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Sep 26, 2006, 7:50:37 PM9/26/06

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Stuck on a little problem...

Prove that log(2)-base10 is irrational, and conclude that there exist infinitely many positive integers k for which the first 11 digits of the base 10 expression of 2^k (counting digits from the left) are 77777777777.

How do you show this is true?

Sep 26, 2006, 8:23:01 PM9/26/06

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For the first part:

Suppose that log_10 2 = a/b, where a and b are integers. Then we have:

10^a = 2^b

But this can't be, because the left side is a multiple of 5 while the right

side is not.

Now for the second part:

Since log_10 2 is irrational, given any two numbers x, y in the interval

(0,1) with x<y, there exist infinitely many k for which the decimal part

of k log_10 2 lies between x and y. So take x to be the deimal part of

log_10 77777777777, and take y to be the decimal part of

log_10 77777777778. Any k for which k log_10 2 lies between x and y

will have 77777777777 as its first eleven digits.

--

Daniel Mayost

In article <4008838.11593146676...@nitrogen.mathforum.org>,

Sep 26, 2006, 9:34:08 PM9/26/06

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Cezanne123 wrote:

> Prove that log(2)-base10 is irrational, and conclude that

> there exist infinitely many positive integers k for which

> the first 11 digits of the base 10 expression of 2^k

> (counting digits from the left) are 77777777777.

Daniel Mayost has already posted a proof, so I thought

I'd point out some interesting extensions that might

not be well known (at least, aside from experts in

Diophantine approximation). Macon/Moser (full reference

below) give fairly elementary proofs for the following

results:

1. Every finite sequence of decimal digits appears as

the first digits of some power of 2.

2. Every finite sequence of decimal digits appears as

the first digits of infinitely many powers of 2.

3. Let k and n be positive integers. Let N(k,n) be the

number of powers of 2 that are less than 2^n and whose

first digits are identical to the digits of k. Then

LIM(n --> oo) N(k,n)/n is equal to (log_10)[(k+1)/k].

Since (log_10)[(k+1)/k] = ln(1 + 1/k) / ln(10), it follows

that this limit is approximately equal to (0.4343)/k.

4. Let m be a positive integer that is not a power of 10.

Then each of #1, 2, 3 holds for powers of m in place

of powers of 2.

Note that the limit for the corresponding version of #3

is independent of m.

They also formulate some simultaneous first digit results

for powers of a fixed finite set of positive integers,

such as if k and k' are given positive integers, then there

are infinitely many values of n (in fact, a positive relative

frequency of such n's, analogous to #3 above) such that the

first digits of 2^n is k AND the first digits of 3^n is k'.

Nathaniel Macon and Leo Moser, "On the distribution of first

digits of powers", Scripta Mathematica 16 (1950), 290-292.

Dave L. Renfro

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