How to convert a bitstream to a base20 number?

Question

Given is a bitstream (continuous string of bits too long to be processed at once) and the result should be a matching stream of base20 numbers.

The process is simple for a small number of bits:

Assuming most significant bit right:

110010011 = decimal 403 (1 * 1 + 1 * 2 + 1 * 16 + 1 * 128 + 1 * 256)
403 / 20 = 20 R 3
20 / 20 = 1 R 0
1 / 20 = 0 R 1
Result is [3, 0, 1] = 3 * 1 + 0 * 20 + 1 * 400

But what if the bits are too much to be converted to a decimal number in one step?

My approach was doing both processes in a loop: Convert the bits to decimal and converting the decimal down to base20 numbers. This process requires the multipliers (position values) to be lowered while walking through the bits, because otherwise, they'll quickly increase too much to be calculated probably. The 64th bit would have been multiplied by 2^64 and so on.

Answer 1

note: I understood the question that a bitstream is arriving of unknown length and during an unknown duration and a live conversion from base 2 to base 20 should be made.

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I do not believe this can be done in a single go. The problem is that base 20 and base 2 have no common ground and the rules of modular arithmetic do not allow to solve the problem cleanly.

(a+b) mod n = ( (a mod n) + (b mod n) ) mod n
(a*b) mod n = ( (a mod n) * (b mod n) ) mod n    
(a^m) mod n = ( (a mod n)^m ) mod n    

Now if you have a number A written in base p and q (p < q) as

A = Sum[a[i] p^i, i=0->n] = Sum[b[i] q^i, i=0->n]

Then we know that b[0] = A mod q. However, we do not know A and hence, the above tells us that

b[0] = A mod q = Sum[a[i] p^i, i=0->n] mod q
               = Sum[ (a[i] p^i) mod q, i=0->n] mod q
               = Sum[ ( (a[i] mod q) (p^i mod q) ) mod q, i=0->n] mod q

This implies that:

If you want to know the lowest digit b₀ of a number in base q, you need to have the knowledge of the full number.

This can only be simplified if q = p^m as

b[0] = A mod q = Sum[a[i] p^i, i=0->n] mod q
               = Sum[ (a[i] p^i) mod q, i=0->n] mod q
               = Sum[ a[i] p^i, i=0->m-1]

So in short, since q = 20 and p = 2. I have to say, no, it can not be done in a single pass. Furthermore, remind yourself that I only spoke about the first digit in base q and not yet the ith digit.

As an example, imagine a bit stream of 1000 times 0 followed by a single 1. This resembles the number 2¹⁰⁰⁰. The first digit is easy, but to get any other digit ... you are essentially in a rather tough spot.