I took a look at Schinzel and Sierpinski, Sur certaines hypotheses concernant les nombres premiers, Acta Arith IV (1958) 185-208, reprinted in Volume 2 of Schinzel's Selecta, pages 1113-1133. In the Selecta, there is a commentary by Jerzy Kaczorowski, who mentions "the G H Hardy and J E Littlewood conjecture implicitly formulated in [33] that $\pi(x+y)\le\pi(x)+\pi(y)$ for $x,y\ge2$." [33] is Partitio Numerorum III. Schinzel and Sierpinski (page 1127 of the Selecta) define $\rho(x)=\limsup_{y\to\infty}[\pi(y+x)-\pi(y)]$, and point to that H-L paper, pp 52-68. They then write (page 1131), "$\bf C_{12.2}.$ L'hypothese de Hardy et Littlewood suivant laquelle $\rho(x)\le\pi(x)$ pour $x$ naturels $\gt1$ equivaut a l'inegalite $\pi(x+y)\le\pi(x)+\pi(y)$ pour $x\gt1,y\gt1$." It should be said that the proof that the first inequality implies the second relies on Hypothesis H, which essentially says that if there is no simple reason why a bunch of polynomials can't all be prime, then they are, infinitely often.
Schinzel and Sierpinski express no opinion as to any degree of belief in the conjecture under discussion.
I don't suppose this actually answers any of the questions, although Kaczorowski's use of the word "implicitly" may be significant.
The modern reference work on the subject seems to be [1], but it spends only a page and a half on the subject of primes in multivariate quadratic polynomials (pp. 396-397). More than half this space is devoted to Iwaniec's 1974 result. The balance mentions Sarnak's application to the Problem of Apollonius and a result of "J. Cho and H. Kim" on counting primes in $\mathbb{Q}[\sqrt{-2}].$ So nothing there.
Pleasants [2] shows that, subject to a Davenport-Lewis [3] condition on the $h^*$ (a complexity measure on the cubic form part), multivariate cubic polynomials have the expected number of primes. Unfortunately this condition requires (as a necessary but insufficient condition) that there be at least 8 variables. Further, it double-counts repeated primes.
Goldoni [4] recently wrote a thesis on this general topic. His new results (Chapter 5) on the $h$ and $h^*$ invariants make it easier to use the results of Pleasants but do not extend them to cubic polynomials with fewer than 8 variables.
Of course I would be remiss in failing to mention the groundbreaking work of Heath-Brown [5], building on Friedlander & Iwaniec [6]. These results will no doubt clear the way for broader research, but so far have not been generalized.
So in short it appears that:
- Nothing further is known about primes represented by quadratic polynomials.
- Apart from $x^3+2y^3$, almost nothing is known about which primes are represented by cubic polynomials, though some results are known for how often such polynomials take on prime values provided $h^*$ and hence the number of variables is large enough.
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
$$\frac{n}{(\log n)^{3/2}}\cdot\frac32\prod_{p\equiv3(4)}\left(1-\frac{1}{p^2}\right)^{-1/2}\left(1-\frac{1}{p(p-1)}\right)$$
but as far as I know the constant still has not been proved even for this special form.
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.)
[1] Friedlander, J. and Iwaniec, H. (2010). Opera de Cribro. AMS.
[2] Pleasants, P. (1966). The representation of primes by cubic polynomials, Acta Arithmetica 12, pp. 23-44.
[3] Davenport, H. and Lewis, D. J. (1964). "Non-homogeneous cubic equations". Journal of the London Mathematical Society 39, pp. 657-671.
[4] Goldoni, L. (2010). Prime Numbers and Polynomials. Doctoral thesis, Università degli Studi di Trento.
[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.
[7] Weber, H. (1882). "Beweis des Satzes, dass, usw". Mathematische Annalen 20, pp. 301-329.
[8] Schering, E. (1909). "Beweis des Dirichletschen Satzes". Gesammelte mathematische Werke, Bd. 2, pp. 357-365.
[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.
Best Answer
This problem was studied by a few, and the ideas involve too much latex to write here. Mainly there are ideas of transforming to crazy weighted sums and then use ERH to bound errors from crazier integrals. It suffices to say that the culmination of this research is the freely available paper:
In that year, they used their ideas to compute the constant for discriminants of up to 72 digits! (assuming ERH) The bottleneck of the process seems to be the calculation of the algebraic invariants class number and regulator. Over a decade has passed, the technology today should be able use the same methods to get up to 100 digits, within reasonable time, and up to 110 digits with a bit more time (apparently, two weeks on a cluster):