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.
Over two thousand years ago, Euclid showed that all even perfect numbers can be represented by:
where p is a prime for which is a Mersenne prime.
That is to say, we have an even perfect number of the form N whenever the Mersenne Number is a prime number. Mersenne was obviously familiar with Euclid's book when coming up with his primes.
Prime, p | Mersenne Prime, | Perfect Number |
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 |
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.
Find all the perfect numbers from 1 to 500.
Example 1
We know that every perfect number can be expressed as where p is a prime
number. We can use this formula to find all the perfect numbers from 1 to 500.
For
For
For
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
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
Since , then 8128 is a perfect number.
Finite Mathematics Diagnostic Tests
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.