How to Identify Prime Numbers Efficiently in Python: A Step-by-Step Guide

Identifying Prime Numbers Efficiently in Python

Finding a series of prime numbers within a given range is a common programming task. To achieve this in Python, we employ a logical sequence of loops and conditional statements to determine primality. However, it's essential to note that some initial attempts might yield incorrect results.

Correcting the Code for Prime Number Identification

An inspection of the original code reveals a critical flaw: it incorrectly prints odd numbers, not primes. This error stems from a missing condition that effectively identifies non-prime numbers. Here's a breakdown of the issue:

<code class="python">for num in range(1, 101):
    for i in range(2, num):
        if num % i == 0:
            break
        else:
            print(num)
            break</code>

To correct this, we need to explicitly check if the number is divisible by any number between 2 and itself. If no divisors are found, it is prime. Here's the improved version:

<code class="python">for num in range(2, 101):
    prime = True
    for i in range(2, num):
        if (num % i == 0):
            prime = False
    if prime:
        print(num)</code>

Optimizing the Code for Efficiency

To enhance performance, it is recommended to only check divisors up to the square root of the given number. If no divisors are found within this range, it can be considered prime. This optimization drastically reduces the number of iterations required:

<code class="python">import math

for num in range(2, 101):
    if all(num % i != 0 for i in range(2, int(math.sqrt(num)) + 1)):
        print(num)</code>

Further Refinements

The code can be made even more efficient by selecting odd numbers only since prime numbers greater than 2 are always odd. The revised code:

<code class="python">import math
print(2)
for num in range(3, 101, 2):
    if all(num % i != 0 for i in range(3, int(math.sqrt(num)) + 1, 2)):
        print(num)</code>

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