Is it known that there are infinitely many primes which can be represented by $n^{3} + 2$ (or similarly any cubic polynomial)?

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    $\begingroup$ No.${}{}{}{}{}$ $\endgroup$ – Will Jagy Jan 29 '13 at 21:00
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    $\begingroup$ You might be interested in Bunyakovsky conjecture, a generalization of Dirichlet's theorem on primes in arithmetic progression. $\endgroup$ – user17762 Jan 29 '13 at 21:02
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    $\begingroup$ It is however known that there are infinitely many primes of the form $x^3+2y^3$ (Heath-Brown). $\endgroup$ – user27126 Jan 29 '13 at 21:03
  • $\begingroup$ See oeis.org/A144953. $\endgroup$ – Robert Israel Jan 29 '13 at 21:23
  • $\begingroup$ What's special about $n^3+2$ in particular? $\endgroup$ – user58512 Jan 29 '13 at 21:25

the only single variable polynomials we have a result like this for are linear ones, due to Dirichlet. (Primes in arithmetic progressions).

A lot is known about primes taken by binary (two variable) quadratic forms due to Fermat, Euler, Gauss and many others.

A modern breakthrough proved that there are infinitely many of the form $x^2 + y^4$ (Friedlander–Iwaniec) and $x^3+2y^3$ (Heath-Brown).

As far as I know essentially nothing else is known.

There there should be infinitely many $x^2+1$ is a long standing open problem.

  • $\begingroup$ It just occurred to me that $\binom{n+m+k}{3} + \binom{n+m}{2} + \binom{m}{1}$ takes on infinitely many primes for the trivial reason that it is a bijection from $\mathbb N^3 \to \mathbb N$. $\endgroup$ – user58512 Jan 29 '13 at 21:10
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    $\begingroup$ Not quite. $\binom{n}{2} + \binom{n}{1}=\frac 12(n^2+n)$ misses $4$ among others. You want $\binom{m+n+1}{2} + \binom{n}{1}$ $\endgroup$ – Ross Millikan Jan 29 '13 at 21:14
  • $\begingroup$ and sums of cubes, etc.. by Warings problem. $\endgroup$ – user58512 Jan 30 '13 at 12:32
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    $\begingroup$ @user58512 The reason that Friedlander-Iwaniec and Heath-Brown results are breakthroughs is that the set of integers of those forms is much sparser than any arithmetic progression or even the primes themselves. In general, $\{\sum_i n_i^{a_i}: n_i \in \mathbb N \}$ is sparse when $c := \sum_i {a_i}^{-1} < 1$, because there are at most $O(x^c)$ such values up to $x$. That's what makes $x^2+y^4$ exciting and $x^2+13y^2$ less so (not trivial, mind you). $\endgroup$ – Erick Wong Feb 1 '13 at 1:26
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    $\begingroup$ @Watson That’s like saying a point and a unit circle both have area $0$ so one can’t be smaller than the other. We can quantify density much more finely than just “$0$” by considering the asymptotic behavior of the counting function. More to the point, the set $\{x^2+y^2\}$ is unboundedly denser than the primes and it even has positive density within the primes. The set $\{x^2+y^4\}$ has density zero with respect to the primes even before taking intersections. As I alluded above, the counting function is only $O(x^{3/4})$ vs the primes’ $O(x/\ln x)$. $\endgroup$ – Erick Wong Dec 3 '18 at 5:17

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