algoadvance

Given an integer n, find the smallest prime palindrome greater than or equal to n.

  1. A prime number is a number that is greater than 1 and has no divisors other than 1 and itself.
  2. A palindrome is a word, number, phrase, or other sequence of characters that reads the same forward and backward.

You need to write a function prime_palindrome(n: int) -> int that returns the smallest prime palindrome greater than or equal to n.

Clarifying Questions

  1. What is the range of the input integer n?
    • The problem doesn’t specify explicitly, but for LeetCode problems, the input is usually within the range 1 <= n <= 10^8.
  2. Is there any preference regarding the time complexity of the solution?
    • The aim is always to have a well-optimized solution. However, given that this involves prime checks and palindrome checks, a brute force solution might be acceptable initially, followed by optimizations if needed.

Strategy

To solve this problem, the following steps can be followed:

  1. Helper Function for Prime Check: Create a function to check if a number is prime.
  2. Helper Function for Palindrome Check: Create a function to check if a number is a palindrome.
  3. Find the Smallest Prime Palindrome: Starting from n, iterate the numbers upwards until a number is found that is both prime and a palindrome.

Step 1: Prime Check

We’ll create a function is_prime that uses trial-division to determine if a number is prime.

Step 2: Palindrome Check

We’ll create a function is_palindrome to check if a number reads the same forward and backward.

Using a loop starting from n, use both helper functions to find the smallest prime palindrome.

Optimization Consideration

  1. Skip Even Length Palindromes: All even-length palindromes are divisible by 11 except the smallest even-length palindromes like 11. This can help reduce checks.
  2. Handle Edge Conditions: Given that very large prime palindromes are rare, it is necessary to ensure efficiency especially for upper bound values.

Time Complexity

Code

def prime_palindrome(n: int) -> int:
    def is_prime(x: int) -> bool:
        if x < 2:
            return False
        for i in range(2, int(x**0.5) + 1):
            if x % i == 0:
                return False
        return True

    def is_palindrome(x: int) -> bool:
        return str(x) == str(x)[::-1]

    # Palindrome generation can be more optimal but let's begin simple and correct
    while True:
        if is_palindrome(n) and is_prime(n):
            return n
        n += 1

# Let's test the provided function
print(prime_palindrome(6))  # Expected output: 7
print(prime_palindrome(8))  # Expected output: 11
print(prime_palindrome(13)) # Expected output: 101

# Note: Further optimization like skipping even-length palindromes can be added.

This initial solution will work correctly needs testing for performance to ensure it runs efficiently for larger values of n. Further steps will involve optimizing palindrome generation and skipping certain non-palindromic ranges intelligently.

Try our interview co-pilot at AlgoAdvance.com