Beyond ideals in the Dickson ring of integral octonions

This is ongoing work on hypercomputation [4]. We revisit the works of Dickson [5] and Mahler [9] on integral octonions. Nonassociativity (in the form of alternativity) leads to the unexpected consequence that the theorem of (at most) 4 squares (Bachet – Lagrange) becomes a theorem of (exactly) 1, 2 or 4 identical squares for a subset of N∗. These numbers measure the norm of the vectors in the Dickson ring of integral octonions for which the associator (with 3 or 7 pairs of distinct imaginary canonical basis vectors in G) is computable as a product in the Dickson ring.