Exact q-adic valuation of the quotient R/k for reciprocals of primes in prime bases | Synapse
April 8, 2026Open Access
Exact q-adic valuation of the quotient R/k for reciprocals of primes in prime bases
Key Points
The aim is to prove the q-adic valuation theorem for quotients of prime reciprocals.
Proof of the q-adic valuation theorem
Computational verification involving 515 cases
Using distinct primes p and b with k defined as ord_p(b)
Defining R as (b^k−1)/p
The theorem v_q(R/k) = v_q(b^k − 1) − v_q(k) is proven
Verification completed for all 515 computations
Confirmed relationships between distinct primes and their q-adic valuations
Abstract
Proof and computational verification (515/515) of the q-adic valuation theorem: vq (R/k) = vq (bᵏ − 1) − vq (k), where p, b are distinct primes, k = ordₚ (b), R = (bᵏ−1) /p, and q ≠ p.