# MacMahon partition function and prime detection (ref arXiv:2405.06451)

In the recent paper arXiv:2405.06451 the authors provide infinitely many characterizations of the primes using MacMahon partition functions: for $$a>0$$ the functions $$M_a(n):=\sum\limits_{0

The simplest is : $$n$$ is prime iff $$(n^2-3n+2)M_1(n)-8M_2(n)=0$$

I'm not really able to compute those functions. Could someone please write out the above characterization in full for the first few $$n=2,3,4,5,6$$ ?

Added:

For $$n=3$$, which is prime, what would make sense is $$M_1(3)=1+3=4$$ since $$3=3×1=1×3$$ and $$M_2(3)=1×1=1$$ since $$3=1×1+1×2$$, and indeed we have the equality $$(3^2-3×3+2)×4-8×1=0$$.

For $$n=4$$, which is composite, since $$4=4×1=2×2=1×4$$ we should have $$M_1(4)=1+2+4=7$$ and since $$4=1×1+1×3=2×1+1×2$$ we should have $$M_2(4)=3+2=5$$ and indeed $$(4^2-3×4+2)×7-8×5=6×7-8×5=2$$ is non zero positive.

Could someone please confirm?

• I think this is even far less efficient than applying Wilson's theorem , but maybe interesting for theoretical purposes. May 15 at 12:30
• @Peter : of course I am not claiming this could be a fast detection method, the number of product and sums obviously grows fast with $n$. What I would like to understand is the partition aspect, to see what those sums are. May 15 at 13:51

## 1 Answer

MacMahon worked out some cases using the function $$\sigma_\nu(n) = \sum_{d \mid n} d^\nu$$ based on the divisors of $$n$$. Here are the first few. $$\begin{gather*} M_1(n) = \sigma_1(n) \\ M_2(n) = \frac{1}{8} [(1-2n)\sigma_1(n) + \sigma_3(n)] \\ M_3(n) = \frac{1}{1920} [(37 - 100n + 40n^2)\sigma_1(n) + (50-20n)\sigma_3(n) + 3 \sigma_5(n)] \end{gather*}$$ The OEIS entries AA002127, A002128, and AA365664 through AA365667 include equivalent formulas using binomial coefficients rather than $$\sigma_k(n)$$.

See also this MathOverflow question and its links.