I think Octonions are special – I am reluctant to launch into a detailed explanation of what they are and why I think that; perhaps I will find the motivation at some point. For now, I am simply going to announce that I was doing some research into a class of them with respect to prime number theory and found myself going down a rabbit hole beginning to formalize theorems in areas that I believe are actually unstudied. These results likely have absolutely no utility, but being an amateur math-lover who has never published a math paper, I thought I would give it a shot! With the power of AI, I actually sound like I know what I’m talking about as it helped critique, test, prove and document my findings in an appropriate fashion.
If you would like to learn more, I’ve posted a preprint:
Preprint: https://doi.org/10.5281/zenodo.21223787
Code: https://github.com/rosolam/Gravesian-Octonion-Verification
Summary: An integer octonion z in the Gravesian order Z^8 — whole-number coordinates — is irreducible if and only if N(z) is an ordinary prime, or N(z) = 2^k with k >= 2 and the norm-2 divisor count σ(z) = 0. This Strong Irreducibility Theorem classifies reducibility in the non-maximal ring that differs from the seven octavian maximal orders only at the prime 2. Aside from prime-norm elements, every composite-norm irreducible is a hole: a point invisible to norm-2 divisors, forming asymptotically 49/64 of each large power-of-two shell. The proof rests on a congruence reduction: divisibility by a norm-m element is a linear condition modulo m, so divisor counts depend only on residues. Three divisor laws emerge: the Parity-Fano Theorem (σ in {0,16,48,112}, governed by the extended Hamming code [8,4,4]); the Odd Divisor Theorem (exactly 16 norm-p divisors per side for odd p); and the Norm-4 Divisor Theorem (19 residue profiles mod 4). Holes factor in maximal orders; product visibility follows a closed Fano product formula. Under the 1344-element automorphism group, hole arithmetic preserves species access in fixed 2:1:1 proportions.
