ForHosting KIT · Developer Utilities

Primitive root calculator

A primitive root modulo a prime p is a generator of the multiplicative group of nonzero residues modulo p.

● BetaFree · in your browser

Use it from WebAPIEmailTelegramApp soon

In plain terms, some integer g between 1 and p minus one has the property that its successive powers g, g squared, g cubed, and so on, eventually hit every residue class coprime to p before repeating. That g is a generator of (Z/pZ)*, and the smallest such positive g is the classic number-theory answer students, cryptographers, and contest solvers look up. This calculator takes a prime modulus p, verifies that p is prime, factors p minus one, and returns the least primitive root together with the group order and the prime-power factorisation of p minus one. The path is fully deterministic: Miller–Rabin with fixed witness sets, trial factorisation, and modular exponentiation with no network, no randomness, and no date dependence. Use it free in the browser for homework and quick checks, or call the API at $0.002 per successful request when you automate fixtures, teaching demos, or crypto-parameter notebooks that need a reproducible generator mod p.

How to use it

Enter your values in the form above. The tool checks them before calculating and shows the result on the same page.

Check your inputs

Use the labels and units shown next to each field. If something is missing or outside the allowed range, the page points to the field to fix.

Use it again or automate it

Use the browser tool for individual checks and the API when you need the same capability in an automated workflow.

What you can do with it

Get an answer now

Enter one set of values and see the result without building a spreadsheet or script.

Compare scenarios

Change one value at a time and rerun the calculation to understand what affects the result.

Automate repeated work

Use the API when the same calculation needs to run inside your product or workflow.

FAQ

How do I use this capability?

Complete the fields above and run it on this page. The form highlights anything that needs attention.

For developers — API access

Everything on this page is available programmatically. This section is for teams who want to wire it into their own systems; everyone else can just use the tool above.

API endpoint

POSThttps://api.kit.forhosting.com/numth/primitive-root

Prefer to automate it? One authenticated POST creates the task; the result comes back by webhook or a signed link. The same capability also runs here on the web, by email and from Telegram — and soon from our app too.

Call it from your stack

curl -X POST https://api.kit.forhosting.com/numth/primitive-root \
  -H "Authorization: Bearer $KIT_KEY" \
  -H "Content-Type: application/json" \
  -d '{"p":"7"}'

const res = await fetch("https://api.kit.forhosting.com/numth/primitive-root", {
  method: "POST",
  headers: {
    "Authorization": `Bearer ${process.env.KIT_KEY}`,
    "Content-Type": "application/json"
  },
  body: JSON.stringify({
    "p": "7"
  })
});
const { task_id } = await res.json();

import os, requests

res = requests.post(
    "https://api.kit.forhosting.com/numth/primitive-root",
    headers={"Authorization": f"Bearer {os.environ['KIT_KEY']}"},
    json={
    "p": "7"
},
)
task_id = res.json()["task_id"]

<?php
$res = file_get_contents("https://api.kit.forhosting.com/numth/primitive-root", false, stream_context_create([
  "http" => [
    "method"  => "POST",
    "header"  => "Authorization: Bearer " . getenv("KIT_KEY") . "\r\nContent-Type: application/json",
    "content" => '{"p":"7"}',
  ],
]));
$task = json_decode($res, true);

body := bytes.NewBufferString(`{"p":"7"}`)
req, _ := http.NewRequest("POST", "https://api.kit.forhosting.com/numth/primitive-root", body)
req.Header.Set("Authorization", "Bearer "+os.Getenv("KIT_KEY"))
req.Header.Set("Content-Type", "application/json")
res, _ := http.DefaultClient.Do(req)

Example request

{
  "p": "7"
}

Example response

{
  "task_id": "tsk_a1b2c3d4e5f6a1b2c3d4e5f6",
  "type": "numth.primitive_root",
  "status": "queued",
  "_links": {
    "result": "/tasks/tsk_…/result"
  }
}

The API is asynchronous: the call returns a task_id immediately and the result arrives by webhook. Polling is capped at 1 req/s per task.

Pricing

Per request$0.002

Published price — no tokens, no invented credits. A failed task is never charged.

Limits

max_digits 15

Errors

HTTP	Code	Meaning
`401`	`unauthorized`	Missing or invalid API key.
`402`	`insufficient_balance`	Your balance doesn't cover the task price.
`404`	`unknown_type`	That task type doesn't exist.
`429`	`rate_limited`	Too many requests. Use the webhook instead of polling.

Read the full KIT documentation →