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.