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This paper shows that if μ1, . . . , μ5 are nonzero real numbers, not all negative, at least one of μj $$ \left( {1\leqslant j\leqslant 5} \right) $$ is irrational, and k is a positive integer, then there exist infinitely many primes p1, . . . , p5, p such that $$ \left[ {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3} \right]=kp $$ . In particular, $$ \left[ {{\mu_1}p_1^3+\cdots +{\mu_5}p_5^3}...
We show that if λ1,λ2,λ3,λ4 are nonzero real numbers, not all of the same sign, η is real, and at least one of the ratios λ1/λj (j=2,3,4) is irrational, then given any real number ω>0, there are infinitely many ordered quadruples of primes (p1,p2,p3,p4) for which $$\bigl|\lambda_1 p_1+\lambda_2 p^2_2+\lambda_3 p^2_3+\lambda_4p^2_4+\eta \bigr|<(\max p_j)^{-\frac{1}{28}+\omega}...
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