Prime Palindrome - Video Solutions

Problem Overview: Given an integer n, return the smallest number greater than or equal to n that is both a palindrome and a prime. A palindrome reads the same forward and backward, while a prime number has exactly two divisors: 1 and itself.

Approach 1: Brute Force Search (Time: O(k * sqrt(n)), Space: O(1))

Start at n and iterate upward one number at a time. For every number, first check if it is a palindrome by converting it to a string and comparing it with its reverse, or by using two pointers from both ends. If the number is a palindrome, run a primality test by checking divisibility from 2 to sqrt(x). The first number that satisfies both conditions is the answer.

This method is straightforward and mirrors the problem definition. However, most integers are neither palindromes nor primes, so the algorithm wastes time checking many candidates. As n grows, the density of primes decreases and brute-force iteration becomes slow.

Approach 2: Optimized Palindrome Generation + Prime Check (Time: ~O(P * sqrt(P)), Space: O(1))

Instead of checking every number, generate only palindrome candidates. Construct palindromes by building the first half of the number and mirroring it to form the second half. This drastically reduces the search space because you skip all non-palindrome integers entirely.

A key observation from math and number theory: every even-length palindrome greater than 11 is divisible by 11, so it cannot be prime. The optimized approach therefore generates only odd-length palindromes (except the special case 11). For each generated palindrome, run a standard primality test using trial division up to sqrt(x). As soon as you find one that is both ≥ n and prime, return it.

This approach reduces the candidate space by orders of magnitude. Instead of scanning every integer, you jump directly between palindrome numbers and only perform primality checks on those candidates.

Recommended for interviews: The optimized palindrome-generation approach is what interviewers typically expect. Brute force demonstrates that you understand palindrome detection and primality testing, but generating palindromes directly shows stronger problem-solving ability and knowledge of number properties.