Problem 255

List $P$ contains all positive
integers from $1$ to $2^{53}$ inclusive, and so $P$ contains each integer of the form $2^k$ where $k$ is a positive integer from $1$ to $53$ inclusive.

Thus, there are $53$ such numbers in
the list $P$.
Since $4^\ell=(2^2)^\ell=2^{2\ell}$, then $4^\ell$ is in list $P$ exactly when $2^{2\ell}\leq2^{53}$ or $2\ell\leq53$ for positive integers $\ell$. The largest positive integer $\ell$ for which $2\ell\leq53$ is $26$ (since $2\times26=52$, and $2\times27=54$).

Thus, for all positive integers $\$\ell \leq$
26$,$$4^\ell\$$ is in list
$P$, and so there are $26$ such numbers.

We demonstrate this relationship between the powers of $2$ and $4$ in the table below.

$2^k$
$2^1$
$2^2$
$2^3$
$2^4$
$2^5$
$2^6$
$\cdots$
$2^{51}$
$2^{52}$
$2^{53}$

$4^{\ell}=2^{2\ell}$

$4^1=2^2$

$4^2=2^4$

$4^3=2^6$
$\cdots$

$4^{26}=2^{52}$

From (b), list $P$ contains
the $26$ numbers $4^r=2^{2r}$ for positive integers $r\leq 26$.

That is, each integer power of $4$ in
$P$ is equal to a power of $2$ whose exponent is an even positive
integer.

Since $8^t=(2^3)^t=2^{3t}$, then each
integer power of $8$ in $P$ is equal to a power of $2$ whose exponent is a positive integer
multiple of $3$.

Thus, any number in $P$ that is an
integer power of both $4$ and $8$ must be equal to a power of $2$ whose positive integer exponent is both
even and a multiple of $3$, and
therefore a multiple of $6$.

$P$ contains all numbers $2^k$ for positive integers $k\leq 53$, and so $P$ contains the following powers of $2$ whose exponent is a multiple of $6$: $2^6$, $2^{12}$, $2^{18}$, $2^{24}$, $2^{30}$, $2^{36}$, $2^{42}$, and $2^{48}$.

For positive integers $r$ and $t$, there are $8$ numbers in $P$ that can be written as both $4^r$ and as $8^t$, and so there are $26-8=18$ numbers in $P$ which can be written as $4^r$ but cannot be written as $8^t$.

We demonstrate this relationship between the powers of $2$, $4$
and $8$ in the table below.

$4^r=2^{2r}$
$4^1$
$4^2$
$4^3=2^6$
$4^4$
$4^5$
$4^6=2^{12}$
$\cdots$
$4^{24}=2^{48}$
$4^{25}$
$4^{26}$

$8^t=2^{3t}$

$8^2=2^{6}$

$8^4=2^{12}$
$\cdots$
$8^{16}=2^{48}$

$4^r$
not $8^t$
$4^1$
$4^2$

$4^4$
$4^5$

$\cdots$

$4^{25}$
$4^{26}$