@Article{LanguascoZaccagnini2012c, Author = "A. Languasco and A. Zaccagnini", Title = "A {Diophantine} problem with a prime and three squares of primes", Journal = "J. Number Theory", Volume = 132, Issue = 12, Pages = "3016--3028", Year = 2012, Abstract = "We prove that if $\lambda_1$, $\lambda_2$, $\lambda_3$ and $\lambda_4$ are non-zero real numbers, not all of the same sign, $\lambda_1 / \lambda_2$ is irrational, and $\alpha$ is any real number then, for any $\varepsilon > 0$ the inequality $\bigl\vert \lambda_1 p_1 + \lambda_2 p_2^2 + \lambda_3 p_3^2 + \lambda_4 p_4^2 - \alpha \bigr\vert \le \bigl( \max_j p_j \bigr)^{-1 / 18 + \varepsilon}$ has infinitely many solutions in prime variables $p_1$, \dots, $p_4$." }