On p.177 of [Number Theory Revealed: A Masterclass](https://rads.stackoverflow.com/amzn/click/com/1470441586) 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](https://en.wikipedia.org/wiki/Landau%27s_problems) so I'm not sure why single-variable polynomials are omitted). For example, there are some classical results, like [Fermat's Christmas theorem](https://en.wikipedia.org/wiki/Fermat%27s_theorem_on_sums_of_two_squares) and [Euler's $6n+1$ theorem](https://mathworld.wolfram.com/Eulers6nPlus1Theorem.html). More recent results include the [Friedlander-Iwaniec theorem](https://en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_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](https://en.wikipedia.org/wiki/Roger_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?](https://math.stackexchange.com/questions/3718836/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.



What is the significance of Friedlander-Iwaniec and related theorems?
I think questions about prime values of polynomials are considered inherently interesting.


All these questions are special cases of Bunyakovsky's conjecture, or, if you want, the Bateman-Horn conjecture. Certainly mathematics is not yet ripe for such problems.


You can also view them as special cases of the general problem of when special sequences of numbers are prime. One generally expects to see transfers of techniques from one special case of this problem to another, but not as much with other areas of number theory and mathematics. So to a large extent I think the results on primes-of-a-special-type problems that are considered most interesting are those that point to way to further solution of primes-of-a-special-type problems.


One could also note that primes of the form $a^2+b^4$ is a special case (in some sense) of primes of the form $a^2 +1$, since infinitely many primes of the form $a^2+1$ implies infinitely many primes of the form $a^2+b^4$. (Of course this is the reverse of the special case direction between the numbers themselves.)


I don't think many who work on these problems are motivated by applications to cryptography. As Ben Smith points out in the comments, there are some applications of Bateman-Horn to cryptography, where we would like to find primes represented by special polynomials for use in certain cryptographic algorithms. Conditional on not just the existence of such primes but the Bateman-Horn conjecture predicting how many there are in a large interval, we can generate such primes in polynomial time by randomly sampling values of the polynomial and primality testing.


However, since these conjectures are very very hard to prove, but we have very strong reasons to believe that they are true, working on them might not help with cryptography very much.


