Simple addition of 1 to a large number does not work? (Python 3.9)

Question

Note: I am not that experienced in Python, therefore my code may not be as good as it could/should be.

I am attempting to create a tool to facilitate calculating the algebraic factors of a certain form of number (see https://en.wikipedia.org/wiki/Aurifeuillean_factorization). This is mostly as a test/learning experience, however I have run into a problem when attempting to calculate the parameter "c", which is defined as 2^(2k+1)+1. The addition step does not work for me. I am simply getting the returned value as 2^129, instead of 2^129+1 as I am looking to get. Is this an issue with Python itself, or am I making some sort of mistake in this.

Code:

import math


def make_aurifeuille_factors(base, exponent):
    if base == 2 and exponent % 4 == 2:
        k = (exponent - 2) / 4
        c = int(1 + 2 ** (2*k + 1))
        d = int(2 ** (k + 1))
        L = c + d
        M = c - d

        return int(k), int(c), int(d), int(L), int(M)


def gcd(a, b):
    return int(math.gcd(a, b))


print(make_aurifeuille_factors(2, 258))

Answer 1

k = (exponent - 2) / 4 makes k a float, which means you potentially introduce numerical error in computations down the line. Use integer division to stay in int world from the start:

def make_aurifeuille_factors(base, exponent):
    if base == 2 and exponent % 4 == 2:
        k = (exponent - 2) // 4
        c = 1 + 2 ** (2*k + 1)
        d = 2 ** (k + 1)
        L = c + d
        M = c - d

        return k, c, d, L, M