# zbMATH — the first resource for mathematics

Diophantine approximation with four squares and one $$k$$-th power of primes. (English) Zbl 1238.11047
Authors’ abstract: We show that if $$\lambda_1,\lambda_2,\lambda_3,\lambda_4,\mu$$ are non-zero real numbers, not all of the same sign, $$\nu$$ is real, and at least one of the ratios $$\lambda_i/\lambda_j$$ is irrational, then for $$0<\sigma<{1\over {3k2^k}}$$ and any positive integer $$k\geq 3$$, the inequality $|\lambda_1 p_1^2+\lambda_2 p_2^2+\lambda_3 p_3^2+\lambda_4 p_4^2+\mu p_5^k + \nu|<(\max p_j)^{-\sigma}$ has infinitely many primes solutions $$(p_1,\ldots,p_5)$$.

##### MSC:
 11D75 Diophantine inequalities 11P32 Goldbach-type theorems; other additive questions involving primes
##### Keywords:
Diophantine approximation; powers of primes
Full Text: