**π‘ Problem Formulation:** Python developers often encounter the challenge of verifying whether a product of prime numbers from an array is a perfect square. A perfect square is a number that is the square of an integer. For example, given an array of prime numbers `[3, 7, 11]`

, the product is `231`

. The goal is to determine if `231`

is a perfect square (`False`

in this case).

## Method 1: Prime Factorization and Group Checking

This method involves prime factorizing each number in the array and ensuring all prime factors occur in pairs. For a number to be a perfect square, all the exponents in its prime factorization must be even. The prime factorization can be done by dividing the number by the smallest prime numbers (2, 3, 5, etc.) until only 1 remains.

Here’s an example:

from collections import Counter def is_perfect_square(primes): def prime_factors(n): i = 2 factors = [] while i * i 1: factors.append(n) return factors prod = 1 for prime in primes: prod *= prime factors = prime_factors(prod) counter = Counter(factors) return all(exp % 2 == 0 for exp in counter.values()) print(is_perfect_square([3, 7, 11]))

Output:

False

This code snippet combines all the prime numbers in the array to compute the product and then breaks down the product into prime factors. By using a `Counter`

from the collections module, it counts the occurrences of each prime factor checking if all of them appear an even number of times, thus confirming if the product is a perfect square.

## Method 2: Using a Mathematical Property

This method makes use of the mathematical property that the square root of a perfect square is an integer. By computing the square root of the product and checking if it’s an integer (its decimal part is zero), one can ascertain if one has a perfect square.

Here’s an example:

import math def is_perfect_square(primes): product = math.prod(primes) return math.isqrt(product) ** 2 == product print(is_perfect_square([3, 7, 11]))

Output:

False

The code multiplies all prime numbers in the array to find the product and then gets the integer square root of the product using the `math.isqrt`

method. Afterward, it squares the integer square root and compares it to the initial product to check if the product is a perfect square.

## Method 3: Incremental Multiplication and Checking

This approach iteratively multiplies each prime number in the array, and immediately after each multiplication, it checks if the current product is a perfect square. The benefit is early termination if a perfect square is found without needing to compute the full product.

Here’s an example:

import math def is_perfect_square(primes): product = 1 for prime in primes: product *= prime if math.isqrt(product) ** 2 != product: return False return True print(is_perfect_square([3, 7, 11]))

Output:

False

In this snippet, the product is calculated by multiplying one prime number at a time, and after each multiplication, it checks whether the current product is a perfect square using the same condition as the previous method. This can potentially save time by stopping early when the product is not a perfect square.

## Method 4: Optimization Using Exponentiation

By recognizing that the product of an array of the same prime numbers raised to even exponents is a perfect square, you can avoid unnecessary calculations. The idea is to pair the primes, raise them to the second power, and then multiply.

Here’s an example:

def is_perfect_square(primes): if len(primes) % 2 != 0: return False product = 1 for i in range(0, len(primes), 2): product *= primes[i] ** 2 return True print(is_perfect_square([3, 3, 5, 5]))

Output:

True

This code assumes the input list is sorted and only contains the same prime number twice. It then calculates the perfect square by raising each unique prime to the power of 2, which assumes an even number of prime occurrences.

## Bonus One-Liner Method 5: Using Python One-Liner

This method provides a quick one-liner using functional programming constructs – `map()`

and `reduce()`

– to multiply all elements and check if the product is a perfect square.

Here’s an example:

from functools import reduce from math import isqrt is_perfect_square = lambda primes: isqrt(reduce(lambda x, y: x*y, primes)) ** 2 == reduce(lambda x, y: x*y, primes) print(is_perfect_square([3, 7, 11]))

Output:

False

This one-liner defines a lambda function that multiplies all the prime numbers in the array to find the product and then checks if the square of the integer square root equals the product.

## Summary/Discussion

**Method 1: Prime Factorization and Group Checking.**This method is thorough and works for any prime array, but can be slower due to the factorization process.**Method 2: Using a Mathematical Property.**Reliable and reasonably efficient for small to medium-sized arrays, but may still involve needless multiplication.**Method 3: Incremental Multiplication and Checking.**Provides early exit capabilities, potentially saving time, but still multiplies each time.**Method 4: Optimization Using Exponentiation.**Fast and efficient, provided that the input list is appropriately structured.**Bonus Method 5: Using Python One-Liner.**Short and clean, ideal for smaller arrays, but less readable and not necessarily performant for larger datasets.