Question

(PrimeNumbers) 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 function that determines whether a number is prime. b. Use this function in a program that determines and prints all the prime numbers between 2 and 10,000 . How many of these numbers do you really have to test before being sure 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. Estimate the performance improvement.

Short answer

Check primality by testing divisibility up to the square root of the number.

Step-by-step solution

Understanding a Prime Number

A prime number is an integer greater than 1 that can only be divided by 1 and itself without leaving a remainder. This means for a number to be prime, no other positive integers apart from 1 and itself should evenly divide it.

Developing a Function to Check Primality

To create a function that determines if a number is prime, start by excluding 1 because it is not a prime number. For a given number, iterate through positive integers starting from 2 up to that number minus 1. If at any point, one of these integers divides the given number evenly, then the number is not prime.

Optimizing the Prime Check Function

Instead of testing divisibility up to the number itself, improve efficiency by testing up to the square root of the number. If a number has a factor larger than its square root, the corresponding factor less than the square root would already have been found.

Writing a Program Using the Primality Function

Write a program that utilizes this function to check each number from 2 to 10,000, printing out those that are prime.

Determining the Testing Limits

According to the mathematical property that divisors are pairs and no divisors exist higher than the square root without a partner below the square root, you need only test numbers up to the square root of the number in question. This reduces the number of necessary tests and thus improves efficiency.

Estimating Performance Improvement

When comparing testing until halfway up to testing the square root, the square root approach substantially reduces test operations. For larger numbers, the difference becomes more pronounced, leading to significant performance improvements in checking for primality.

Key concepts

Primality Testing

Primality testing is a method for determining whether a given number is prime. A prime number is defined as an integer greater than 1 that is divisible only by 1 and itself. When creating a function to check for primality, one should focus on reducing unnecessary computations. The most naive approach would test divisibility by all integers less than the number in question, but this can be quite inefficient.
An optimized function should use the property that divisors are paired above and below the square root. Therefore, to test if a number, say \( n \), is prime, you only need to check for divisibility using numbers from 2 up to \( \sqrt{n} \). This is because if \( n \) were divisible by some number larger than its square root, then the co-factor would necessarily be smaller, and already found in earlier checks. This significantly cuts down the number of calculations and is thus a more efficient approach for primality testing.

Algorithm Optimization

Algorithm optimization is critical for enhancing the efficiency of primality testing. This involves refining the logic of the function to minimize the computational steps while maintaining accuracy. Instead of using a basic loop that tests all integers up to \( n - 1 \), the optimized approach uses up to \( \sqrt{n} \). This transformation leverages mathematical insights about factor pairs and eliminates redundant calculations.
Optimization does not stop at adjusting limits. Further improvements could include:

• Eliminating even numbers from checks past the integer 2, as they cannot be prime.
• Using known small primes to skip unnecessary tests on larger numbers.
• Implementing more advanced methods like the Sieve of Eratosthenes for bulk prime calculations.

By continually refining the algorithm, performance efficiencies can be realized, particularly when dealing with large datasets or numbers.

Programming Efficiency

Programming efficiency in the context of primality testing can dramatically affect the runtime of an application, especially when needing to determine numerous primes or verify the primality of very large numbers. Efficient programming involves not only algorithmic improvements but also considerations related to code structure, the choice of data structures, and language-specific optimizations.
Efficiency can be achieved through:

• Choosing appropriate data types to handle large integers without overflow.
• Minimizing computational complexity by using efficient loops and conditionals.
• Profiling the code to identify bottlenecks and streamline sections that are computationally intensive.
• Ensuring that the code is readable and maintainable, as this is often a precursor to easier optimization in the future.

By focusing on these factors, developers can create programs that are not only effective in solving the problem but also efficient in terms of resource consumption, leading to faster and more scalable applications.