Will Two Power?

It’s such a simple thing: repeatedly doubling a number: 1, 2, 4, 8, 16, 32, 61, 128… And yet it yields such riches, reminiscent of DNA or a literary text:

2^0 = 1
2^1 = 2
2^2 = 4
2^3 = 8
2^4 = 16
2^5 = 32
2^6 = 64
2^7 = 128
2^8 = 256
2^9 = 512
2^10 = 1024
2^20 = 1048576
2^30 = 1073741824
2^40 = 1099511627776
2^50 = 1125899906842624
2^60 = 1152921504606846976
2^70 = 1180591620717411303424
2^80 = 1208925819614629174706176
2^90 = 1237940039285380274899124224
2^100 = 1267650600228229401496703205376
2^200 = 1606938044258990275541962092341162602522202993782792835301376

Although, by Benford’s law*, 1 is the commonest leading digit, do all numbers eventually appear as the leading digits of some power of 2? I conjecture that they do. indeed, I conjecture that they do infinitely often. If the function first(n) returns the power of 2 whose leading digits are the same as the digits of n, then:

first(1) = 2^0 = 1
first(2) = 2^1 = 2
first(3) = 2^5 = 32
first(4) = 2^2 = 4
first(5) = 2^9 = 512
first(6) = 2^6 = 64
first(7) = 2^46 = 70368744177664
first(8) = 2^3 = 8
first(9) = 2^53 = 9007199254740992
first(10) = 2^10 = 1024

And I conjecture that this is true of all bases except bases that are powers of 2, like 2, 4, 8, 16 and so on. A related question is whether the leading digits of any 2^n are the same as the digits of n. Yes:

2^6 = 64
2^10 = 1024
2^1542 = 1.54259995… * 10^464
2^77075 = 7.70754024… * 10^23201
2^113939 = 1.13939932… * 10^34299
2^1122772 = 1.12277217… * 10^337988

That looks like a look of calculation, but there’s a simple way to cut it down: restrict the leading digits. Eventually they will lose accuracy, because the missing digits are generating carries. With four leading digits, this happens:

1: 0001
2: 0002
4: 0004
8: 0008
16: 0016
32: 0032
64: 0064
128: 0128
256: 0256
512: 0512
1024: 1024
2048: 2048
4096: 4096
8192: 8192
16384: 1638…
32768: 3276…
65536: 6552…

But working with only fifteen leading digits, you can find that 1122772 = the leading digits of 2^1122772, which has 337989 digits when calculated in full.


Previously pre-posted (please peruse):

Talcum Power


*Not Zipf’s law, as I originally said.

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