I read this article titled Extending Given Digits to make Primes or Perfect Powers by Sury B, which appeared in the Resonance periodical (October 2010, Indian Statistical Institute, Bangalore). In this question, we are concerned with constructing Perfect Powers by extending digits of a given number. Refer to the linked article for the procedure to follow to extend digits of a given number to create another number that is a perfect power. The main result is that every integer can be extended with more digits to construct a perfect power.
The big idea is to compress data using this technique and the outline is as follows:
The sequence of bytes that we want to compress can be treated as a large integer. Say we have such data encoded as a large integer $n$. We extend the digits of $n$ such that,
$10^kn + c = a^m$ for some $a, k, m, c \in Z$, where $c \equiv a^m \mod 10^k$.
Note that we are only interested in $n$. It can be recovered through integer division of $a^m$ by $10^k$ since $n = \lfloor a^m / 10^k \rfloor$
In order for this to be an effective data compression method, the number of bits to represent $k, a, m$ should be less than the number of bits needed to represent $n$. In order for this scheme to be effective in representing large integers up to a bound $b$, the average bits required should approach the theoretical Shannon Limit.
- Can we show that this scheme is effective (or prove that it isn't)?
This MSE article is for perfect squares with specified starting digits. A square number can be represented with only half the bits (i.e., we can send $x$ instead of $x^2$)
This MSE question (posed by me) is for perfect power values of polynomials that uses a slightly different technique to get a perfect power value