Definition of a Perfect Number
A perfect number is a positive integer that is equal to the sum of its proper positive divisors. Proper divisors are all positive divisors of a number, excluding the number itself. For example, for the number 6, its proper divisors are 1, 2, and 3. When you add these together (1 + 2 + 3), the sum is 6, making 6 a perfect number.
Historical Significance and Early Discovery
The concept of perfect numbers dates back to ancient Greek mathematics, notably studied by Euclid. They were considered special due to their unique properties and were often associated with mystical or theological significance. Early mathematicians explored these numbers as a way to understand the fundamental patterns and structure within numbers, influencing the development of number theory.
An Illustrative Example: The Number 28
To further illustrate, consider the number 28. Its proper positive divisors are 1, 2, 4, 7, and 14. If we sum these divisors (1 + 2 + 4 + 7 + 14), the result is 28. Since the sum of its proper divisors equals the number itself, 28 is classified as a perfect number, following 6 as the second in the sequence.
Key Properties and Unsolved Questions
All currently known perfect numbers are even. A significant unsolved problem in mathematics is whether any odd perfect numbers exist. Euclid and Euler established a powerful connection between even perfect numbers and Mersenne primes (prime numbers of the form 2^p - 1), proving that an even number is perfect if and only if it can be expressed as 2^(p-1) * (2^p - 1) where (2^p - 1) is a Mersenne prime.