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 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

Check divisibility up to \( \sqrt{n} \) for prime checking efficiency. Test 9,998 numbers for primes below 10,000.

Step-by-step solution

Understand the Definition of a Prime Number

A prime number is a natural number greater than 1 that is not a product of two smaller natural numbers. This means it can only be divided by 1 and itself without leaving a remainder. For instance, numbers like 2, 3, and 5 are prime, while numbers like 4 and 6 are not, because they can be divided evenly by other numbers.

Determine if a Number is Prime

To check if a number is prime, create a method that checks divisibility by numbers other than 1 and itself. Loop from 2 to the square root of the number and check if the number is divisible by any of these. If yes, the number is not prime.

Write the Prime Checking Method

Create a method called `isPrime` that takes an integer as a parameter and returns a boolean. For a number `n`, loop from 2 to \( ext{int}( ext{sqrt}(n)) \). If `n % i == 0` for any `i`, return false. Otherwise, return true.

Display All Prime Numbers Less Than 10,000

Use the `isPrime` method in a loop to check each number from 2 to 9,999. Print each number that returns true from `isPrime`. This loop will check all numbers up to 9,999.

Calculate Total Numbers Tested

To find all prime numbers below 10,000, test numbers from 2 to 9,999. This means checking 9,998 numbers in total.

Explain Square Root Optimization

Instead of testing divisors up to \( n/2 \), testing up to \( \sqrt{n} \) is sufficient. If `n` is divisible by a number greater than its square root, then it must also be divisible by a number smaller than the square root. Thus, only factors up to \( \sqrt{n} \) need to be checked.

Rewrite and Test Both Methods

Implement two versions of the program: one that tests up to \( n/2 \) and another that tests up to \( \sqrt{n} \). Compare for efficiency and confirm both find the same prime numbers. The \( \sqrt{n} \) method will perform fewer checks.

Key concepts

Divisibility

In mathematics, divisibility is a fundamental concept. A number is considered divisible by another if there is no remainder after division. When exploring prime numbers, understanding divisibility assists in determining whether a number is prime.
Prime numbers are only divisible by 1 and themselves. Consider the number 15, which can be divided by 1, 3, 5, and 15 without leaving a remainder. Since it has divisors other than 1 and itself, 15 is not a prime number.
To check a number's divisibility efficiently, use the modulus operation. For any two integers, say `a` and `b`, `a % b` or the remainder operation, helps determine divisibility. If `a % b` equals zero, then `a` is divisible by `b`.
This concept is key for algorithms that check for primes by evaluating divisibility from 2 up to a certain point, like the square root of the number in question.

Square Root Optimization

One of the key strategies to determine if a number is prime is the method of using square root optimization. This technique greatly reduces the number of checks needed to determine primality.
The logic behind this technique is simple yet profound. If a number, `n`, is divisible by some integer larger than its square root, it must also be divisible by an integer smaller than its square root. Hence, checking for divisibility up to the square root is not only adequate but efficient.
For example, if you are assessing whether 29 is a prime number, you need only check divisibility utilizing integers up to the square root of 29, which is approximately 5.4. In practical terms, check divisibility using integers 2 through 5. This approach significantly reduces computational efforts and is often preferred in algorithms assessing prime numbers.

Algorithm Efficiency

Algorithm efficiency is a measure of the computational resources required by an algorithm. In the context of prime number determination, efficiency directly impacts performance, especially as numbers grow larger.
For finding prime numbers, efficient algorithms are crucial. By leveraging square root optimization, you reduce the number of necessary checks, enhancing algorithm efficiency. Instead of testing all potential divisors up to a number, testing up to its square root reduces the complexity.
The traditional method of checking all numbers up to `n/2` can be simplified to checking only up to `\( \sqrt{n} \)`. This technique improves the program's performance, decreasing the growth rate of necessary operations from a linear to a much lower order. Such efficiency gains are especially significant when working with large datasets or performing multiple operations.

Number Theory

Number theory explores properties and behaviors of numbers, particularly integers. This area of mathematics is essential for understanding concepts like primes and divisibility.
Central to number theory is the concept of prime numbers. Primes are the building blocks of whole numbers since any number can be expressed as a product of prime numbers. For example, 20 can be broken down into its prime factors, 2 and 5, as \( 2^2 \times 5 \).
This branch of mathematics informs methods for determining prime numbers. Techniques like square root optimization are based on deep number theoretical insights. These approaches allow us to comprehensively explore numerical relationships while maintaining algorithmic efficiency.
In sum, number theory not only deals with fundamental questions about numbers but also provides practical solutions applicable in computer science, encryption, and random number generation.