ ₁ , . . . , μ ₅ are nonzero real numbers, not all negative, at least one of μ _j $ \left( {1\leqslant j\leqslant 5} \right) $ is irrational, and k is a positive integer, then there exist infinitely many primes p ₁ , . . . , p ₅ , p such that $ \left {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3} \right]=kp $ . In particular, $ \left {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3} \right] $ represents infinitely many primes.