Perfect Number Checker
Check if a number is perfect, abundant, or deficient by analyzing the sum of its proper divisors.
Enter n (1 to 100,000,000) to check if it's a perfect number
A perfect number is a positive integer that equals the sum of its proper divisors (all positive divisors excluding the number itself). This concept dates back to ancient Greek mathematics, where perfect numbers were considered to have mystical significance.
Classification:
- Perfect: Sum of proper divisors = n
- Abundant: Sum of proper divisors > n
- Deficient: Sum of proper divisors < n
First Perfect Numbers: 6, 28, 496, 8128, 33550336. These are increasingly rare—there are only 51 known perfect numbers, and all of them are even.
Euclid-Euler Theorem: Every even perfect number has the form 2^(p-1) × (2^p - 1), where 2^p - 1 is a Mersenne prime. For example, 6 = 2^1 × 3 (where 3 = 2^2 - 1 is prime), and 28 = 2^2 × 7 (where 7 = 2^3 - 1 is prime).
Open Problems: It is unknown whether any odd perfect numbers exist. If one does exist, it must be greater than 10^1500 and have at least 101 prime factors. The existence of infinitely many even perfect numbers is also unproven, as it depends on whether there are infinitely many Mersenne primes.
Historical Significance: Perfect numbers were studied by Euclid around 300 BCE and later by Euler in the 18th century. The ancient Greeks associated them with harmony and completeness, as the number perfectly balances with its parts.
What is a perfect number?
A perfect number is a positive integer that equals the sum of its proper divisors (all divisors except itself). For example, 6 is perfect because 1 + 2 + 3 = 6, and 28 is perfect because 1 + 2 + 4 + 7 + 14 = 28.
How many perfect numbers are known?
Currently, 51 perfect numbers are known. All known perfect numbers are even, and it remains an open question whether any odd perfect numbers exist. The largest known perfect number has over 49 million digits.
What is the connection between perfect numbers and Mersenne primes?
The Euclid-Euler theorem states that every even perfect number has the form 2^(p-1) × (2^p - 1), where 2^p - 1 is a Mersenne prime. Conversely, every Mersenne prime generates a unique even perfect number. This is why the search for perfect numbers is closely tied to the search for Mersenne primes.
What are abundant and deficient numbers?
A number is abundant if the sum of its proper divisors exceeds the number (like 12, where 1+2+3+4+6=16>12), and deficient if the sum is less than the number (like 8, where 1+2+4=7<8). Perfect numbers are the rare cases where the sum exactly equals the number.