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Write $N(m, n; c)$ for the number of $m\times n$ zero-one matrices where each zero is adjacent to precisely $c$ others, where by "adjacent" I mean up/down/left/right but not diagonally. (Notice that we need $0 \leq c \leq 2$ here, as $c = 3,4$ has only the trivial solution where every entry is one.)

We have

$$\frac{N(3, 15; 2)}{N(3, 7; 0)} = \frac{10882}{10897} \approx 0.99862347,$$

which agrees with the observed proton/neutron mass ratio to eight decimal places, which is about as accurately as it's been measured.

Now, before anyone decides I'm a crank, let me hastily mention that I'm not suggesting that this has any physical meaning. Here's how I obtained those numbers:

  • I took the rational number between 0.99862347 and 0.99862348 with the smallest denominator. This turned out to be 10882/10897.
  • I looked up the numerator 10882 and denominator 10897 in OEIS.
    • 10897 has about 30 hits, but many fall into a few basic categories -- for instance, there are tables of $N(-, 7; 0)$ and $N(3,-; 0)$ and also of $N(-,-;0)$, and a few cases where it looks like it may be counting essentially the same thing; then there are several things that are present because $10897 = 17\cdot 641$ has a very large prime factor relative to its size; there are a few unrelated things; and a few are sort of "numerological" (i.e. depend on the base-10 expansion of the number).
    • 10882 has 23 hits, which are more varied, but also happens to have an interpretation as a number $N(m,n;c)$ for small values of $n,m,c$.

My question is whether there's a heuristic reason we'd expect the numbers $N(m,n;c)$ to appear disproportionately likely when applying such a process. Is there a relation to Farey numbers, for instance?

It seems unlikely that something like this would happen purely by coincidence, especially since the numbers $N(m,n;c)$ do not appear to be particularly dense.

Daniel McLaury
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  • Although the numbers $N(m,n;c)$ may not be dense, the oeis has many other number families that, together, are fairly dense among small integers. – vadim123 Aug 20 '15 at 06:02

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