Figuring out base 4

4^n = m
n = 4    3   2   1   0
m = 256  64  16  4   1

numbers: 15,39,63,157,255,1023

After working through breaking down 15, I figured it out. Keep subtracting 4^n from the number N until N < 4^n. Every time you subtract, add 1 to place value <n>, when N < 4^n, subtract 1 from n, and continue.

Edit: I wrote an implementation in python: http://pastebin.com/fp5ApbeS

15
15 < 256
0
15 < 64
00
15 < 16
000
15 >= 4
15-4 = 11
0001
11-4 = 7
0002
7-4 = 3
0003
3 < 4
3 >= 1
3-1 = 2
00031
2-1 = 1
00032
1-1 = 0
>>> 00033

Or, 15 = 4^2 (100 base 4) - 1 (=33 base 4) => 00033 (which is how I knew 15 in base 4 in the first place)

39
00
39-16 = 23
001
23-16 = 7
002
7-4 = 3
0021
3-1 = 2
00211
2-1 = 1
00212
1-1 = 0
00213

63
63 = (4^3)-1 = 00333

157
0
157-64 = 93
01
93-64 = 29
02
29-16 = 13
021
13-4 = 9
0211
9-4 = 5
0212
5-4 = 1
0213
1-1 = 0
02131

255
255 = (4^4)-1 = 03333

1023
1023 = (4^5)-1 = 33333

15 = 00033
39 = 00213
63 = 00333
157 = 02131
255 = 03333
1023 = 33333

Oh, how interesting. 157 (2131) has a very similar base 4 output to 39 (213). This means that 157 should be equal to (39<<1) + 1, or (39 * 4) + 1. And (63 * 4) + 3 should equal 255. Give me a sec to test that...

>>> 39 * 4 + 1
157
>>> 63 * 4 + 3
255

Yep. I love it when models of reality have predictive capability, and those predictions are consistent with reality. The Scientific Method (in this case, Number Theory) wins again.