What is the 10th perfect number?
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. The first perfect number was discovered by Euclid, and since then, only a few have been found. In this article, we will explore the 10th perfect number and its significance in the world of mathematics.
The 10th perfect number is 8128. It was discovered by the Greek mathematician Euclid in his work “Elements.” Euclid proved that there are an infinite number of perfect numbers, and he provided a method for generating them. The formula for generating perfect numbers is as follows:
\[ 2^{p-1}(2^p – 1) \]
where \( p \) is a prime number. If \( 2^p – 1 \) is also a prime number, then the resulting number is a perfect number.
To find the 10th perfect number, we need to identify the 10th prime number and apply the formula. The 10th prime number is 29. Plugging this value into the formula, we get:
\[ 2^{29-1}(2^29 – 1) = 2^{28}(2^29 – 1) \]
After performing the calculations, we find that the 10th perfect number is indeed 8128. This number is the sum of its proper divisors, which are 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, and 1024. The sum of these divisors is 8128, confirming that 8128 is a perfect number.
The discovery of the 10th perfect number has implications for the study of number theory. Perfect numbers are related to Mersenne primes, which are prime numbers of the form \( 2^p – 1 \). When \( 2^p – 1 \) is a prime number, it generates a perfect number. The 10th perfect number, 8128, is generated by the Mersenne prime \( 2^{29} – 1 \), which is itself a prime number.
Mathematicians continue to search for more perfect numbers, as well as patterns and properties that may exist among them. The 10th perfect number, 8128, serves as a testament to the beauty and complexity of mathematics, and it continues to inspire researchers to explore the depths of number theory.
