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We study the Goldbach problem for primes represented by the polynomialx  2   +   y  2   +  1   . 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   n   satisfying certain necessary local conditions are representable as the sum of two primes of the formx  2   +   y  2   +  1   . This improves a result of Matomäki, which tells us that almost all even   n   satisfying a local condition are the sum of one prime of the formx  2   +   y  2   +  1    and one generic prime. We also solve the analogous ternary Goldbach problem, stating that every large odd   n   is the sum of three primes represented by our polynomial. As a byproduct of the proof, we show that the primes of the formx  2   +   y  2   +  1    contain infinitely many three‐term arithmetic progressions, and that the numbers    α  p    (  mod    1  )   , with   α   irrational and   p   running through primes of the formx  2   +   y  2   +  1   , are distributed rather uniformly.

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