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I'm interested in finding out whether there exists a prime partition of a given positive integer $N>1$ such that the partition has specific number of parts.

For instance, as given in another question, $N=12$ has seven partitions: $\{2, 2, 2, 2, 2, 2\}$, $\{2, 2, 2, 3, 3\}$, $\{3, 3, 3, 3\}$, $\{2, 2, 3, 5\}$, $\{2, 5, 5\}$, $\{2, 3, 7\}$, and $\{5, 7\}$. So there exist partitions of length 2, 3, 4, 5, 6 and no partition of length 1 (obviously).

So given an integer $N$ and a partition length ($\leq N/2$), is there an effective way of constructing or verifying the existence of a prime partition of the given length (particularly for large values of $N$, so simply computing primes up to $N$ and searching is not an option)?

I've seen the formulae on computing the total number of prime partitions (e.g. Prime Partition and http://programmingpraxis.com/2012/10/19/prime-partitions/), but I see no clear way of relating this to specific partition sizes.

Thanks.

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In general this seems like a very difficult problem. Indeed, when partition length$=2$ and $N>2$ is even, this question is an instance of the longstanding Goldbach's Conjecture.

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  • $\begingroup$ Absolutely, and I believe I read somewhere that this strong version of Goldbach's conjecture has been verified up to ~$10^{18}$, so that's certainly useful. But it's not immediately clear how it would help for arbitrary partition length. $\endgroup$ – user3638633 May 21 '14 at 22:01

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