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In Mathematical terms, a perfect number is a positive number and is equivalent to the sum of its factors excluding itself. For example, 6 has divisors 1, 2 and 3 (not considering 6), and 1 + 2 + 3 is equal to 6. Therefore six is a perfect number example.
The sum of factors of a number, barring the actual number, is its aliquot sum, so a perfect number is equivalent to its aliquot sum. Comparably, a perfect number is a number that is half of the sum of its positive factors, including the number itself.
History of the perfect numbers
It wasn’t realised when the Perfect Numbers were found. It is believed that the Egyptians may have known about it previously. Although the mathematicians in ancient times knew about the presence of Perfect Numbers, Greeks were the ones who were fascinated by it, particularly Pythagoras and some of his adherents.
They observed the number six as intriguing (more for its magical and unusual properties than any numerical importance). The number 6 is the number of appropriate elements; for example, six equals one + two + three. It is the littlest Perfect Number, the following being twenty-eight.
However, the Pythagoreans were keen on mysterious and unique behaviours, so they researched a little about them. Somewhere close to 300 BC, the actual primary outcome was made.
There are various ways of characterising Perfect Numbers; the ancient definitions concerned the aliquot parts. The creator characterises:
A perfect number is a number that equals the sum of the factors of the number, not considering the number itself.
How can we find/determine the Perfect Numbers?
Perfect numbers are the unique numbers that are lesser-known and referred to the students when contrasted with other different kinds of numbers. This part will figure out how to find perfect numbers with ease.
There is no single defined relational formula to track down the perfect numbers. The initial four perfect numbers can be generated with the help of the formula 2p−1(2p − 1), where p is a prime number. Prime numbers of the structure 2p − 1 are called Mersenne Primes.
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Proposition
If however many numbers as those start from the unit are set out persistently in twofold extent until the amount of all turned into a prime, then, at that point, the result of the total and the last number make a perfect number. Twofold Proportion implies that each number in succession is twofold the former number. For instance, 1+2+4=7 is a prime number.
As indicated by Proposition 1, the total sum × last number is equal to the perfect number
7 × 4= 28, and indeed, 28 is a perfect number.
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Proposition 2
(N(N+1))/2 is a perfect number if N is a Mersenne prime. Mersenne prime is one less than the power of two. For instance, Let’s accept N as 31, which is one less than 24. Then, at that point, (N(N+1))/2=(31(31+1))/2=(N(N+1))/2=(31(31+1))/2=496. And 496 is a perfect number example.
Table of The Perfect Numbers
|
Prime, p |
Mersenne Prime, 2p -1 |
Perfect Number, 2p-1(2p -1) |
|
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 |
Examples Of The Perfect Numbers
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There are many perfect numbers in the number system.
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There exist only two perfect numbers from 1 to 100, six and twenty-eight. 6, 28, 496 are some more perfect numbers.
Some facts about the perfect number
- The kth perfect number has k digits.
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The perfect numbers are even. The fact that the odd perfect numbers exist is still not known.
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The perfect numbers end with 6 and 8 alternatively.
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The sum of all the factors of a perfect number is double the perfect number.
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6 is the smallest perfect number.
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6 is the only square-free perfect number.
Conclusion
In the end, we now know that a unique set of numbers in our number system are the perfect numbers, and they are lesser-known to the students when contrasted with other different kinds of numbers. A perfect number is a number that equals the sum of its factors, not considering the number itself. Examples of perfect numbers from 1 to 100 are 6 and 28. We also know that there is not just a single formula or method to find the perfect numbers. The initial four perfect numbers can be generated with the help of the formula 2p−1(2p − 1), where p is a prime number.
