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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
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