How can integers be converted into strings in arbitrary bases?

Conversion of Integers to Strings in Arbitrary Bases

This question seeks a method to convert integers into strings in any base. Unlike Python's int(str, base) function, which allows creation of integers from strings of a specified base, the desired approach is its inverse, whereby strings are constructed from integers.

A Simple Solution

Intuitively, people tended to focus on converting to small bases (e.g., less than the English alphabet's length). However, the question demands a universal solution applicable to any base from 2 to infinity.

A straightforward solution is presented below:

def numberToBase(n, b):
    if n == 0:
        return [0]
    digits = []
    while n:
        digits.append(int(n % b))
        n //= b
    return digits[::-1]

This function takes an integer n and a base b as inputs and returns a list of digits representing n in base b. It works by iteratively dividing n by b and collecting the remainders (i.e., digits) in a list. The list is then reversed to obtain the correct order of digits.

For example, to convert 67854**15 - 102 to base 577:

print(numberToBase(67854 ** 15 - 102, 577))

Output:

[4, 473, 131, 96, 431, 285, 524, 486, 28, 23, 16, 82, 292, 538, 149, 25, 41, 483, 100, 517, 131, 28, 0, 435, 197, 264, 455]

This result can be further converted into any other desired base.

Key Points

• The simplicity of the solution lies in the realization that a number in a specific base can be represented by a list of digits where each digit represents a power of the base.
• The function returns a list of digits to accommodate any base, even those larger than 36 (the alphabetic base in Python).
• Understanding the concept of numbers in different bases is crucial for grasping the working of this method.

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