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# Perfect Numbers

A perfect number N is defined as any positive integer where the sum of its divisors except the number itself equals the number. The first few of these, which were known and respected by the ancient Greeks, are 6, 28, 496, and 8128.

## Perfect number formula

Over two thousand years ago, Euclid showed that all even perfect numbers can be represented by:

$N={2}^{p-1}\left({2}^{p}-1\right)$ where p is a prime for which ${2}^{p}-1$ is a Mersenne prime.

That is to say, we have an even perfect number of the form N whenever the Mersenne Number ${2}^{p}-1$ is a prime number. Mersenne was obviously familiar with Euclid's book when coming up with his primes.

## Perfect number table

 Prime, p Mersenne Prime, ${2}^{p}-1$ Perfect Number $N={2}^{p-1}\left({2}^{p}-1\right)$ 2 3 6 3 7 28 5 31 496 7 127 8128 13 8191 33550336 17 131071 8589869056 19 524287 137438691328 31 2147483647 2305843008139952128 61 2305843009213693951 2658455991569831744654692615953842176

## How many perfect numbers are there?

There are 51 known perfect numbers. There are only two perfect numbers from 1 through 100, which are 6 and 28. The most recently discovered perfect number was discovered in 2018 and it has 49,724,095 digits. So far, all known perfect numbers are even. It is not known if there are any odd perfect numbers, but none have been found so far.

## Solved examples of perfect numbers

Find all the perfect numbers from 1 to 500.

Example 1

We know that every perfect number can be expressed as $N={2}^{p-1}\left({2}^{p}-1\right)$ where p is a prime

number. We can use this formula to find all the perfect numbers from 1 to 500.

For $n=2,{2}^{2-1}\left({2}^{2}-1\right)=2\left(4-1\right)=6$

For $n=3,{2}^{3-1}\left({2}^{3}-1\right)={2}^{2}\left(8-1\right)=4×7=28$

For $n=5,{2}^{5-1}\left({2}^{5}-1\right)={2}^{2}\left(32-1\right)=16×31=496$

And as we know from the table above, the perfect numbers from 1 to 500 are 6, 28, and 496.

Example 2

Check whether the following numbers are perfect numbers or not.

a. 282

The factors of 282 are 1, 2, 3, 6, 47, 94, 141, and 282

The proper divisors of 282 are 1, 2, 3, 6, 47, 94, and 141

$1+2+3+6+47+94+141=294$

Since 294 (does not equal) 282, then 282 is not a perfect number.

b. 8128

The factors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, and 8128

The proper divisors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, and 4064

$1+2+4+8+16+127+254+508+1016+2032+4064=8128$

Since $8128=8128$ , then 8128 is a perfect number.

## Get help learning about perfect numbers

Remembering the formulas (functions) used to find perfect numbers can be confusing. It can also be tricky to come up with all the factors of the higher numbers to check and see if they are indeed perfect numbers. A lot of tedious math can be involved, and not all students are excited about performing these operations.

One of the best ways your student can get a handle on these and other concepts is by working with a private tutor. Tutoring can help motivate your student when they are having a tough time motivating themselves. Also, private tutors have the time to work with your student on the exact concepts that are challenging your student, while skimming over the concepts they easily pick up. If your student is having a difficult time understanding perfect numbers the way that the problems are approached in their classroom, a tutor can try different approaches and use different techniques to solve problems until they find a method that clicks for your student.

Contact the Educational Directors at Varsity Tutors today to see how tutoring can help your student learn about perfect numbers and more. We look forward to helping you find a suitable tutor for your student who matches their unique needs.

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