The best examples of identifying perfect numbers: 3 practical examples and beyond

Let’s begin with the most famous examples of identifying perfect numbers: 3 practical examples that every number theory student should know cold. A perfect number is a positive integer that equals the sum of its proper divisors (all positive divisors except the number itself).

Example of a perfect number: 6

Take 6. Its positive divisors are 1, 2, 3, and 6. The proper divisors are 1, 2, and 3.

• Sum of proper divisors: 1 + 2 + 3 = 6

Because the sum of its proper divisors equals the number, 6 is a perfect number. This is the smallest and cleanest example of identifying a perfect number.

Example of a perfect number: 28

Now look at 28. Its positive divisors are 1, 2, 4, 7, 14, and 28. Proper divisors: 1, 2, 4, 7, 14.

• Sum of proper divisors: 1 + 2 + 4 + 7 + 14 = 28

Again, the sum matches the original number. That makes 28 another perfect number. Notice that the divisors come in pairs: (1, 28), (2, 14), (4, 7). This pairing trick will matter when we try bigger examples.

Third classic example: 496

Now let’s level up with 496. Listing all divisors by brute force is tedious, but we can still manage it by being systematic.

Factor 496:

• 496 = 16 × 31 = 2⁴ × 31

The positive divisors are all products of powers of 2 (from 2⁰ to 2⁴) with either 1 or 31:

• With 1: 1, 2, 4, 8, 16
• With 31: 31, 62, 124, 248, 496

Proper divisors exclude 496, so we sum:

• 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248

Group them cleverly:

• (1 + 31) = 32
• (2 + 62) = 64
• (4 + 124) = 128
• (8 + 248) = 256
• And 16 alone

Now add:

• 32 + 64 = 96
• 96 + 128 = 224
• 224 + 256 = 480
• 480 + 16 = 496

The sum of proper divisors is 496, so 496 is our third textbook perfect number. These three — 6, 28, and 496 — are the standard best examples of identifying perfect numbers: 3 practical examples that show the pattern clearly.

Going further: more examples of identifying perfect numbers beyond the first 3

Those first three are friendly. The next ones are not. But if you understand how to identify 6, 28, and 496, you can stretch the same logic to larger cases.

Fourth example: 8,128

The next even perfect number is 8,128. Listing its divisors by hand is possible but painful. Instead, we use a powerful pattern that connects perfect numbers to a special kind of prime.

All even perfect numbers have the form:

N = 2ⁿ⁻¹ (2ⁿ − 1), where 2ⁿ − 1 is prime.

A prime of the form 2ⁿ − 1 is called a Mersenne prime. This result goes back to Euclid and was later proven in full by Euler. You can read a clear statement of this connection on the University of Tennessee at Martin’s number theory pages: https://www.utm.edu/departments/math/ (navigate to their resources on perfect and Mersenne numbers).

For 8,128, we use n = 7:

• 2⁷ − 1 = 128 − 1 = 127, which is prime.
• Plug into the formula: N = 2⁷⁻¹ (2⁷ − 1) = 2⁶ × 127 = 64 × 127 = 8,128.

Because 127 is prime, this formula guarantees 8,128 is a perfect number. This is a very efficient example of identifying a perfect number without summing dozens of divisors.

Fifth and sixth examples: 33,550,336 and 8,589,869,056

Using the same formula, we get more real examples of identifying perfect numbers:

• For n = 5: 2⁵ − 1 = 31 (prime) → N = 2⁴ × 31 = 16 × 31 = 496 (already seen).
• For n = 7: 2⁷ − 1 = 127 (prime) → N = 2⁶ × 127 = 8,128 (already seen).
• For n = 13: 2¹³ − 1 = 8,191 (prime) → N = 2¹² × 8,191 = 33,550,336.
• For n = 17: 2¹⁷ − 1 = 131,071 (prime) → N = 2¹⁶ × 131,071 = 8,589,869,056.

So the first six even perfect numbers are:

• 6
• 28
• 496
• 8,128
• 33,550,336
• 8,589,869,056

These are some of the best examples of identifying perfect numbers: 3 practical examples you can fully verify by hand (6, 28, 496), plus three larger ones you can trust via the Mersenne prime pattern.

How to systematically identify perfect numbers in practice

So far, we’ve focused on concrete examples. Now let’s turn that into a repeatable method. When someone asks for examples of identifying perfect numbers: 3 practical examples, what they usually want is a playbook they can reuse.

Strategy 1: Direct divisor-sum method (good for small numbers)

For small integers (say up to a few thousand), the straightforward way to test if a number is perfect is:

1. List all positive divisors smaller than the number.
2. Add them up.
3. See if the sum equals the original number.

To do this efficiently:

• Only search for divisors up to the square root of the number.
• Every time you find a divisor d, you also get a partner divisor n/d.

For example, to test 28:

• √28 is a bit more than 5, so check 1, 2, 3, 4, 5.
• Divisors: 1, 2, 4.
• Partner divisors: 28, 14, 7.
• Proper divisors: 1, 2, 4, 7, 14.
• Sum: 28 → perfect.

This method gives you direct, transparent examples of identifying perfect numbers, but it quickly becomes slow for large candidates.

Strategy 2: Using the Mersenne prime formula (good for big perfect numbers)

For larger perfect numbers, nobody is manually summing divisors. Instead, serious searches rely on the Euclid–Euler theorem:

Every even perfect number has the form N = 2ⁿ⁻¹ (2ⁿ − 1) where 2ⁿ − 1 is prime.

So the real challenge becomes: find values of n for which 2ⁿ − 1 is prime. Those primes are rare and hard to spot. The Great Internet Mersenne Prime Search (GIMPS), run by a nonprofit project at https://www.mersenne.org, has been using distributed computing for decades to hunt for these primes.

Once you know 2ⁿ − 1 is prime, identifying the corresponding perfect number is immediate using the formula above. That’s how we get modern gigantic perfect numbers, with millions of digits.

Modern trends (2024–2025): perfect numbers in the age of big computation

Perfect numbers are not just a historical curiosity. As of 2024–2025, they’re still tightly connected to some of the biggest open questions in mathematics and to large‑scale computing projects.

Current status of known perfect numbers

• Every known perfect number is even.
• Every known perfect number comes from a known Mersenne prime.
• As of early 2025, there are only a few dozen known Mersenne primes, each giving one even perfect number.

GIMPS, mentioned above, continues to discover new Mersenne primes using volunteers’ computers worldwide. Each time a new Mersenne prime appears, it immediately produces a new perfect number via the Euclid–Euler formula. These are some of the most dramatic real examples of identifying perfect numbers in the modern era: you don’t see the full number written out; you see its structure and its exponent.

For a bit of historical and theoretical context, the American Mathematical Society’s resources at https://www.ams.org/ and various university number theory pages (for example, MIT’s math department at https://math.mit.edu/) give accessible introductions to Mersenne primes and perfect numbers.

The mystery of odd perfect numbers

Here’s where things get interesting. All the examples of identifying perfect numbers: 3 practical examples you see in textbooks are even. So are all the huge ones discovered with computers. No one has ever found an odd perfect number.

What we know today:

• No odd perfect number has been found despite extensive searching.
• If an odd perfect number exists, it must be extremely large and have a very restrictive prime factorization.

Researchers have proved many conditions that an odd perfect number would have to satisfy, but no one has either found one or proved that they cannot exist. This makes perfect numbers a live research topic in 2024–2025, not just a historical footnote.

Putting it all together: spotting patterns in examples of identifying perfect numbers

Let’s step back and look at the patterns across our main examples of identifying perfect numbers: 3 practical examples plus the bigger ones.

• 6 = 2 × 3

  • Mersenne form: 2¹ × (2² − 1) = 2 × 3
  • Proper divisors: 1, 2, 3 → sum 6.

• 28 = 4 × 7

  • Mersenne form: 2² × (2³ − 1) = 4 × 7
  • Proper divisors: 1, 2, 4, 7, 14 → sum 28.

• 496 = 16 × 31

  • Mersenne form: 2⁴ × (2⁵ − 1) = 16 × 31
  • Proper divisors sum to 496.

• 8,128 = 64 × 127

  • Mersenne form: 2⁶ × (2⁷ − 1) = 64 × 127.

• 33,550,336 = 4,096 × 8,191

  • Mersenne form: 2¹² × (2¹³ − 1).

• 8,589,869,056 = 65,536 × 131,071

  • Mersenne form: 2¹⁶ × (2¹⁷ − 1).

In every case, the structure is the same: a high power of 2 multiplied by a prime of the form 2ⁿ − 1. This pattern is exactly what lets you jump from small, hand‑checked examples to gigantic, computer‑verified ones.

So when you hear someone talk about the best examples of identifying perfect numbers: 3 practical examples, keep this connection in mind: those three are just the visible tip of a very deep iceberg built from Mersenne primes.

FAQ: short answers about examples of perfect numbers

What are some simple examples of identifying perfect numbers?

The go‑to simple examples of identifying perfect numbers are 6, 28, and 496. For each one, you list its proper divisors and check that their sum equals the original number. These are the classic 3 practical examples taught in early number theory courses.

Is 28 the only example of a perfect number under 100?

Yes. Under 100, the only perfect numbers are 6 and 28. Numbers like 12, 18, or 48 might look promising because they have many divisors, but the sum of their proper divisors does not equal the number itself, so they are not perfect.

What is an example of a large perfect number?

A widely cited example of a large perfect number is 8,128, which equals 2⁶ × 127. Even larger real examples include 33,550,336 and 8,589,869,056. These are identified using the formula N = 2ⁿ⁻¹ (2ⁿ − 1), where 2ⁿ − 1 is a Mersenne prime.

Are there any known odd perfect numbers?

No. Despite centuries of work and modern computer searches, no odd perfect number has ever been found. If one exists, it would have to be extremely large and satisfy strict conditions on its prime factors. This remains an open problem in number theory.

Where can I learn more about the theory behind perfect numbers?

For a deeper theoretical background, you can explore number theory resources from universities and professional societies, such as the American Mathematical Society at https://www.ams.org/ or math department pages at major universities like https://math.mit.edu/. These sources discuss Mersenne primes, perfect numbers, and related open problems in more detail.