Question

What is the largest value you can represent using a 256-bit unsigned integer

Answer 1

The largest 256-bit value have all 256 bits set to one (1):

255 0
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 1 1 1 1 1

Expressing a binary value (base 2) as decimal (base 10) is done by adding the nth bit multiplied by nth power of two, for every bit, starting from zero.

255 0
1*2^255 + 1*2^254 + 1*2^253 + ... + 1*2^1 + 1*2^0

But that's quite a complex expression, is there a easier way to to this?
If we look closly at the at the beginning of the summation
1*2^0 = 1 = 2^1 - 1
1*2^0 + 1*2^1 = 1 + 2 = 3 = 2^2 - 1
1*2^0 + 1*2^1 + 1*2^3 = 1 + 2 + 4 = 7 = 2^3 - 1

This means the entire summation can be simplified to 2^256 - 1.

Answer 2

Which is a really big number.

Answer 3

(2 to the 256 power) - 1