What are the first 5 perfect numbers?
Perfect numbers have fascinated mathematicians for centuries. They are positive integers that are equal to the sum of their proper divisors, excluding the number itself. In other words, if you add up all the positive divisors of a perfect number except the number itself, you will get the number back. This unique property has led to a rich history of study and discovery in the field of mathematics. In this article, we will explore the first five perfect numbers and their significance in the world of mathematics.
The first perfect number is 6, which is also known as the smallest perfect number. It can be found by summing the divisors of 6, which are 1, 2, 3, and 6. By excluding 6 from the sum, we get 1 + 2 + 3 = 6. This discovery was made by the ancient Greek mathematician Euclid, who proved that all even perfect numbers can be expressed in the form 2^(p-1) (2^p – 1), where 2^p – 1 is a prime number, known as a Mersenne prime.
The second perfect number is 28, which was also known to the ancient Greeks. It can be found by summing the divisors of 28, which are 1, 2, 4, 7, 14, and 28. Excluding 28 from the sum, we get 1 + 2 + 4 + 7 + 14 = 28. This number is significant because it is the first perfect number after 6 and is the sum of the first four prime numbers.
The third perfect number is 496, which was discovered by the Chinese mathematician Zhang Heng in the 3rd century. The divisors of 496 are 1, 2, 4, 8, 16, 31, 62, 124, 248, and 496. Excluding 496 from the sum, we get 1 + 2 + 4 + 8 + 16 + 31 + 62 + 124 + 248 = 496. This number is significant because it is the first perfect number that is not the sum of consecutive primes.
The fourth perfect number is 8128, which was discovered by the English mathematician Euclid in the 3rd century. The divisors of 8128 are 1, 2, 4, 8, 16, 32, 64, 127, 254, 508, 1016, 2032, 4064, and 8128. Excluding 8128 from the sum, we get 1 + 2 + 4 + 8 + 16 + 32 + 64 + 127 + 254 + 508 + 1016 + 2032 + 4064 = 8128. This number is significant because it is the first perfect number that is not the sum of a single prime and its powers.
The fifth perfect number is 33550336, which was discovered by the French mathematician Euclid in the 3rd century. The divisors of 33550336 are 1, 2, 4, 8, 16, 32, 4281, 8562, 17124, 34248, 68496, 136992, 273984, 547968, 1095936, 2191864, 4383728, 8767456, and 33550336. Excluding 33550336 from the sum, we get 1 + 2 + 4 + 8 + 16 + 32 + 4281 + 8562 + 17124 + 34248 + 68496 + 136992 + 273984 + 547968 + 1095936 + 2191864 + 4383728 + 8767456 = 33550336. This number is significant because it is the largest perfect number that can be expressed as the sum of two distinct Mersenne primes.
In conclusion, the first five perfect numbers are 6, 28, 496, 8128, and 33550336. These numbers have played a crucial role in the development of number theory and have provided mathematicians with a deeper understanding of the properties of integers. As more perfect numbers are discovered, the mystery and beauty of these special numbers continue to captivate mathematicians around the world.
