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Perfect Number

Perfect Number - Examples, History, What are perfect numbers, List of Perfect Numbers Table and Best Way to Find a Perfect Number


A perfect number is a positive integer that equals the sum of its positive divisors, excluding the number itself. The lowest perfect number is six, which is equal to the sum of the numbers one, two, and three. The numbers 28, 496, and 8128 are also perfect. It is unknown when perfect numbers were originally examined, and the initial studies may date back to when numbers first piqued people's interest. Given the way the Egyptians calculated, it is quite plausible, though uncertain, that they would have come upon such numbers organically.

Perfect Numbers in History

There isn't a lot of information available on how perfect numbers were discovered. They were possibly found by the Egyptians. Even though perfect numbers existed, only the Greeks were interested in learning more about them. Pythagoras and his disciples investigated perfect numbers for their magical qualities. The smallest perfect number that was discovered was 6, mostly for its apparent mystical characteristics. It's still unclear if infinite perfect numbers exist.

The Pythagoreans were mostly concerned with the occult qualities of these numbers, and they did little with them in terms of mathematics. The first real result was achieved approximately 300 BC when Euclid composed his Elements. Euclid's book contains several number theory conclusions, despite his focus on geometry (Burton, 1980).

What are perfect numbers, and how can you find them?

Perfect numbers are positive integers that, except for the number itself, are equal to the sum of their components. Perfect numbers, in other terms, are positive integers that are the sum of their appropriate divisors. The lowest perfect number is 6, which has the following divisors: one, two, and three

List of Perfect Numbers in a Table

Since ancient times, we have known the four perfect numbers 6, 28, 496, and 8128. Let's look at their divisors and totals in the table below:

Perfect Number Sum of Divisors
6 1 + 2 + 3
28 1 + 2 + 4 + 7 + 14
496 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248
8128 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064


What Is the Best Way to Find a Perfect Number?

Perfect numbers are a particular sort of number that pupils are less familiar with than other types of numbers. This section will teach you how to determine perfect numbers.

No one preposition can be used to find ideal numbers

The formula 2p1(2p 1) generates the first four perfect numbers, with p being a prime integer. Mersenne Primes are prime numbers of the type 2p 1 named after the 17th century monk, Marin Mersenne.

Helpful Hints

  • There are kth digits in the kth perfect number.
  • The ideal numbers are all the same. It's still a mystery whether odd perfect numbers exist.
  • All perfect integers alternately finish in 6 and 8.
  • When all of a perfect number's divisors are added together, the result is twice the perfect number.
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