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


Triangular Number
Hexagonal Number
28 j=0 implies n=8(0)+2=2; n=2 implies Tn=(1/2)*2*(2+1)=3; Tn=3 implies P=1+9(3)=28 sum from n=1 to 7 n=1+2+...+7=28
496 j=1 implies n=8(1)+2=10; n=10 implies Tn=(1/2)*10*(10+1)=55; Tn=55 implies P=1+9(55)=496 sum from n=1 to 31 n=1+2+...+31=496
Not a
j=0 implies n=8(2)+2=18; n=18 implies Tn=(1/2)*18*(18+1)=171; Tn=171 implies P=1+9(171)=1540 Not applicable
8128 j=5 implies n=8(5)+2=42; n=42 implies Tn=(1/2)*42*(42+1)=903; Tn=903 implies P=1+9(903)=8128 sum from n=1 to 127 n=1+2+...+127=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 P=1_9Tn where Tn is the triangular number Tn=(1/2)n(n+1) such that n=8j+2 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.



  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. Encyclopedia Britannica. Merriam-Webster. Last Accessed 12/3/2018. number. Buy the book

Cite this article as:

McAdams, David E. Perfect Number. 4/28/2019. All Math Words Encyclopedia. Life is a Story Problem LLC.

Image Credits

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.)
12/1/2018: Removed broken links, updated license, implemented new markup. (McAdams, David E.)
10/25/2008: Initial version. (McAdams, David E.)

All Math Words Encyclopedia is a service of Life is a Story Problem LLC.
Copyright © 2018 Life is a Story Problem LLC. All rights reserved.
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License