THE GOLDBACH PROBLEM FOR PRIMES THAT ARE SUMS OF TWO SQUARES PLUS ONE

Abstract

We study the Goldbach problem for primes represented by the polynomial . The set of such primes is sparse in the set of all primes, but the infinitude of such primes was established by Linnik. We prove that almost all even integers satisfying certain necessary local conditions are representable as the sum of two primes of the form . This improves a result of Matomäki, which tells us that almost all even satisfying a local condition are the sum of one prime of the form and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the form contain infinitely many three-term arithmetic progressions, and that the numbers , with irrational and running through primes of the form , are distributed rather uniformly.

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