Problem 3 - A Gentle Introduction to the Art of Mathematics

Question

What would be the largest prime one would sieve with in order to find all primes up to \(400 ?\)

Step-by-Step Solution

Answer

The largest prime is 19.

Step 1: Understand the Problem

To find all prime numbers up to 400, we need to determine the largest prime number we would use as a 'sieve'. This involves understanding the process of the Sieve of Eratosthenes.

Step 2: Find the Square Root

Calculate the square root of 400 to identify the threshold. \( \sqrt{400} = 20 \) This means we only need to check for prime numbers less than or equal to 20.

Step 3: Identify Primes Up to the Square Root

List all primes up to 20: \( 2, 3, 5, 7, 11, 13, 17, 19 \)

Step 4: Determine the Largest Prime

Among the primes up to 20, the largest prime number is 19.

Key Concepts

finding prime numbers

To tackle problems like finding all prime numbers up to a given number, we often use the Sieve of Eratosthenes. This is an efficient algorithm that helps identify prime numbers within a range. Here's a simplified explanation:

• Start with a list of numbers from 2 to the desired maximum number, in this case, 400.
• Begin with the first number in the list (2) and mark all its multiples as non-prime.
• Move to the next number that is not marked and repeat the process.
• Continue the process until you've processed numbers up to the square root of the maximum number.

This approach systematically eliminates non-primes, leaving only prime numbers in the list.

square root calculation

Calculating the square root is a crucial step in the Sieve of Eratosthenes. Knowing the square root of the maximum number (in this case, 400) tells us up to which number we need to check for primes.
For instance, the square root of 400 is calculated as follows:
\[ \sqrt{400} = 20 \] This simplifies our task because it tells us that we only need to use prime numbers up to 20 to mark the multiples. Therefore, understanding and calculating the square root can save a lot of work and make the sieve method more efficient.

prime identification

After calculating the square root, we need to identify the prime numbers up to that point. For our example, we need all the primes up to 20.
Here's how to identify them:

• List numbers from 2 to 20.
• Check each number to see if it has any divisors other than 1 and itself. If it doesn't, it's a prime number.

As we go through the list, we find 2, 3, 5, 7, 11, 13, 17, and 19 are all primes.
The largest prime number in this list is 19, and it is the largest prime we would use in the Sieve of Eratosthenes method to find all primes up to 400.