Finding any digit (base 10) of a binary number $2^n$

Question

I was doing some Math / CS work, and noticed a pattern in the last few digits of $2^n$.

I was working in Python, in case anyone is wondering.

The last digit is always one of 2, 4, 8, 6; and has a period of 4:

n = 1, str(2 ^ 1)[-1] = 2
n = 2, str(2 ^ 2)[-1] = 4
n = 3, str(2 ^ 3)[-1] = 8
n = 4, str(2 ^ 4][-1] = 6

I was interested, so I looked further. The second to last digit also has a repeating pattern, this time with a period of 20:

n =  2, str(2 ^  2)[-2] =  0
n =  3, str(2 ^  3)[-2] =  0
n =  4, str(2 ^  4)[-2] =  1
n =  5, str(2 ^  5)[-2] =  3
n =  6, str(2 ^  6)[-2] =  6
n =  7, str(2 ^  7)[-2] =  2
n =  8, str(2 ^  8)[-2] =  5
n =  9, str(2 ^  9)[-2] =  1
n = 10, str(2 ^ 10)[-2] =  2
n = 11, str(2 ^ 11)[-2] =  4
n = 12, str(2 ^ 12)[-2] =  9
n = 13, str(2 ^ 13)[-2] =  9
n = 14, str(2 ^ 14)[-2] =  8
n = 15, str(2 ^ 15)[-2] =  6
n = 16, str(2 ^ 16)[-2] =  3
n = 17, str(2 ^ 17)[-2] =  7
n = 18, str(2 ^ 18)[-2] =  4
n = 19, str(2 ^ 19)[-2] =  8
n = 20, str(2 ^ 20)[-2] =  7
n = 21, str(2 ^ 21)[-2] =  5

Is there a way I can generalize this so I can find any digit of any $2^n$ with out having to actually calculate the value of $2^n$?

If not, is there a way to find the period of the pattern, given an index (from the back of the integer) to look at?

@MatthewD flag it for a diamond mod, you can't do it yourself. — Kevin, Dec 12 '11 at 21:24
Google found this: http://www.exploringbinary.com/cycle-length-of-powers-of-two-mod-powers-of-ten/ — lhf, Dec 12 '11 at 21:33
@lhf Thank you!!! (I'll be making a trip to the math department tomorrow for an explanation though...) — Matthew Denaburg, Dec 12 '11 at 21:45

score 4 · Answer 1 · edited Apr 13 '17 at 12:20

4

The reason for the observer behavior is periodicity of $n \mapsto 2^n \bmod 10$, or $n \mapsto 2^n \bmod 10^k$ for any fixed $k$. This wikipedia article, as well as this, among others, math.SE answer are relevant.

Specifically, for a fixed positive integer $k$, $m \mapsto 2^m \bmod 10^k$ is quasi-periodic with period $p = \phi(5^k) = 4 \cdot 5^{k-1}$, i.e. for all $m \geq k$, $2^m \equiv 2^{m+p} \mod 10^k$.

Thus the last digit has period $4$, the one before last, has period $20$, the third one from the end has period $100$ and so on.

edited Apr 13 '17 at 12:20

Community

1

answered Dec 12 '11 at 22:56

Sasha

70,631

Let me see if I understand the first paragraph: when you said that $n \mapsto 2^n \bmod 10^k$, k is the index of the digit from the end of $2^n$? And you lost me completely in the second paragraph. I haven't had any number theory yet. – Matthew Denaburg Dec 13 '11 at 01:12

Finding any digit (base 10) of a binary number $2^n$

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