Covered but excluded primes

New to this? It is explained from scratch, assuming nothing: English · Español

For the multiplicative function f given on prime powers by f(q^e) = q^(2e) − q^e + 1, the prime 3 has incoming arrows in the covering digraph of f and nevertheless divides no element at all of S(f) = { n : rad(n) | f(n) }. Having an incoming arrow is therefore necessary but not sufficient for a prime to divide an element of S(f).

Throughout, f is multiplicative with f(q^e) = F(q^e) for a fixed F ∈ Z[x] with F(0) = ±1 — which is exactly the condition p ∤ f(p^a). Write q → p when p | f(q^e) for some e ≥ 1.

Theorem 1 (criterion). q → p holds iff some ord_p(r) divides ord_p(q), with r running over the roots of F mod p. Consequently p has an incoming arrow iff F has a root mod p.

Theorem 2 (density). The density of primes with no incoming arrow is the proportion of elements of Gal(F) acting on the roots of F with no fixed point. This is the Frobenius density theorem (1896) applied to this object; the density statement is classical and is not claimed here.

Theorem 3 (one step is not enough). For F = x² − x + 1, the prime 3 has an incoming arrow and divides no element of S(f).

Theorem 4 (where failures can live). If p is unramified in the splitting field of F and F has a root mod p, then some prime q that itself has a root of F mod q satisfies q → p. Hence every failure of the kind in Theorem 3 occurs at a prime where F mod p has a repeated root — finitely many for each F.

F = x² − x + 1, i.e. f(q^e) = q^(2e) − q^e + 1.

A number n ∈ S(f) is built only from primes that are themselves covered inside n: every vertex of its digraph needs an incoming arrow from another prime of n. Modulo 3 the polynomial x² − x + 1 becomes (x+1)², whose root 2 has order 2, so by Theorem 1 an arrow q → 3 forces q ≡ 2 (mod 3). But a prime p ≠ 3 has a root of F only when 6 | p − 1, hence p ≡ 1 (mod 3). The two classes are disjoint, so every prime that could cover 3 is itself uncovered and cannot appear in n. Therefore 3 ∤ n for every n ∈ S(f).

No dependencies, standard library only:

git clone https://github.com/jorgell23-sys/covered-but-excluded-primes
cd covered-but-excluded-primes
python verify.py

It prints one PASS/FAIL line per check — 81 of them — and exits non-zero if any fails. Check [3] is the theorem above, [4] is the ramification bound over a sweep of 310 polynomials, and check [7] is external: it recomputes { n : rad(n) | σ(n) } from scratch and compares it against the 49 terms published independently as OEIS A175200.

• It does not claim the density statement of Theorem 2 as new. That is Frobenius (1896); what is new here is applying it to S(f) and reading the excluded primes off F alone.
• It does not prove that the two bounds in {p on a cycle} ⊆ D(f) ⊆ C_∞ always coincide. They coincided in every case measured — 310 polynomials against the 168 primes below 1000, with zero gaps — but that is a measurement, recorded as a conjecture.
• It says nothing about how many elements S(f) has below x. A hypothesis about that was declared, tested out of sample and refuted; see RESULT.md.
• σ, σ₂ and φ* have no excluded primes at all; the phenomenon needs a function whose profile forbids arrows.

The explanation for non-specialists is at https://jorgell23-sys.github.io/covered-but-excluded-primes/ (and in Spanish at /es/).

Ellena Godoy, Jorge (2026). Covered but excluded primes: which primes divide an element of { n : rad(n) | f(n) }. Zenodo. https://doi.org/10.5281/zenodo.22565634

The DOI above is the concept DOI and always resolves to the latest version.

Jorge Ellena Godoy — jorgell23@gmail.com

System design and research direction are the author's. The mathematical results were produced by an automated system (Claude, Anthropic) under that direction. All computations were verified by two independent implementations and cross-checked against published work. The author is responsible for the correctness of everything published here.

MIT for the code, CC BY 4.0 for text and data. See LICENSE.