Project Euler Problem 51 Solution

Project Euler Problem 51 Statement

The prime $41$, can be written as the sum of six consecutive primes:

This is the longest sum of consecutive primes that adds to a prime below one-hundred.

The longest sum of consecutive primes below one-thousand that adds to a prime, contains $21$ terms, and is equal to $953$.

Which prime, below one-million, can be written as the sum of the most consecutive primes?

import Euler, itertools

def main():
    N, K, L = map(int, input().split())

    lower, upper = 10**(N-1), 10**N - 1
    is_prime = Euler.prime_sieve(upper)
    
    # Generate all N-digit primes
    primes = [p for p in range(lower, upper + 1) if is_prime[p]]
    
    for prime in primes:
        s = str(prime)
        unique_digits = set(s)
        
        for digit in unique_digits:
            # Find all positions of the current digit
            positions = [i for i, ch in enumerate(s) if ch == digit]
            
            # Only proceed if there are at least K positions to replace
            if len(positions) >= K:
                # Sort positions in reverse to prioritize rightmost digits
                sorted_positions = sorted(positions, reverse=True)
                
                # Generate all combinations of K positions to replace
                for selected in itertools.combinations(sorted_positions, K):
                    family = []
                    
                    for replacement_digit in '0123456789':
                        temp = list(s)
                        
                        # Replace the selected positions with the replacement digit
                        for pos in selected:
                            temp[pos] = replacement_digit
                        
                        # Skip numbers with leading zeros
                        if temp[0] == '0':
                            continue
                        new_num = int(''.join(temp))  # Convert back to integer
                        
                        # Check if the new number is prime
                        if new_num >= lower and is_prime[new_num]:
                            family.append(new_num)
                    
                    # Check if the family size meets or exceeds the required size
                    if len(family) >= L:
                        print(*sorted(family)[:L])  # Print the first L primes in the family in sorted order
                        return
main()	

HackerRank version

Solves all test cases for Hackerrank Project Euler Problem 51

HackerRank Project Euler 51: Find the smallest $N-digit$ ($2≤ N ≤ 7$) prime which, by replacing $k-digits$ ($1≤ K ≤ N$) of the number (not necessarily adjacent digits) with the same digit, is part of an $L$ ($1≤ L ≤ 8$) prime value family.

Solution

This was a journey! This Python script is designed to solve Project Euler's Problem 51, which involves finding prime numbers that form a specific "family" when certain digits are replaced. This problem requires identifying N-digit primes where replacing exactly K digits results in at least L prime numbers within that family. The program has internal documentation that augments this discussion. What follows is a breakdown of how the script accomplishes this task:

Generating and Iterating Through Primes

With the sieve ready, the script filters out all N-digit prime numbers and stores them in the primes list. It then iterates through each prime number in this list. For each prime, it converts the number to a string to facilitate digit manipulation and identifies all unique digits present in the number.

Identifying and Replacing Digits

For each unique digit in the prime number, the script locates all positions (indices) where this digit appears. If the number of such positions is at least K, it proceeds to generate all possible combinations of these positions taken K at a time using itertools.combinations. This ensures that the script considers every possible way to replace exactly K digits in the prime number.

Generating and Validating the Prime Family

For each combination of positions to replace, the script attempts to create a "family" of numbers by substituting the selected digits with every possible digit from '0' to '9'. It carefully avoids generating numbers with leading zeros, which would result in numbers with fewer than N digits. Each newly formed number is then checked against the prime sieve to determine if it is prime. Valid primes are added to the family list.

Checking for the Desired Family Size

After generating all possible replacements for a given combination of digits, the script checks if the size of the family meets or exceeds the required size L. If such a family is found, it prints the first L primes in the family in sorted order and terminates the program.

Optimization and Efficiency

The script optimizes the search by prioritizing replacements of digits that appear multiple times and by processing combinations in reverse order (rightmost digits first). This strategy increases the likelihood of quickly finding a valid prime family without exhaustively searching all possibilities. Additionally, using a prime sieve allows for rapid prime validation, which is crucial given the potentially large number of candidate numbers generated through digit replacements.

In summary, the script methodically generates N-digit prime numbers, explores all viable digit replacement combinations, and efficiently identifies a prime family that satisfies the specified conditions of replacing K digits to achieve at least L prime numbers.

Last Word

• The prime_sieve function will be imported instead of hard coded into the source code. This provides flexibility to include different sieves as required to solve a problem.