Master

Perfect Numbers

Master perfect numbers with interactive lessons and practice problems! Designed for students like you!

Learn

Get the basics

Practice

Apply your skills

Master

Achieve excellence

Perfect Numbers

Study Guide

Key Definition

A perfect number $N$ is a positive integer where the sum of its divisors except the number itself equals the number.

Important Notes

The first known perfect numbers are 6, 28, 496, and 8128.
All known perfect numbers are even, as proven by Euclid and Euler.
Perfect numbers can be expressed by $N = 2^{p-1} (2^p - 1)$ where $p$ is a prime and $2^p - 1$ is a Mersenne prime.
As of 2023, there are 51 known perfect numbers (OEIS A000396).
The most recently discovered perfect number has 49,724,095 digits.
It is still an open question whether any odd perfect numbers exist.

Mathematical Notation

$+$ represents addition

$-$ represents subtraction

$\times$ represents multiplication

$\div$ represents division

$\sigma(n)$ denotes the sum of all positive divisors of $n$

Use $σ(n)=∑_{d|n}d$ to denote the sum of divisors of $n$.

Why It Works

Euclid proved that if $2^p - 1$ is prime then $N = 2^{p-1}(2^p - 1)$ is a perfect number by showing its divisors sum to $N$. Euler proved the converse: every even perfect number must be of this form. Together, these results, known as the Euclid–Euler theorem, fully classify even perfect numbers in terms of Mersenne primes.

Remember

All known perfect numbers are even and related to Mersenne primes.

Quick Reference

Perfect Number Formula:$N = 2^{p-1} (2^p - 1)$

Understanding Perfect Numbers

Choose your learning level

Watch & Learn

Video explanation of this concept

Beginner

Start here! Easy to understand

Beginner Explanation

A perfect number equals the sum of its proper divisors. For example, the divisors of 6 are 1, 2, and 3, and 1 + 2 + 3 = 6, so 6 is perfect.

Practice Problems

Test your understanding with practice problems

Quick Quiz

Single Choice Quiz

Beginner

Which of the following is a perfect number?

Real-World Problem

Question Exercise

Intermediate

Teenager Scenario

Imagine you are organizing a math club. You want to explain what a perfect number is to your friends. How would you explain $N = 2^{p-1} (2^p - 1)$?

Thinking Challenge

Thinking Exercise

Intermediate

Think About This

Is there a pattern to the known perfect numbers? How does the formula $N = 2^{p-1} (2^p - 1)$ influence this?

Challenge Quiz

Single Choice Quiz

Advanced

Which of the following expressions represents a perfect number when $p$ is a prime?

Recap

Watch & Learn

Review key concepts and takeaways