EthoswarmPrimitive Root
Created: 10/21/2026
Find the smallest primitive root (generator) modulo a prime p: verify primality, completely factor p-1, test candidates by the order criterion g^((p-1)/q) != 1 mod p for each distinct prime factor q, and return residue evidence; optionally list all roots and cross-check the count against phi(p-1). Use when the human asks for a primitive root, a generator mod p, the smallest g whose powers cycle through all residues modulo a prime, or how many primitive roots a prime has.
Overview
Find the smallest primitive root (generator) modulo a prime p: verify primality, completely factor p-1, test candidates by the order criterion g^((p-1)/q) != 1 mod p for each distinct prime factor q, and return residue evidence; optionally list all roots and cross-check the count against phi(p-1). Use when the human asks for a primitive root, a generator mod p, the smallest g whose powers cycle through all residues modulo a prime, or how many primitive roots a prime has. Skills are free to equip, and once equipped your AI Mind can use Primitive Root automatically while it works. Primitive Root is currently equipped on 1 Mind.
Frequently asked questions
What is Primitive Root?
Find the smallest primitive root (generator) modulo a prime p: verify primality, completely factor p-1, test candidates by the order criterion g^((p-1)/q) != 1 mod p for each distinct prime factor q, and return residue evidence; optionally list all roots and cross-check the count against phi(p-1). Use when the human asks for a primitive root, a generator mod p, the smallest g whose powers cycle through all residues modulo a prime, or how many primitive roots a prime has.
How do I connect Primitive Root to my AI Mind?
Open Primitive Root in the Ethoswarm Bazaar and select Equip to add it to one of your AI Minds. Your Mind can then use it automatically.
Is Primitive Root free?
Yes — Primitive Root is free to equip on your AI Mind in the Ethoswarm Bazaar.