Motivation: A friend of mine told me that the number gained by interpreting the binary code of is quite often a prime. A slight copyright themed discussion later and after developing an algorithm to make any visual medium a prime:
While(notPrime):
mutate
The question arose if there is a wasteful lossless encoding of the form:
Number Data Encodinginformation
e.g.:
Encode "a" as
10110000101111
which is:
1 (number sought to make it prime)
01100001(ascii for "a")
0111(encoding of length of String)
1(always last number to ensure the number is odd)
Math Question:
given a number $n$ coprime to $b$ is there a prime of the form $$p=n+\sum_{i=k}^m a_ib^i$$ where $a_i\in\{0,\ldots,b-1\}$, $m$ is arbitrary and $k=\lceil log_b(n) \rceil$?
As Yves and Mastrem pointed out: One can find infinitly many of those primes because of Dirichlets Theorem one just has to realize that
$n+\sum_{i=k}^ma_ib^i = n + b^k \sum_{i=0}^m\tilde{a}_ib^i$
and that trivially $a$ is coprime to $b^k$ as it is coprime to $b$.