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Say absolute values of $a,b,c$ is $O(log^{k}{n})$ for some positive constant $k$. Given positive integer $n$ that is reasonably large, we cannot always find integers $a,b,c$ such that $|a{b^{c}} - n|$ is very close to $n$ say within $O(log(n))$ since there are only $log^{O(k)}{n}$ such $a,b,c$ combinations(answer from Petrov).

What is the lower bound on $n$ as a function of $k$ upto which I can find such $a,b,c$?

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Size here is a confusing term. Do you mean the absolute values of a and of b are both less than some fixed power of log n? Unless something bizarre is happening, you will need fewer than log n values for c, and if b is small, then probably fewer than k values for c. Gerhard "Ask Me About System Design" Paseman, 2011.08.11 – Gerhard Paseman Aug 11 '11 at 22:29
you can not do it for all $n$, since if,2,\dots,n we consider all possible $n\in\\{1,2,\dots,N\\}$, then we have $O(\log^{3k}N)$ triples $(a,b,c)$ and only $O(\log N)$ suitable $n$ for each of them. – Fedor Petrov Aug 11 '11 at 22:31
I get $O(log^{2k+2}N)$ triples, but I take your excellent point, Fedor. Gerhard "Ask Me About System Design" Paseman, 2011.08.11 – Gerhard Paseman Aug 11 '11 at 23:01
How about for the second question? @Gerhard: yes size here is absolute values. – user16007 Aug 12 '11 at 1:47
For $a=0$, we have $|ab^c - n| = n $, which is certainly "very close to $n$". – js21 Sep 5 '12 at 20:25

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