💡 Problem Formulation: We need to determine if the sum of all prime numbers within a given array is itself a prime number or not. Given an input array such as [2, 3, 5, 8, 13], the output for the sum of prime numbers (2 + 3 + 5 + 13 = 23) should be a confirmation whether 23 is prime (true or false).
Method 1: Iterative Prime Check
This method consists of two primary operations: identifying prime numbers within the array and subsequently checking if their sum is prime. We accomplish this by defining a function to check for primality and then iterating through the array to sum up the prime numbers. The primality of the resulting sum is then verified by the same primality function.
Here’s an example:
def is_prime(num):
if num <= 1:
return False
for i in range(2, int(num ** 0.5) + 1):
if num % i == 0:
return False
return True
def sum_of_primes(array):
total = sum(filter(is_prime, array))
return is_prime(total)
array = [2, 3, 5, 8, 13]
print(sum_of_primes(array))Output: True
The code snippet defines a function is_prime() to test the primality of a given number, and a function sum_of_primes() which filters the array using this primality check. It sums the prime numbers and checks if this sum is prime, returning a boolean result.
Method 2: List Comprehensions with Primality Function
This approach leverages Python’s list comprehensions to create an elegant solution. We still require a function to check for a prime number, but the process of finding the sum of primes is handled within a single line of code, and its primality is checked and returned.
Here’s an example:
def is_prime(num):
if num <= 1:
return False
for i in range(2, int(num ** 0.5) + 1):
if num % i == 0:
return False
return True
array = [2, 3, 5, 8, 13]
sum_of_primes = sum(x for x in array if is_prime(x))
print(is_prime(sum_of_primes))Output: True
Using a list comprehension, the code succinctly calculates the sum of the prime numbers in the array through a single-line expression. It then checks if this sum is prime, outputting the result.
Method 3: Sieve of Eratosthenes with Summation
Instead of checking each number for primality, we can generate a list of all primes up to the maximum number in the array using the Sieve of Eratosthenes. The prime numbers in the array are then summed, and the total is checked for primality.
Here’s an example:
def sieve(n):
primes = [True] * (n + 1)
p = 2
while p * p <= n:
if primes[p]:
for i in range(p * p, n + 1, p):
primes[i] = False
p += 1
return [p for p in range(2, n+1) if primes[p]]
def sum_of_primes(array, primes):
total = sum(x for x in array if x in primes)
return is_prime(total) # Using is_prime from previous examples
array = [2, 3, 5, 8, 13]
primes = sieve(max(array))
print(sum_of_primes(array, primes))Output: True
This code uses the Sieve of Eratosthenes in the sieve() function to precalculate prime numbers and then iterates through the array to sum the numbers that appear in the primes list. It finishes by checking the sum’s primality.
Method 4: Using NumPy for Array Operations
This method requires the NumPy library to handle array operations efficiently. We apply a mask to the array with our primality check to filter and sum prime elements. As with the other methods, we then check if the resulting sum is prime.
Here’s an example:
import numpy as np
def is_prime(num):
if num <= 1:
return False
for i in range(2, int(num ** 0.5) + 1):
if num % i == 0:
return False
return True
array = np.array([2, 3, 5, 8, 13])
prime_mask = np.vectorize(is_prime)(array)
sum_of_primes = array[prime_mask].sum()
print(is_prime(sum_of_primes))Output: True
The np.vectorize() function allows us to apply the is_prime() function to each element in the NumPy array. The boolean mask prime_mask is used to filter and sum the prime elements. As usual, we check if the sum is prime.
Bonus One-Liner Method 5: Functional Programming Approach
This one-liner method combines functional programming with the use of filter(), map(), and a lambda function to create a succinct solution for finding the sum of primes and checking its primality in a single expression.
Here’s an example:
array = [2, 3, 5, 8, 13] print((lambda x: all(x % i != 0 for i in range(2, int(x**0.5) + 1)))(sum(filter(lambda x: all(x % i != 0 for i in range(2, int(x**0.5) + 1)), array))))
Output: True
This one-liner uses a lambda function that checks if a number is prime and applies it both to filter the primes in the array and to check the primality of their sum. It’s a compact, but less readable solution.
Summary/Discussion
- Method 1: Iterative Prime Check – Thorough and transparent method. Strengths: Easy to understand and debug. Weaknesses: Can be slower for large arrays or large numbers due to repeated primality checks.
- Method 2: List Comprehensions – Clean and Pythonic approach. Strengths: Concise and utilizes Python’s expressive syntax. Weaknesses: Similar to Method 1, it can perform inefficiently for larger datasets.
- Method 3: Sieve of Eratosthenes – Optimizes the prime check by precalculating. Strengths: Efficient for multiple prime checks. Weaknesses: Overhead of calculating a list of primes up front; not optimal for small arrays or max numbers.
- Method 4: Using NumPy – Takes advantage of fast array operations. Strengths: Potentially faster for large arrays; leverages optimized C libraries. Weaknesses: Requires an external library; less transparent than pure Python.
- Method 5: Functional Programming – Ultra-compact method. Strengths: Clever use of functional programming constructs. Weaknesses: Readability may suffer; Python may not optimize such expressions effectively.