TITLE: What is the significance of Friedlander-Iwaniec and related theorems?
QUESTION [11 upvotes]: On p.177 of Number Theory Revealed: A Masterclass by Andrew Granville, the author states that "One can ask for prime values of polynomials in two or more variables." (though he later mentions Landau's $n^2+1$ conjecture so I'm not sure why single-variable polynomials are omitted). For example, there are some classical results, like Fermat's Christmas theorem and Euler's $6n+1$ theorem. More recent results include the Friedlander-Iwaniec theorem, which states that there are infinitely many prime numbers of the form $a^2+b^4$ for integers $a$ an $b,$ and the result of Heath-Brown on the infinitude of primes of the form $x^3+2y^3$ for integers $x$ and $y.$ Granville's book mentions other recent advances by Maynard and collaborators of Heath-Brown and Iwaniec.
Initially, I wondered why the Friedlander-Iwaniec theorem is considered to be a breakthrough, so I asked Why does the infinitude of primes of a certain form matter? on math.stackexchange, but it was closed for being "opinion-based". Granville's book cleared up a part of this question for me because it states that the Friedlander-Iwaniec theorem was the first example in which the polynomial is "sparse" in taking on integer values, and that later results of the same type based on norm forms were inspired by Friedlander-Iwaniec (although I don't know to what extent the proofs of the former are related to the proof of the latter).
My question is this: is the area of exploration quoted at the beginning of this post an island in that it is considered to be an inherently interesting question requiring no further justification, or are there external motivations for pursuing it? Some satisfactory motivations (this list is by no means exhaustive) might be that other problems reduce to it, that the techniques developed in the course of solving these problems are widely applicable elsewhere, real world applications (e.g., cryptography), or maybe these are instances of much more general problems for which "Mathematics is not yet ripe" in the words of Erdős.

REPLY [10 votes]: Suppose you suspect that among some subset $\mathcal{S}$ of the natural numbers, there ought to be infinitely many primes. How would you confirm that this is indeed the case?
Clearly, the "bigger" $\mathcal{S}$ is (measured in terms of natural density), the easier the question should be. The biggest set is of course $\mathbb{N}$ itself, and it was a highly non-trivial theorem of Euclid that confirmed this. What about the next most obvious "big" sets, the arithmetic progressions? Indeed, consider the set $\mathcal{S}(a; q)$ by
$$\displaystyle \mathcal{S}(a; q) = \{qx + a : x \in \mathbb{N}\}.$$
In the special case $q = 4, a = 3$ one can use Euclid's argument again to show that there are infinitely many primes in the set (but far from the correct density). A much more difficult argument shows that the case $q = 4, a = 1$ also gives infinitely many primes. But we expect that there are infinitely many primes in $\mathcal{S}(a; q)$ whenever $\gcd(a,q) = 1$. This is not known until the work of Dirichlet, almost 2000 years after Euclid!
The most extensive generalization of the theorem of Dirichlet known is the Chebotarev density theorem, which shows that in any number field there are infinitely many primes which split completely (over simplifying here).
So far, all of the sets considered have log density one. If we put $\mathcal{S}(X) = \# \{n \leq X : n \in \mathcal{S} \}$ then the log density is the infimum among all non-negative real numbers $\delta$ such that $\mathcal{S}(X) < X^{\delta}$ for all $X$ sufficiently large. Log density one sets can still be very very hard to count primes in: indeed, we expect the set of primes $p$ such that $p + 2$ is also prime to be infinite, and this set (if infinite) is expected to have log density one.
Thus, producing an infinite set of primes which have log density less than one is a formidable task. Both the Friedlander-Iwaniec theorem and Heath-Brown's theorem are examples where this can be achieved. These should be considered highly exceptional in the sense that similar, perhaps even easier looking polynomials are not amenable to their methods! Perhaps the most infamous is the polynomial $F(x,y) = 4x^3 - y^2$, which counts discriminants of elliptic curves in short Weierstrass form.