Question

The Goldbach conjecture asserts that every even number is the sum of two prime numbers. Write a program that gets a number from the user, checks to make sure that it is even, and then finds two prime numbers that add up to the number.

Short answer

Create a function to check for prime numbers, loop through candidates, and find a pair of primes summing to the input number.

Step-by-step solution

Input the Number

Prompt the user to enter an even number. Ensure that the input is converted to an integer.

Verify the Number is Even

Check if the number entered by the user is even. This can be done by using the condition `number % 2 == 0`. If the number is not even, display an error message and stop the program.

Check Prime Function

Define a helper function that takes a number as input and returns True if it is prime, otherwise False. This function will check divisibility from 2 up to the square root of the number.

Loop to Find Prime Pairs

Iterate over numbers from 2 to the number entered by the user. For each number, check if both it and its complement (entered number - current number) are prime using the prime-checking function defined previously.

Output the Prime Pair

As soon as a pair of prime numbers is found that add up to the given number, print this pair and terminate the loop.

Key concepts

Prime Numbers

Prime numbers are the building blocks of mathematics, much like atoms are to chemistry. These special numbers are defined as natural numbers greater than 1 that have no divisors other than 1 and themselves. This means they cannot be evenly divided by any other number without leaving a remainder. Some of the smallest prime numbers include 2, 3, 5, 7, and 11. It's interesting to note that 2 is the only even prime number. All other even numbers can be divided by 2, which is why they are not prime.
Understanding prime numbers is essential when exploring mathematical theories such as the Goldbach conjecture. This conjecture hypothesizes that every even integer greater than 2 can be expressed as the sum of two prime numbers. Whether you're writing algorithms or exploring number theory, primes are integral to many mathematical concepts.

Even Number Validation

Validating whether a number is even is a simple yet crucial step in solving problems like the Goldbach conjecture. An even number is any integer that can be divided by 2 without leaving a remainder. In programming terms, you can verify a number's 'evenness' using the modulus operator: `number % 2 == 0`. This expression will return `True` if the number is even, indicating no remainder.
Ensuring that a user input or any number is even is important for the integrity of a program. Accidentally using an odd number could lead to incorrect results, especially in algorithms designed to find something specific about even numbers, such as pairs of primes that add up to them. Thus, validation is a vital step before proceeding with more complex calculations.

Prime Number Checking

Checking if a number is prime requires a bit more effort compared to validating evenness. A classic approach involves testing divisibility, ensuring that no integer between 2 and the square root of the number can evenly divide it. This method is efficient because if a number is divisible by a number larger than its square root, it must also be divisible by a smaller number.
For implementation, you can write a function where, for each number `n` you check divisibility starting from 2 up to the integer value of its square root, using a loop. If none of these numbers divide `n` without a remainder, then `n` is prime. This kind of function becomes a pivotal tool when designing algorithms that require checking many numbers for primality, especially when solving problems in number theory.

Algorithm Design

Algorithm design is the art of creating a step-by-step solution to a problem. In the context of the Goldbach conjecture, the algorithm must not only handle user input and validation but also systematically search for two prime numbers that add up to the given even number. It involves several components:

• Input: Prompt and capture an even number input from the user.
• Validation: Ensure the number is even for the conjecture to hold.
• Prime-checking: Deploy a function to verify the primality of numbers.
• Looping: Use a loop to try combinations of numbers to find the prime pairs.

Each part of the algorithm is crucial. For instance, efficiently checking prime numbers reduces the computational load. The use of loops aids in checking numerous combinations swiftly. Finally, output the result to provide the prime numbers that satisfy the condition for the given even number. With careful design and testing, such an algorithm becomes a powerful tool in computational mathematics.