Question

(Perfect Numbers) An integer is said to be a perfect number if the sum of its factors, including 1 (but not the number itself), is equal to the number. For example, 6 is a perfect number, because \(6=1+2+3 .\) Write a function perfect that determines whether parameter number is a perfect number. Use this function in a program that determines and prints all the perfect numbers between 1 and 1000 . Print the factors of each perfect number to confirm that the number is indeed perfect. Challenge the power of your computer by testing numbers much larger than 1000 .

Short answer

Perfect numbers between 1 and 1000 are 6, 28, 496. Each is a sum of its divisors.

Step-by-step solution

Understanding Perfect Numbers

A perfect number is a positive integer that is equal to the sum of its proper divisors, excluding itself. For example, 6 is a perfect number because its factors 1, 2, and 3 sum to 6.

Write a Function to Check Perfect Numbers

Write a function named `is_perfect` that takes an integer as input and returns `True` if it is a perfect number. Inside the function, initialize a variable to store the sum of factors. Iterate over possible divisors from 1 to the number/2, and if the number is divisible by the divisor, add it to the sum. Finally, check if the sum equals the number itself.

Implement the `is_perfect` Function

```python def is_perfect(number): sum_of_factors = 0 for i in range(1, number): if number % i == 0: sum_of_factors += i return sum_of_factors == number ```

Find Perfect Numbers up to 1000

Use the `is_perfect` function to check each number from 1 to 1000. Collect and print each perfect number found and its factors to confirm its perfection.

Print Perfect Numbers and Their Factors

```python for num in range(1, 1001): if is_perfect(num): factors = [i for i in range(1, num) if num % i == 0] print(f"{num} is a perfect number with factors: {factors}") ``` This loop will print each perfect number and its factors up to 1000.

Short Program to Test Larger Numbers

Optionally, modify the range in the loop to test numbers larger than 1000 to challenge the computer's computational ability. Adjust the loop range appropriately.

Key concepts

Number Theory

Number theory is a fascinating branch of mathematics focused on the properties and relationships of numbers, particularly integers. A key concept within number theory is perfect numbers. A perfect number is defined as a positive integer that equals the sum of its proper divisors, excluding itself. This topic has intrigued mathematicians for centuries due to its simplicity yet deep mathematical implications.

To understand perfect numbers, let's look at an example: 6. The divisors of 6 are 1, 2, 3, and 6. However, when considering perfect numbers, we exclude the number itself. The proper divisors of 6 are 1, 2, and 3, and indeed, their sum is 6. Thus, confirming 6 is a perfect number. The concept of perfect numbers ties into other areas such as divisibility, prime numbers, and later into more complex structures like algebraic number theory.

Algorithm Design

Algorithm design involves creating a step-by-step method to solve a particular problem efficiently. For finding perfect numbers, we employ an algorithm that checks each number for its proper factors and verifies their sum.

The general steps are:

• Initialize a sum variable to zero to keep track of the sum of divisors.
• Iterate over numbers from 1 up to (but not including) the number in question.
• If a number divides evenly, add it to the sum.
• After the loop, check if the sum equals the original number.
• If true, the number is perfect.

This approach efficiently identifies perfect numbers within a specified range. When designing this algorithm, considerations such as computational efficiency and clarity of implementation guide the process.

Python Programming

Python is a powerful, high-level programming language renowned for its readability and straightforward syntax. Python makes it easy to implement algorithmic solutions due to its extensive library support and active community.

For solving the perfect numbers problem, you can use Python to write a concise solution. By defining a function like `is_perfect`, you encapsulate the logic needed to check for perfect numbers. Python's simple loop structures and conditionals allow the function to iterate over potential divisors and perform the necessary arithmetic operations efficiently.

Here's a brief look at how Python can be used: ```python def is_perfect(number): sum_of_factors = 0 for i in range(1, number): if number % i == 0: sum_of_factors += i return sum_of_factors == number ``` This code snippet demonstrates Python's ability to handle simple numerical comparisons and iterations, crucial to solving such problems in number theory.

Factorization

Factorization is the process of breaking down a number into its constituent factors or divisors, excluding the number itself. In the context of perfect numbers, factorization helps identify whether a number is perfect by examining its divisors.

Consider the number 28, which is another perfect number. Its factors are 1, 2, 4, 7, 14, and 28. Excluding 28, we sum the other factors: 1 + 2 + 4 + 7 + 14 = 28. The sum matches the original number, confirming 28 as a perfect number.

In algorithm design, factorization plays a critical role. The divisor loop in the perfect number algorithm iterates only half way up the number, thanks to factorization principles. This significantly reduces computation time, illustrating how understanding mathematical concepts can optimize programming tasks.