What Is a Perfect Number?
Since the dawn of mathematics, people have been fascinated with patterns that can be found in the number system. In exploration of these patterns, mathematicians discovered that some numbers actually equal the sum of their divisors (excluding the number itself). Since their first discovery over 2,000 years ago, 48 of these special numbers, called perfect numbers, have been identified. Now, before you roll your eyes at the history lesson I just gave you, I promise it's actually pretty cool once you see some of these perfect numbers in action.
A perfect number is a positive number that equals the sum of its divisors, excluding itself. This is also known as its aliquot sum. At this time, it is unknown how many perfect numbers truly exist in our number system. While we have discovered 48 perfect numbers, the fact that there are an infinite number of prime numbers leads us to believe that there could be an infinite number of perfect numbers. This might seem a little confusing now since we haven't really talked about how prime numbers relate to perfect numbers, but don't worry. It will all make sense once we discuss the formula used to find this pattern and how it applies to finding perfect numbers. It is also interesting to note that all of the perfect numbers that have been discovered are even numbers. Like with the uncertainty in how many perfect numbers exist, it is also uncertain if there are any odd perfect numbers that have just not been found yet.
The first four numbers date back to the time of Euclid over 2,000 years ago. These numbers are 6, 28, 496, and 8,128. This chart shows each perfect number and the sum of its divisors.
Looking at the four perfect numbers above, you might begin to notice a pattern. Starting with the divisor of 1, each perfect number begins with a series of powers of 2.
Remember, 1 is the same as 20.
So, looking at our first perfect number, 6, we see 2o, 21, and then 3.
In our second perfect number, 28, we see 20, 21, 22, and then 7.
In our third perfect number, 496, we see 20, 21, 22, 23, 24, and then 31.
This pattern holds true for every perfect number. Each perfect number will begin with a series of powers of 2 followed by a prime number. Now, you might notice something about this prime number. If you look closely, you will see that when you double your last power of 2 and then subtract 1, your result is the prime number!
Take our second perfect number, 28, for example. We see that our last power of 2 is 4, and the prime number that follows is 7. Using the pattern we noted above, doubling our power of 2 (4 x 2) equals 8, minus 1, equals seven.
Don't believe me yet? Let's try it with our perfect number 496. We see that our last power of 2 is 16 and the prime number that follows is 31. When we double our power of 2 (16 x 2) and then subtract 1, we get a solution of 31.
We can then take this pattern and turn it into a formula to solve for a perfect number. We will double a power of 2 and then subtract 1. These numbers are called Mersenne primes. This is just one part of our formula for perfect numbers. This gives us our prime number, but we need to also have a portion of our formula to account for the powers of 2.
Our power of 2 can be written as:
Combining these two parts, we get our completed formula:
Perfect Number = 2(n-1) * ((2(n)) - 1)
We can use our first four perfect numbers to prove that this formula works.
- Starting with the first prime number of 2, we get our first perfect number of 6.
- The second prime number of 3 gives us our second perfect number of 28.
- The third prime number of 5 gives us our third perfect number of 496.
- The fourth prime number of 7 gives us our forth perfect number of 8,128.
Now do you see why mathematicians believe that there could be an infinite number of perfect numbers? Since perfect numbers rely on prime numbers, and there are an infinite number of prime numbers, then why can't there be an infinite number of perfect numbers, too?
What Is the Fifth Perfect Number?
Using our formula, we can solve for our fifth perfect number. The prime number that follows 7 is 13. So we can plug 13 into our formula to find its perfect number:
Perfect Number = 2(13-1) * ((2(13)) - 1)
Perfect Number = 2(12) * ((2(13)) - 1)
Perfect Number = 4,096 * (8,192 - 1)
Perfect Number = 4,096 * 8,191
Perfect Number = 33,550,336
In this lesson, you learned that a perfect number is a number that equals the sum of its divisors, excluding the number itself. We discovered the pattern found in divisors and how it translates into the formula to solve for perfect numbers:
Perfect Number = 2(n-1) * ((2(n)) - 1)
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