#866

Medium

LeetCode Problem Workspace

Prime Palindrome

The Prime Palindrome problem asks for the smallest prime palindrome greater than or equal to a given integer.

Problem Statement

Given an integer n, return the smallest prime palindrome greater than or equal to n. A prime number is a number greater than 1 with no divisors other than 1 and itself. A palindrome number is one that reads the same forwards and backwards. Your solution should efficiently compute the smallest prime palindrome greater than or equal to the input.

The problem requires identifying prime numbers and ensuring they are palindromes, which means checking both divisibility and symmetry. While the problem has constraints up to 10^8, you need to consider optimization techniques for prime checking and palindrome validation to achieve an efficient solution.

Examples

Example 1

Input: n = 6

Output: 7

Example details omitted.

Example 2

Input: n = 8

Output: 11

Example details omitted.

Example 3

Input: n = 13

Output: 101

Example details omitted.

Constraints

1 <= n <= 108

Solution Approach

Prime Checking

First, you'll need an efficient method to check if a number is prime. A common approach is to use trial division up to the square root of the number. This step ensures the number has no divisors other than 1 and itself.

Palindrome Validation

Next, check if the number is a palindrome by converting it to a string and comparing it to its reverse. This allows you to confirm if the number reads the same from both directions.

Increment and Search

Start from the input n and incrementally check each number for primality and palindrome properties. Once a valid prime palindrome is found, return that number as the result. Consider early stopping strategies if performance becomes a concern.

Complexity Analysis

Metric	Value
Time	Depends on the final approach
Space	Depends on the final approach

The time complexity depends on how efficiently you check for primality and palindrome properties. For prime checking, trial division up to the square root is typically O(sqrt(n)), while palindrome checking is O(d), where d is the number of digits. Given the constraints, the overall time complexity will scale depending on the efficiency of these operations.

What Interviewers Usually Probe

Look for the candidate’s understanding of prime number theory and palindrome detection.
Pay attention to optimization strategies for both checking and searching for prime palindromes.
Evaluate their approach to handling large inputs efficiently under the constraint of 10^8.

Common Pitfalls or Variants

Common pitfalls

Failing to optimize the prime checking process, leading to slow solutions.
Not validating palindrome properties correctly, resulting in incorrect outputs.
Using brute force without considering performance optimizations when searching for the smallest prime palindrome.

Follow-up variants

Finding the nth prime palindrome instead of just the smallest.
Limiting the solution to prime palindromes within a certain range.
Generalizing to find prime palindromes of even or odd length only.

FAQ

What is the approach for finding the smallest prime palindrome?

Start by checking if the number is prime and a palindrome, and incrementally search for the smallest valid number greater than or equal to the given n.

How do you efficiently check if a number is prime?

Use trial division up to the square root of the number, which is the most efficient method for small to moderate-size numbers.

What are some common mistakes in solving the Prime Palindrome problem?

Common mistakes include inefficient prime checking, failing to correctly validate palindromes, and not optimizing the search process for large values of n.

How does the GhostInterview tool help with prime palindrome problems?

It provides targeted problem-solving scenarios, guides you through optimization steps, and helps you understand key patterns like prime checking and palindrome validation.

What are the time and space complexities for this problem?

The time complexity depends on the efficiency of prime checking and palindrome validation, while space complexity is typically O(1) if only a constant amount of space is used for processing the number.

terminal

Solution

Solution 1

#### Python3

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class Solution:
    def primePalindrome(self, n: int) -> int:
        def is_prime(x):
            if x < 2:
                return False
            v = 2
            while v * v <= x:
                if x % v == 0:
                    return False
                v += 1
            return True

        def reverse(x):
            res = 0
            while x:
                res = res * 10 + x % 10
                x //= 10
            return res

        while 1:
            if reverse(n) == n and is_prime(n):
                return n
            if 10**7 < n < 10**8:
                n = 10**8
            n += 1

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