The only perfect number of the form x^n + y^n

Define a perfect number. Do x, y and n have to be integers?

The only perfect number of the form x^n + y^n

One solution is 1^3 + 3^3 = 28, a perfect number.

A perfect number is a positive integer that is equal to the sum of its proper divisors. Currently, there are no known perfect numbers of the form x^n + y^n, where x, y, and n are positive integers greater than 1.

However, it is worth noting that there are perfect numbers of the form 2^(p-1)*(2^p - 1), where p and 2^p - 1 are both prime numbers. For example, when p = 2, the formula gives the perfect number 6 (2^(2-1)*(2^2 - 1) = 2*3 = 6). When p = 3, the formula gives the perfect number 28 (2^(3-1)*(2^3 - 1) = 4*7 = 28).

It is an open question in mathematics whether there are any odd perfect numbers, and no odd perfect numbers have been found so far.

To find the only perfect number of the form x^n + y^n, we can break down the problem into steps:

Step 1: Understand the concept
A perfect number is a positive integer that is equal to the sum of its proper divisors (excluding the number itself). In this case, we are looking for a perfect number that can be expressed as x^n + y^n, where x and y are positive integers, and n is a positive integer.

Step 2: Explore special cases
Let's consider some special cases to get a better understanding of the problem:
- When n = 1, the equation becomes x + y. This is not a perfect number because x and y are positive integers, and the sum of proper divisors for any positive integer is always greater than the number itself.
- When n = 2, the equation becomes x^2 + y^2. This is also not a perfect number. In fact, there are no perfect numbers of this form, as proven by Euler.

Step 3: Deduce a generalized solution
Based on the special cases, we can deduce that there are no perfect numbers of the form x^n + y^n where n is greater than 2. This is because for n > 2, the exponential terms x^n and y^n will grow much faster than the linear terms x and y, making it highly unlikely for the equation to hold.

Therefore, the only perfect number of the form x^n + y^n is when n = 2, which is a well-known case. The first (and only) perfect number of this form is 28, where x = 5 and y = 1.

In summary, there are no other perfect numbers of the form x^n + y^n apart from 28 when n equals 2.