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[5] Heath-Brown, D. R. (2001). Primes represented by $x^3 + 2y^3$. Acta Mathematica 186, pp. 1-84.

[6] Friedlander, J. and Iwaniec, H. (1997). Using a parity-sensitive sieve to count prime values of a polynomial. Proceedings of the National Academy of Sciences 94, pp. 1054-1058.

[5] Heath-Brown, D. R. (2001). Primes represented by $x^3 + 2y^3$. Acta Mathematica 186, pp. 1-84; Wayback Machine.

[6] Friedlander, J. and Iwaniec, H. (1997). Using a parity-sensitive sieve to count prime values of a polynomial. Proceedings of the National Academy of Sciences 94, pp. 1054-1058.

[9] Motohashi, Y. (1969). On the distribution of prime numbers which are of the form $x^2+y^2+1$. Acta Arithmetica 16, pp. 351-364.

Edit: Apparently Bredihin [10] proved the infinitude of primes of the form $x^2+y^2+1$ some years before Motohashi. He only gave a slight upper-bound on their density, though: $O(n/(\log n)^{1.042}).$ (Motohashi improved the exponent to 1.5 in a later paper.)

[9] Motohashi, Y. (1969). On the distribution of prime numbers which are of the form $x^2+y^2+1$. Acta Arithmetica 16, pp. 351-364.

[10] Bredihin, B. M. (1963). Binary additive problems of indeterminate type II. Analogue of the problem of Hardy and Littlewood (Russian). Izv. Akad. Nauk. SSSR. Ser. Mat. 27, pp. 577-612.

On the historical side, of course Fermat is responsible for the proof of the case $x^2+y^2$. I have references that say that Weber [7] and Schering [8] handled the case of (primitive) binary quadratic forms with nonsquare discriminants, but I haven't read the papers. Motohashi [9] proved that there are $\gg n/\log^2 n$ primes of the form $x^2+y^2+1$ up to $n$, apparently (?) the first such result with a constant. He conjectured that the true number was

but as far as I know the constant has not been proved even for this special form.

[2] P. A. B. Pleasants (1966). The representation of primes by cubic polynomials, Acta Arithmetica 12, pp. 23-44.

On the historical side, of course Fermat is responsible for the proof of the case $x^2+y^2$. I have references that say that Weber [7] and Schering [8] handled the case of (primitive) binary quadratic forms with nonsquare discriminants, but I haven't read the papers. Motohashi [9] proved that there are $\gg n/\log^2 n$ primes of the form $x^2+y^2+1$ up to $n$, apparently (?) the first such result with a constant term. He conjectured that the true number was

but as far as I know the constant still has not been proved even for this special form.

[2] Pleasants, P. (1966). The representation of primes by cubic polynomials, Acta Arithmetica 12, pp. 23-44.