Largest Left-Truncatable Prime in a given base
Given an integer N representing the base of a number, the task is to find the largest left-truncatable prime in the given base N.
Input: N = 3
Left-truncatable prime in base N(= 3) are given below:
(12)3 = (5)10
(212)3 = (23)10
Therefore, the largest left-truncatable prime in base N(= 3) is (23)10.
Input: N = 5
Approach: The idea is to generate all left-truncatable prime numbers in the given base N and print the largest left-truncatable prime number based on the following observations:
If a number containing (i) digits is a left-truncatable prime number, then the numbers formed from the last (i – 1) digits must be a left-truncatable prime number.
Therefore, to make a left-truncatable prime number of digits i, first find all the left-truncatable prime numbers of (i – 1) digits.
- Initialize an array, say candidates, to store the all possible left truncatable prime numbers in the given base N.
- Iterate over the range [0, infinity] using variable i and insert all the left truncatable prime numbers of digits i. If no left truncatable number consisting of i digits is found, then return from the loop.
- Finally, print the maximum element present in the array candidates.
Below is the implementation of the above approach:
Time Complexity: O(k * log3N), where k is the rounds performed in Miller-Rabin primality test
Auxiliary Space: O(X), where X is the total count of left-truncatable prime in base N