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The first digit of
Submitted by Steven Fuqua, March 18 1997, and originally answered by Valerio De Angelis.
The sequence of powers of two
produces the sequence 
of first digits of powers of two (base ten). So,
,
,
,
,
and so on. Does the number seven ever occur in the sequence
?
Which occurs most
frequently, seven or eight? Determine explicitly the probability
that 1 occurs as a first digit and repeat for each of
2, 3, 4, ..., 9.
[Technically, the probability that the number five,
for instance, occurs in
is the
limit
].
Hint sequence by Valerio De Angelis
Hint 1
The solution of this problem will require the following result:
Theorem A
Let 
be an
irrational number. Then the sequence
fractional part of
is uniformly distributed
over the interval 
, in
the sense that for every interval
I,
.
Intuitively, this means that the sequence does not 'prefer' any
portion of , and the
proportion of times 
spends in a given interval is equal to the length of the interval.
Theorem A is a standard result, proved in many textbooks. See for example Hardy and Wright, "An Introduction to the Theory of Numbers", Theorem 445, pp. 390-391. (This reference provided by Allen Stenger).
Assuming Theorem A true, our problem can be solved without too much effort. On the other hand, a proof of Theorem A requires some work, but also provides a good opportunity for illustrating a number of important notions and techniques in Real Analysis (such as the Fourier series of continuous functions, uniform convergence, and approximation methods for piecewise continuous functions).
Click here for a proof of Theorem A.
Click here for more hints on how to solve the problem assuming Theorem A.