Problem 2 - A Gentle Introduction to the Art of Mathematics

Question

Use the sieve of Eratosthenes to find all prime numbers up to 100 . $$ \begin{array}{cccccccccc} 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\ 11 & 12 & 13 & 14 & 15 & 16 & 17 & 18 & 19 & 20 \\ 21 & 22 & 23 & 24 & 25 & 26 & 27 & 28 & 29 & 30 \\ 31 & 32 & 33 & 34 & 35 & 36 & 37 & 38 & 39 & 40 \\ 41 & 42 & 43 & 44 & 45 & 46 & 47 & 48 & 49 & 50 \\ 51 & 52 & 53 & 54 & 55 & 56 & 57 & 58 & 59 & 60 \\ 61 & 62 & 63 & 64 & 65 & 66 & 67 & 68 & 69 & 70 \\ 71 & 72 & 73 & 74 & 75 & 76 & 77 & 78 & 79 & 80 \\ 81 & 82 & 83 & 84 & 85 & 86 & 87 & 88 & 89 & 90 \\ 91 & 92 & 93 & 94 & 95 & 96 & 97 & 98 & 99 & 100 \end{array} $$

Step-by-Step Solution

Answer

The prime numbers up to 100 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

Step 1 - Create a List

List all numbers from 2 to 100. These numbers will be used to find the prime numbers.

Step 2 - Strike Multiples of 2

Starting from 2, strike out all multiples of 2 (4, 6, 8, etc.) in the list. The number 2 itself remains because it is a prime number.

Step 3 - Strike Multiples of 3

Find the next number that has not been struck. This number is 3. Strike out all multiples of 3 (6, 9, 12, etc.) in the list. The number 3 itself remains because it is a prime number.

Step 4 - Strike Multiples of 5

Find the next number that has not been struck. This number is 5. Strike out all multiples of 5 (10, 15, 20, etc.) in the list. The number 5 itself remains because it is a prime number.

Step 5 - Strike Multiples of 7

Find the next number that has not been struck. This number is 7. Strike out all multiples of 7 (14, 21, 28, etc.) in the list. The number 7 itself remains because it is a prime number.

Step 6 - Continue Until the Square Root of 100

Continue the process with the next number that has not been struck and strike out its multiples. Continue this process until you reach the square root of 100, which is 10.

Step 7 - Remaining Numbers Are Prime

The remaining numbers in the list that have not been struck out are all prime numbers.

Key Concepts

Prime Numbers

Prime numbers are fascinating and critical in mathematics, particularly in number theory. A prime number is any number greater than 1 that only has two distinct positive divisors: 1 and itself. For instance, 2, 3, 5, 7, and 11 are prime numbers because they cannot be divided evenly by any other numbers except 1 and themselves.
Prime numbers are building blocks in mathematics - much like atoms in chemistry. They play a crucial role in many mathematical concepts and applications, such as cryptography and primality testing. To understand why sieve algorithms like the Sieve of Eratosthenes are useful, we first need to recognize the properties of prime numbers.
Knowing these properties helps us develop efficient methods to find primes and solve problems that rely on prime factorization.

Step-by-Step Algorithm

The Sieve of Eratosthenes is an ancient and effective algorithm used to find all prime numbers up to a given limit. Here’s how you can apply it step-by-step:

Step 1: **Create a List**
Write down all numbers from 2 to the given limit (in this case, 100).

Step 2: **Strike Multiples of 2**
Begin with the first prime number, 2. Strike out all multiples of 2 (4, 6, 8, etc.).

Step 3: **Strike Multiples of 3**
Move to the next number in the list that hasn’t been struck out, which is 3. Strike out all multiples of 3 (6, 9, 12, etc.).

Step 4: **Strike Multiples of 5**
Continue to the next unmarked number, which is 5. Strike out all multiples of 5 (10, 15, 20, etc.).

Step 5: **Strike Multiples of 7**
Proceed to 7, and strike out all multiples of 7 (14, 21, 28, etc.).

Step 6: **Continue Until the Square Root of the Limit**
Keep repeating the process for the next unmarked number until you reach the square root of 100, which is 10 in this case.

Step 7: **Remaining Numbers Are Prime**
The numbers left in the list that aren’t struck out are the prime numbers. Following each of these steps in sequence will efficiently yield all the prime numbers up to 100.

Number Theory

Number theory is a branch of mathematics devoted to the study of integers and integer-valued functions. A major aspect of number theory is understanding the properties and relationships of numbers, including prime numbers.
In number theory, prime numbers are crucial because they are the basic building blocks of integers. Every integer greater than 1 can be uniquely factored into prime numbers, known as its prime factorization. For example, 28 can be factored as 2 x 2 x 7.
Algorithms like the Sieve of Eratosthenes are essential in number theory for identifying prime numbers within a specific range. These primes help in various applications, including coding theory, encryption algorithms, and can even simplify complex mathematical proofs.
The elegance and simplicity of prime numbers make them an intriguing topic of continuous research and discovery in the field of mathematics.