{"id":"palomar/PALOMAR-2026-10-21-000021","url":"https://github.com/wcook04/plectis-erdos-lean/tree/9ab395cda99d13f8fcd16cf655fc17458b3eaa38","json":"https://data.palomar-registry.org/entries/PALOMAR-2026-10-21-000021-v1.json","repo":"https://github.com/wcook04/plectis-erdos-lean","tags":["math.NT","msc:11A25","msc:11B83","number-theory","totient-function","linear-algebra","arithmetic-functions"],"score":0,"title":"All-base bases and integral relations for the totient kernel","trust":"high","domain":"math.NT","source":"palomar","authors":["Will Cook"],"reviews":0,"subject":"math.NT","summary":"For every integer base k at least two and depth e at least one, the rational span of the sections n ↦ φ(k^j n+r) through level e has dimension k^e+1. An explicit retained family is a basis, every omitted section is an integer multiple of a retained section, and these reductions form a basis for all integral relations, with rank Σ_(1≤j<e) k^j. The entry also gives the full dyadic basis and an affine independence theorem. The retained positive residues satisfy k ∤ r, a weaker condition than gcd(k,r)=1. Martin’s earlier theorem already implies affine independence, and Coons proved…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"verified":true,"discussed":0,"published":"2026-10-21","reactions":{},"formalized":true,"formalizes":[{"title":"Bases and Integral Relations for the k-Kernel of Euler’s Totient","identifier":"https://github.com/wcook04/plectis-erdos/blob/147f3f1da400a84dcb3013ea6cef75dc8f382b09/paper/249/erdos-249-binary-totient-series.pdf","relationship":"formalizes"}]}