brute_force.py

import random
import itertools

# List of primes for which we wish to find imaginary sum representations
sqrtList = [2, 3, 5]

# Number of terms in the imaginary sum
total_terms = 3


# Helper function that ingests the numbers representing a term in the sum
# and outputs the corresponding, formatted string
def to_string(
	coefNumerator, coefDenominator, powerNumerator, powerDenominator, plusMinus
):
	string = (
		f"({coefNumerator}/{coefDenominator})i^({powerNumerator}/{powerDenominator})"
	)

	if plusMinus == 1:
		string = "-" + string

	return string


# Generate (representations for) all terms/powers of i for which we wish to use
# in the sums
terms = set()
for coefNumerator in range(0, 7):
	for coefDenominator in range(1, 7):
		for powerNumerator in range(0, 7):
			for powerDenominator in range(1, 7):
				termNotFinalized = (coefNumerator / coefDenominator) * pow(
					1j, powerNumerator / powerDenominator
				)
				for plusMinus in range(2):
					if plusMinus == 0:
						terms.add(
							(
								termNotFinalized,
								(
									coefNumerator,
									coefDenominator,
									powerNumerator,
									powerDenominator,
									plusMinus,
								),
							)
						)
					else:
						terms.add(
							(
								-termNotFinalized,
								(
									coefNumerator,
									coefDenominator,
									powerNumerator,
									powerDenominator,
									plusMinus,
								),
							)
						)

# Generate all combinations based on the above terms
combos = itertools.combinations_with_replacement(terms, total_terms)

"""
Loop through all combinations and evaluate them. If they
are sufficiently close (within rounding error) to the
square root of a prime at which we are looking, we have
found a valid imaginary representation of our prime!
"""
for subset in combos:
	# Add up all the terms in our combination
	total = 0
	for term in subset:
		total += term[0]

	# For each prime we want to examine, check the imaginary representation
	for num in sqrtList:
		# Check the imaginary representation
		if abs(total.imag) < 1e-10 and abs(pow(num, 1 / 2) - total.real) < 1e-10:
			print(f"Found for the square root of {num}.")

			# Format the string representation of the imaginary representation
			term_string = ""
			for i, term in enumerate(subset):
				term_string += to_string(*term[1])
				if i < len(subset) - 1:
					term_string += " + "

			# Output the string
			print(term_string)
			print()