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Let 1 < k < 33/29. We prove that if λ₁, λ₂ and λ₃ are non-zero real numbers, not all of the same sign and such that λ₁/λ₂ is irrational, and ϖ is any real number, then for any ε > 0 the inequality $|\lambda ₁p₁+\lambda ₂p²₂+\lambda ₃p{₃}^k+\varpi |\le {\left(ma{x}_j{p}_j\right)}^{-(33-29k)/\left(72k\right)+\epsilon }$ has infinitely many solutions in prime variables p₁, p₂, p₃.

Linnik proved, assuming the Riemann Hypothesis, that for any $ϵ\&gt;0$, the interval $[N,N+{log}^{3+ϵ}N]$ contains a number which is the sum of two primes, provided that $N$ is sufficiently large. This has subsequently been improved to the same assertion being valid for the smaller gap $C{log}^{2}N$, the added new ingredient being Selberg’s estimate for the mean-square of primes in short intervals. Here we give another proof of this sharper result which avoids the use of Selberg’s estimate and is therefore more in the spirit of Linnik’s...