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It is known (Senge and Straus, 1971, see also C.L.Stewart, 1980) that for every natural $a$, not a power of 10, and every natural $s$, there are only finitely many $k$ such that the sum of decimal digits of $a^k$ does not exceed $s$. So let $f(s)$ be the largest $k$ with this property. What is the growth rate of $f$? In particular, is it always at most linear?

Update 1: Conjecture. $\liminf S_{10}(a^k)/k > \log_2(a)$. This would imply that $f$ is bounded by a linear function. Here $S_{10}(u)$ is the sum of decimal digits of $u$.

Update 2: As far as I know the best proved estimate of $f$ is double exponential (C.L. Stewart).

Update 3: Since somebody erased my comments below, I add it here. The problem is related to the exponential Diophantine equation $10^{k_1}+...+10^{k_s}=a^k$. See the book Shorey, T. N.; Tijdeman, R. Exponential Diophantine equations. Cambridge Tracts in Mathematics, 87. Cambridge University Press, Cambridge, 1986.

Update 4: Possibly a better conjecture than in Update 1. The distribution of digits in $a^k$ should be very close to uniform. Hence $\liminf S_{10}(a^k)/k\approx 4.5\log_{10} a$.

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@Mark: Just for my own education, could you identify in which Shorey-Tijdeman paper this is established? They have written quite a few papers together... –  Joseph O'Rourke Sep 16 '10 at 16:05
    
By the way, for $a=2$ or $5$, there is an old result of A. Schinzel, which is included in SierpiƄski's "250 problems in elementary number theory". If I remember the proof correctly, this result shows that in this case $f$ grows at most linearly. –  Mark Sapir Sep 16 '10 at 16:52
    
@Joseph: I changed the references. –  Mark Sapir Sep 18 '10 at 16:34
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The references in full: C L Stewart, On the representation of an integer in two different bases, J Reine Angew Math 319 (1980) 63-72, MR 81j:10012; H G Senge, E G Straus, PV-numbers and sets of multiplicity, Proceedings of the Washington State University Conference on Number Theory (1971) 55-67, MR 47 #8452. –  Gerry Myerson Sep 20 '10 at 1:07
    
@Gerry: Yes, it is correct according to MathSci. Also you may find in Stewart's paper a double exponential estimate for $f$. I have not been able to find better estimates in the literature although there exist much more recent papers on the subject generalizing Stewart's results, in particular. Calculations suggest a linear bound. –  Mark Sapir Sep 20 '10 at 2:28
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