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
10^{300} have been checked without finding
a perfect number.
Even perfect numbers greater than 6 take the form
where T_{n} 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
References
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
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
Cite this article as:
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.
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.)