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. perfect number. Encyclopedia Britannica. Merriam-Webster. Last Accessed 3/12/2009. number.

Cite this article as:

McAdams, David E. Perfect Number. 5/5/2011. All Math Words Encyclopedia. Life is a Story Problem LLC.

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10/25/2008: Initial version. (McAdams, David E.)

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