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**Andre** Assume the following assertions:

$\displaystyle a$, $\displaystyle b$ and $\displaystyle c$ are positive integers.

$\displaystyle b > 1$

$\displaystyle b^{\lceil\log_b(2^a)\rceil} \geq 2^a$

$\displaystyle 8c \leq a$

Given $\displaystyle a$ and $\displaystyle b$, is there a way to test that $\displaystyle \lceil\log_b(2^{8c})\rceil < \lceil\log_b(2^{8(c+1)})\rceil$ is true for all possible values of $\displaystyle c$, without trying each one?

Example 1:

$\displaystyle a = 32$

$\displaystyle b = 85$

The set of possible values for $\displaystyle c$ is $\displaystyle \{1, 2, 3, 4\}$. $\displaystyle 5$ is not valid because $\displaystyle 8{\times}5 > 32$. The condition that $\displaystyle \lceil\log_b(2^{8c})\rceil < \lceil\log_b(2^{8(c+1)})\rceil$ is satisfied by all possible values of $\displaystyle c$.

Example 2:

$\displaystyle a = 984$

$\displaystyle b = 268$

The set of possible values for $\displaystyle c$ is $\displaystyle \{1, 2, 3, ..., 123\}$. This example does not pass the test; it fails when $\displaystyle c = 122$ because:

$\displaystyle \lceil\log_{268}(2^{8{\times}122})\rceil = \lceil\log_{268}(2^{8(122+1)})\rceil$

Any help will be greatly appreciated. Thanks!

Andre

P.S. If this is not in the most appropriate thread, please direct me.