# Next smallest prime palindrome

Given a positive integer **N** where . The task is to find the smallest prime palindrome greater than or equal to N.

**Examples:**

Input: 8 Output: 11 Input: 7000000000 Output: 10000500001

**Approach:**

The **Naive **approach is to loop from **N + 1** until we found next **smallest prime palindrome** greater than or equal to **N**.

**Efficient Approach:**

Lets say **P = R** is a the next **smallest prime-palindrome** greater than or equal to **N**.

Now since **R **is a palindrome, the first half of the digits of **R **can be used to determine **R **up-to two possibilities. Let **k **be the first half of the digits in **R**. For eg. if** k = 123**, then **R = 12321** or **R = 123321**.

Thus we iterate through each **k **upto **10 ^{5}** and create the associated palindrome

**R**, and check whether

**R**is a

**prime**or not.

Also we will handle the odd and even palindromes separately, and break when we fount our result.

**Below is the implementation of above approach:**

## Python3

`# Python3 implementation of above approach ` `import` `math ` ` ` `# Function to check whether a number is prime ` `def` `is_prime(n): ` ` ` `return` `n > ` `1` `and` `all` `(n ` `%` `d ` `for` `d ` `in` `range` `(` `2` `, ` `int` `(math.sqrt(n)) ` `+` `1` `)) ` ` ` `# function to generate next smallest prime palindrome ` `def` `NextprimePalindrome(N): ` ` ` ` ` `for` `k ` `in` `range` `(` `1` `, ` `10` `*` `*` `6` `): ` ` ` ` ` `# Check for odd-length palindromes ` ` ` `s ` `=` `str` `(k) ` ` ` `x ` `=` `int` `(s ` `+` `s[` `-` `2` `::` `-` `1` `]) ` `# eg. s = '1234' to x = int('1234321') ` ` ` ` ` `if` `x >` `=` `N ` `and` `is_prime(x): ` ` ` `return` `x ` ` ` ` ` `# Check for even-length palindromes ` ` ` `s ` `=` `str` `(k) ` ` ` `x ` `=` `int` `(s ` `+` `s[` `-` `1` `::` `-` `1` `]) ` `# eg. s = '1234' to x = int('12344321') ` ` ` ` ` `if` `x >` `=` `N ` `and` `is_prime(x): ` ` ` `return` `x ` ` ` `# Driver code ` `N ` `=` `7000000000` ` ` `# Function call to print answer ` `print` `(NextprimePalindrome(N)) ` ` ` `# This code is written by ` `# Sanjit_Prasad ` |

*chevron_right*

*filter_none*

**Output:**

10000500001

Time Complexity: O(N*sqrt(N)) where N is upper limit and the sqrt(N) term comes from checking if a candidate is prime.

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