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Topic: power of 2 starting with 7 (Read 491 times) |
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gkwal
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power of 2 starting with 7
« on: Jul 16th, 2007, 9:24pm » |
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What proportion of the powers of two begin with a "7"?
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towr
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Re: power of 2 starting with 7
« Reply #1 on: Jul 17th, 2007, 1:59am » |
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Guessing, none at all. But, ok, a guess isn't that interesting, even if it were correct.
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pex
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Re: power of 2 starting with 7
« Reply #2 on: Jul 17th, 2007, 2:28am » |
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on Jul 17th, 2007, 1:59am, towr wrote:| Guessing, none at all ... |
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246 = 70368744177664
256 = 72057594037927936
266 = 73786976294838206464
276 = 75557863725914323419136
286 = 77371252455336267181195264
296 = 79228162514264337593543950336
And I guess there will be more such runs. Wild guess: wouldn't it just even out to 1/10?
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towr
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Re: power of 2 starting with 7
« Reply #4 on: Jul 17th, 2007, 4:01am » |
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Using the fractional part of 10log N and the fact that 10log 2 is irrational; it seems relatively simply to show the answer is 10log 8- 10log 7.
And that it also holds when considering powers of other numbers for which the 10log is irrational.
That might have taken me quite a while to come to, if not for http://www.urticator.net/essay/4/487.html
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