Question

An integer is said to be prime if it is divisible by only 1 and itself. For example, 2,3,5 and 7 are prime, but 4,6,8 and 9 are not: a) Write a method that determines whether a number is prime. b) Use this method in an application i that determines and displays all the prime numbers less than \(10,000 .\) How many numbers up to 10,000 do you have to test to ensure that you have found all the primes? c) Initially, you might think that \(n / 2\) is the upper limit for which you must test to see whether a number is prime, but you need only go as high as the square root of \(n .\) Why? Rewrite the program, and run it both ways.

Short answer

The `is_prime` method determines primality by testing divisibility up to the square root. We test 9,998 numbers to find all primes below 10,000.

Step-by-step solution

Understanding Prime Numbers

A prime number is defined as a natural number greater than 1 that is not a product of two smaller natural numbers. The only divisors of a prime number are 1 and itself. For example, 2 is a prime number because no other whole numbers multiply to 2 except 1 and 2. In contrast, 4 can be divided evenly by 2, so it is not prime.

Writing a Method to Check for Primality

To write a method to check if a number is prime, we iterate from 2 up to the square root of the number, checking if any number in this range can divide the given number without a remainder. If such a number is found, the number is not prime. Otherwise, it is prime. The method can be implemented as follows: ```python import math def is_prime(n): if n <= 1: return False for i in range(2, int(math.sqrt(n)) + 1): if n % i == 0: return False return True ```

Using the Method to Find Primes under 10,000

We will use the `is_prime` method to iterate over numbers from 2 to 9,999 and display numbers that return `True` from the `is_prime` method. By doing this, we will find all the primes below 10,000. ```python for num in range(2, 10000): if is_prime(num): print(num) ```

Determining the Number of Tests

To find all prime numbers less than 10,000, we must test every number from 2 to 9,999 to ensure we've identified all primes. Thus, the number of tests is 9,998 (as we are not testing number 1).

Explanation of Square Root Approach

Initially, one might test divisibility by all numbers up to n/2 to determine primality. However, testing up to the square root of n is sufficient. This is because if a number n is divisible by another number p, then n = p * q where q must be greater than p if n > p^2. If both were greater, their product would exceed n. Thus, any factor larger than the square root would already have been tested as part of a smaller factor pair.

Key concepts

Primality Testing

Primality testing is the process of determining whether a given number is a prime. A prime number is only divisible by 1 and itself, meaning it has no other factors. This characteristic is useful in various fields, especially in cryptography. To test if a number is prime, you can follow a straightforward method. Start by checking if the number is less than 2, since numbers less than 2 are not prime. Then, try dividing the number by every integer up to a certain limit. If you find any number that divides evenly (no remainder), the number in question is not prime. Otherwise, it is prime. This logical approach helps efficiently decide whether numbers like 2, 3, 5, or even larger numbers, are primes or not.

Square Root Method

When testing for prime numbers, reducing computation time is crucial, especially for large numbers. The square root method aids in this by reducing the range of potential divisors. Instead of checking divisibility up to the number itself or even half of it, you only need to test up to the square root. This method works because any factors greater than the square root would imply a pair of smaller factors that multiply to the number. If no divisor is found by the time you reach the square root, the number is prime. This significantly improves efficiency, especially when dealing with large numbers, making the check much faster and computationally cheaper.

Iterative Algorithms

An iterative algorithm repeatedly executes a series of instructions to progress towards a solution. In the context of primality testing, an iterative algorithm explores divisibility for a number by checking each potential factor incrementally. For instance, in a simple Python program, you'd use a `for` loop to iterate through numbers, checking for divisibility until reaching your testing limit (like the square root of the number). This systematic approach ensures that each possible factor is considered, aiding in the determination of a number's primality. Such algorithms are essential because they offer a clear, repeatable process for testing many numbers efficiently.

Number Theory

Number theory is a branch of pure mathematics devoted to studying the properties and relationships of numbers, particularly integers. It provides the foundation for understanding prime numbers, one of its central topics. Through various theorems and properties, number theory illuminates how numbers interact and offers methodologies for discovering and verifying prime numbers. In practical terms, it gives insight into devising efficient primality tests and understanding why they work. Number theory isn't just theoretical; it has real-world applications in computer science, coding theory, and cryptography, where understanding primes is critical for encryption algorithms.