# Perfect Number

Pronunciation: /ˈpɜr.fɪkt ˈnʌm.bər/ Explain

A perfect number is a number whose divisors, excluding the number itself, add up to the number. 6 is a perfect number. The divisors of 6 are 1, 2, 3, and 6. Leaving the 6 out, the sum is 1 + 2 + 3 = 6. The second perfect number is 28: 1 + 2 + 4 + 7 + 14 = 28. The third perfect number is 496: 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248. The next few perfect numbers are 8128, 33550336, 8589869056, 137438691328, 2305843008139952128.

#### Properties of Perfect Numbers

Perfect
Number

Triangular Number
Hexagonal Number
28  496  Not a
perfect
number Not applicable
8128  Table 1
• It is not known if any odd perfect numbers exist. All odd numbers up to 10300 have been checked without finding a perfect number.
• Even perfect numbers greater than 6 take the form where Tn is the triangular number such that where j is a natural number. However not all numbers generated by this algorithm are perfect numbers. See table 1.
• All even perfect numbers are also hexagonal numbers that are the sum of a set of consecutive integers starting at 1. An example is 28 = 1 + 2 + 3 + 4 + 5 + 6 + 7. See table 1.
• The sum of the reciprocals of the divisors all the even perfect numbers is 2.

### Generate Perfect Numbers

This sections uses a program to generate perfect numbers. Click on the 'Generate' button to check one value to see if it is a perfect number. Click again to see the next, and so on. The results of the generation will appear below.

#### Results

1. McAdams, David E.. All Math Words Dictionary, perfect number. 2nd Classroom edition 20150108-4799968. pg 137. Life is a Story Problem LLC. January 8, 2015. Buy the book
2. perfect number. merriam-webster.com. Encyclopedia Britannica. Merriam-Webster. Last Accessed 12/3/2018. http://www.merriam-webster.com/dictionary/perfect number. Buy the book

McAdams, David E. Perfect Number. 4/28/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/p/perfectnumber.html.

### Revision History

4/28/2019: Changed equations and expressions to new format. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)