Question

A number is said to be palindrome if it is invariant under reversion; that is, the number is the same if its digits are reversed. For example, 3456543 is palindromic. Write a program that checks each of the first 10,000 prime numbers and prints those that are palindromic.

Short answer

Identify palindromic primes among the first 10,000 primes by checking each for primality and palindromicity, then print them.

Step-by-step solution

Define a Function to Check Palindrome

To begin, we need a function that checks if a number is a palindrome. A palindrome number should read the same when reversed. The function will take a number as input, convert it to a string, and compare it to its reverse. If they are identical, the function returns true (the number is palindromic).

Define a Function to Check Primality

We also need a function that determines if a number is prime. This function should check divisibility from 2 up to the square root of the number. If the number is not divisible by any of these, it's prime, and the function returns true.

Loop Through Natural Numbers

Create a loop to iterate through natural numbers starting from 2 (the first prime) and keep a count of prime numbers identified. We will end the loop once we have found 10,000 prime numbers.

Check Each Number for Primality

Within the loop, use the prime-checking function on each number. If the number is prime, increase the count of found prime numbers.

Check for Palindrome

For each prime number found, check if it's a palindrome using the palindrome-checking function. If the number is both prime and palindromic, store or print the number.

Print or Output the Results

Once the loop has been processed and 10,000 prime numbers have been checked, output the stored palindromic numbers.

Key concepts

Prime Numbers

Prime numbers are a fundamental concept in mathematics. These numbers are greater than 1 and have no divisors other than 1 and themselves. This means they cannot be divided evenly by any other number. For example, the numbers 2, 3, 5, and 7 are all prime numbers.

Identifying prime numbers is crucial, especially in the context of programming and algorithm development. To determine if a number is prime, we can use an efficient method. One common technique involves testing divisibility only up to the square root of the number. This is because a larger factor of the number must be a multiple of a smaller one that has already been checked.

When checking primality within a program, efficiency is key, especially when dealing with large numbers, such as the first 10,000 primes. Once a number is confirmed as a prime, it can then be further analyzed or used for different applications, such as checking for palindromic properties.

Algorithm Development

Algorithm development is an essential skill in programming. It involves crafting a step-by-step method to solve a problem or perform a task efficiently. When developing an algorithm, clarity and efficiency are top priorities.

In the context of our palindrome program, we start by identifying two core tasks: checking if a number is prime and checking if it is a palindrome. The solution is to develop separate functions for each task. This modular approach allows you to reuse these functions as needed.

Following the steps used in the solution:

• First, create a function that ensures a number reads the same forwards and backwards, thus identifying palindromes.
• Next, include a function that confirms if a number is prime by assessing divisibility against a set range of numbers.
• Finally, establish a mechanism, like a loop, to iterate through natural numbers and check each one for primality and then palindromic properties.

By breaking down the problem into smaller, manageable parts, algorithm development becomes less daunting and the resulting code becomes both readable and maintainable.

Programming Concepts

Programming is essentially a toolbox filled with concepts and techniques that help solve complex problems efficiently. For a palindrome program, several core concepts are employed, highlighting fundamental programming techniques.

One of the first concepts is the creation of functions. Functions allow you to encapsulate a block of code that performs a specific task, such as checking if a number is a palindrome or if it is prime. This makes the code modular and organized.

Loops are another important programming concept. They enable repeated execution of a block of code, which is particularly useful in iterating through a large set of numbers, like the first 10,000 primes. A loop facilitates checking each number using the previously defined functions, efficiently handling large computational processes.

Lastly, concepts such as conditionals ("if" statements) are pivotal. They allow the program to make decisions, such as determining whether a number should be counted or printed based on its primality and palindromic status.

Understanding and applying these concepts will significantly enhance your programming abilities and problem-solving skills.