It seems to me that you are identifying certain interlinks and patterns that exist in mathematics.
There are obviously many alternative ways to describe numbers using these alternative routes through these mathematical interlinks.
Bertrand Russell made a similar link to describe numbers via set theory.
My personal view is that mathematics is a codification of the ways in which physical things can be arranged, and their mutual interactions described.
(i.e. Nominalist)
Kurt Gödel proved the "incompleteness" of mathematics as exemplified by the phrase "This statement is unprovable"
However I would argue that the statement is actually meaningless, since it cannot be applied to any "real" situation,
it is just an arrangement of words that does not relate to anything "real"
I believe (but can't prove) that a similar argument could be leveled at any of the incompleteness examples.
We can perhaps divide mathematics into maths that codifies "reality" and maths that doesn't
Maths that doesn't codify reality might be interesting in an abstract way, but is not otherwise useful.
Being indivisible, the primes will (fairly clearly) have a natural relationship to the formation of the nuclei of chemical elements
(in particular how they may be formed from the combinations of some elements, but not from combinations of others).