---
tag: 0KMVFSHFGZ
kind: statement
role: theorem
name: Strong induction
aliases: [complete-induction, course-of-values-induction]
msc: [03F30]
uses: [0ASJVEFA9J, 0Y9H52BT03]
sources:
  - url: https://en.wikipedia.org/wiki/Mathematical_induction
    locator: Complete (strong) induction
---

## Statement

Let $P(n)$ be a property of natural numbers $n$.
Suppose that for each natural number $n$, if $P(k)$ holds for every natural number $k < n$, then $P(n)$ holds.
Then $P(n)$ holds for every natural number $n$.

## Remarks

The hypothesis for $n = 0$ asks for a proof of $P(0)$ with no assumption, because no natural number is less than $0$.
The proof applies ordinary [[induction|0ASJVEFA9J]] to the property "$P(k)$ holds for every $k < n$".
A standard use is the proof that each integer $n \ge 2$ is a product of primes.

## Explanations

### L1 human

In ordinary induction, you may use only the case just before $n$.
In strong induction, you may use all the cases before $n$.
This helps when $n$ breaks into smaller pieces that are not just $n - 1$, for example when $n = a b$ with $a$ and $b$ smaller than $n$.