Introduction

In this blog post, we will explore how to solve the LeetCode problem "2761" using AI. LeetCode is a popular platform for preparing for coding interviews, and with the help of AI tools like Chatmagic, we can generate solutions quickly and efficiently - helping you pass the interviews and get the job offer without having to study for months.

Problem Statement

You are given an integer n. We say that two integers x and y form a prime number pair if: 1 <= x <= y <= n x + y == n x and y are prime numbers Return the 2D sorted list of prime number pairs [xi, yi]. The list should be sorted in increasing order of xi. If there are no prime number pairs at all, return an empty array. Note: A prime number is a natural number greater than 1 with only two factors, itself and 1. Example 1: Input: n = 10 Output: [[3,7],[5,5]] Explanation: In this example, there are two prime pairs that satisfy the criteria. These pairs are [3,7] and [5,5], and we return them in the sorted order as described in the problem statement. Example 2: Input: n = 2 Output: [] Explanation: We can show that there is no prime number pair that gives a sum of 2, so we return an empty array. Constraints: 1 <= n <= 106

Explanation

Here's the approach, complexity analysis, and Python code for the prime number pair problem:

• Sieve of Eratosthenes: Use the Sieve of Eratosthenes to efficiently precompute all prime numbers up to n. This avoids repeated primality tests.
• Iterate and Check: Iterate through numbers from 2 up to n // 2. For each number x, check if both x and n - x are prime using the precomputed prime table.

• Sorted Pairs: If both are prime, add the pair [x, n - x] to the result list. The iteration ensures the list is sorted by the first element.

• Runtime Complexity: O(n log log n) for Sieve of Eratosthenes + O(n/2) for iteration and checking. Dominates to O(n log log n).

• Storage Complexity: O(n) for storing the prime table.

Code

    def prime_number_pairs(n: int) -> list[list[int]]:
    """
    Given an integer n. We say that two integers x and y form a prime number pair if:

    1 <= x <= y <= n
    x + y == n
    x and y are prime numbers

    Return the 2D sorted list of prime number pairs [xi, yi]. The list should be sorted in increasing order of xi.
    If there are no prime number pairs at all, return an empty array.

    Note:
    A prime number is a natural number greater than 1 with only two factors, itself and 1.

    For example:
    prime_number_pairs(10) == [[3, 7], [5, 5]]
    prime_number_pairs(2) == []
    """

    if n < 2:
        return []

    is_prime = [True] * (n + 1)
    is_prime[0] = is_prime[1] = False

    for i in range(2, int(n**0.5) + 1):
        if is_prime[i]:
            for j in range(i * i, n + 1, i):
                is_prime[j] = False

    result = []
    for x in range(2, n // 2 + 1):
        if is_prime[x] and is_prime[n - x]:
            result.append([x, n - x])

    return result