Although Euclid concentrated on Geometry, many number theory results can be found in his text Burton, Perfect numbers are the positive integers that are equal to the sum of its factors except for the number itself. In other words, perfect numbers are the positive integers which are the sum of its proper divisors. The smallest perfect number is 6, which is the sum of its proper divisors: 1, 2 and 3. Do you know that when the sum of all the divisors of a number is equal to twice the number, the number has a separate name?
Such numbers are called complete numbers. In fact, all the perfect numbers are also complete numbers. The four perfect numbers 6, 28, and are known to us since ancient times. Let's see their divisors and their sum through the table given below:. Perfect numbers are the special type of numbers that are less known to students as compared to the other types of numbers.
In this section, you will learn how to find perfect numbers easily. If as many numbers as those beginning from the unit be set out continuously in double proportion until the sum of all becomes a prime, then the product of the sum and the last number makes a perfect number.
Double Proportion means that each number in a sequence is double the preceding number. Mersenne prime is a prime that is one less than the power of 2. For example, Let's take N as 31 which is 1 less than 2 4. Here are a few topics that can be referred to for more information on topics related to perfect numbers.
The proper factors of 28 are 1, 2, 4, 7 and The sum of proper factors is According to the definition of perfect numbers, 28 is a perfect number. According to Euclid's proposition, if 2 p -1 is a prime number, then 2 p-1 2 p -1 is a perfect number. Can you make a list of 8 perfect numbers using this proposition? So no perfect primes, and no perfect prime powers. What can be perfect? Notice that each of the last three numbers is a multiple of 7: We can factor out that 7 to reveal some hidden structure:.
This means that we can actually write. Why does this work? Well, the divisors of a number come from its prime factors. Consider 28 again, which is the product of 2 2 and 7, and think about the multiplication table below:. Along the top are the powers of 2 that evenly divide 28, and down the side are the powers of 7 that evenly divide Notice what happens when we fill out this multiplication table.
We get all the divisors of When we multiply this out using the distributive property, this also produces all the divisors of 28 and then adds them up:. Euclid used this fact 2, years ago to create a formula for finding perfect numbers, with help from a special kind of prime and a clever argument about products and divisors. In doing so, he took the first step toward determining what every even perfect number has to look like.
This is a consequence of the geometric series formula we discussed earlier. But 2 k is only divisible by powers of 2. You may have noticed that all these perfect numbers are even. But the question of what odd perfect numbers might be like if they exist remained open. And it remains open today. It would have to have at least nine distinct prime factors, the second-largest of which would have to be greater than 10, And it would have to have a remainder of 1 when divided by 12 or a remainder of 9 when divided by It might seem strange to prove results about numbers that might not even exist.
But every new rule narrows the search a little more. These spoofs are like generalizations of perfect numbers, and so anything true about a spoof would have to be true about a perfect number as well.
Understanding odd spoofs would be especially useful, since any rule discovered for odd spoofs could be added to the existing rules for odd perfect numbers, increasing the chances of finding contradictory criteria and tightening the overall search space. In taking up the challenge, mathematicians have broadened the concept of a spoof and have discovered a new class of numbers to study.
For the most part, investigations into these spoof perfect numbers are done simply for the joy of mathematical exploration. But we don't say anything about triangular numbers, numbers such as 3, 6 and 10 that can be arranged into triangles, even though they're almost as important as squares.
There are a lot of different kinds of numbers out there that are just as accessible as the few we are teaching, but people don't know about them. They encourage kids to create and improvise, which in turn makes for a much richer literacy. Math instructors here are competent in terms of knowing what is in textbooks, but they don't have the ability to improvise with numbers, to play with them.
We expect them to write about what they've read and to communicate effectively.
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