Perfect Numbers: The Case of Number 28

Perfect Numbers in Mathematics

A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding the number itself. This concept has fascinated mathematicians for centuries, with the number 28 being one of the smallest and most well-known perfect numbers. The sum of the proper divisors of 28 is 1 2 4 7 14 28, validating its status as a perfect number.

Mathematical Verification

To verify whether a number is perfect, we can use the sum of its proper divisors. For the number 28, the proper divisors are 1, 2, 4, 7, and 14. The sum of these divisors is:

1 2 4 7 14 28

Since the sum of the proper divisors equals the number itself, 28 is indeed a perfect number. This method can be applied to verify other perfect numbers as well. For instance, another well-known perfect number is 6, with proper divisors 1, 2, and 3, and their sum is 1 2 3 6.

Formula and Mersenne Primes

Perfect numbers, particularly even perfect numbers, can be expressed using Mersenne primes. The formula is given by:

P 2^{p-1} * (2^p - 1)

For the number 28, we have ( p 3 ) because ( 2^3 - 1 7 ), which is a prime number. Using this, we can calculate:

2^{3-1} * (2^3 - 1) 2^2 * 7 4 * 7 28

This confirms that 28 is a perfect number through the application of the formula.

Sequence of Perfect Numbers

Perfect numbers are rare and follow a certain sequence. The first few perfect numbers are:

1. 6 2. 28 3. 496 4. 8128 5. 33,550,336 (over 33 million) 6. 8,589,869,056 (over 8 billion) 7. 1,374,386,913,282 (over 137 billion) 8. 2,305,843,008,139,952,128 (over 2 quintillion) 9. 2,658,455,991,569,831,744,654,692,615,953,842,176 (over 658 decillion) 10. 191,561,942,608,236,107,294,793,378,084,303,638,130,997,321,548,169,216 (over 191 sexdecillion)

It is observed that the sequence of perfect numbers follows a specific pattern associated with Mersenne primes. The number 28 is particularly interesting as it is the second perfect number, and the next perfect number, 8,589,869,056, is the 6th perfect number where 6 is itself a perfect number.

Bi-Perfect and Tri-Perfect Numbers

A bi-perfect number is a perfect number whose sequence number is also a perfect number. For example, 28 is a bi-perfect number because its position in the sequence (2) is a perfect number. Similarly, 8,589,869,056 is a bi-perfect number because it is the 6th perfect number, and 6 is a perfect number.

The concept of tri-perfect numbers involves a perfect number whose position in the sequence of perfect numbers is also a bi-perfect number. However, as of now, it is unknown if such numbers exist. The first tri-perfect number, if it exists, would be the 8,589,869,056th perfect number, but its existence remains in question.

Further Reading and Resources

To explore the topic of perfect numbers further, you can refer to the following resources:

Perfect Numbers - This website provides a comprehensive list of the known perfect numbers and their properties. The Largest Known Perfect Number - This page explains the latest known perfect numbers and the methods used to discover them.

The study of perfect numbers continues to be a fascinating area of mathematics, offering insights into the nature of numbers and their relationships.