<s>[INST] Show that $\frac 1x+\frac 1y =(\frac 27)^a$ does not produce integer solutions for $a > 3$. Show that $\frac 1x+\frac 1y =(\frac 27)^a$ does not produce integer solutions for $a > 3$.
I have shown it is not possible for $a=4$, but not for any a greater than $4$.
 [/INST] Rewrite the equation into
$$7^a\cdot(x+y)=2^a\cdot xy$$
From this, we conclude that there is some positive integer $k$ with
\begin{align}
xy&=k\cdot 7^a\tag{1}\label{eq1}\\
x+y&=k\cdot 2^a\tag{2}\label{eq2}
\end{align}
Now, from equation $\eqref{eq1}$, we conclude that if $p\neq 7$ is prime  and $p|k$, then $p|x$ or $p|y$.
On the other hand, equation $\eqref{eq2}$ implies that if $p\neq2$ is prime and $p$ divides both $x$ and $k$, then $p|y$, and similarly exchanging the roles of $x$ and $y$.
Together, this means that any prime $p\neq 2,7$ in the factorization of $k$ also factors both $x$ and $y$. Say $p$ is one such prime and let $e_x$ be the exponent of $p$ in the factorization of $x$, and similarly for $e_y$. From equation $\eqref{eq1}$, $p$ has an exponent of $e_x+e_y$ in the factorization of $k$, but from equation $2$ that exponent is $\max\{e_x,e_y\}$.
It follows that no prime $p\neq 2,7$ is in the factorization of $k$, so $k,x$ and $y$ are all of the form $2^s7^t$. We may then rewrite this system. In the obvious notation, equation $\eqref{eq1}$ yields
\begin{align}
s_x+s_y&=s_k\\
t_x+t_y&=t_k+a
\end{align}
Equation $\eqref{eq2}$ then yields
$$2^{s_x}7^{t_x}+2^{s_y}7^{t_y}=2^{s_k+a}7^{t_k}$$
We may assume without loss of generality that $s_x\geq s_y$. Then:
$$2^{s_x-s_y}7^{t_x}+7^{t_y}=2^{s_k-s_y+a}7^{t_ k}=2^{s_x+a}7^{t_ k}$$
so if $s_x>s_y$, the RHS is odd while the LHS is odd. It follows that $s_x=s_y$ and the equation may be simplified to
$$7^{t_x}+7^{t_y}=2^{s_x+a}7^{t_k}$$
Now, we may assume without loss of generality that $t_x\geq t_y$. Then
$$7^{t_x-t_y}+1=2^{s_x+a}7^{t_k-t_y}=2^{s_x+a}7^{t_x-a}$$
We must then have $t_x=a$, for otherwise the RHS is either fractional, or a multiple of $7$ (while the LHS is not). This then implies $t_y=t_k$. Hence
$$7^{a-t_k}+1=2^{s_x+a}$$
If either $a$ were even or $s_x$ were positive and even, we could factor $2^{s_x+a}-1$ as
$$\Big({\left(2^{s_x}\right)}^{a/2}-1\Big)\Big({\left(2^{s_x}\right)}^{a/2}+1\Big)$$
Since these factors differ by $2$, at most one can be a multiple a $7$, and this would imply that the smallest is $1$, so that $\frac{as_x}2=1$. Since $a\geq 4$, this is impossible, and $a$ must be odd and $s_x$ must be either odd or $0$.
If $s_x$ were odd, then $a+s_x$ is even, and we may repeat the factorization argument:
$$\Big(2^{(s_x+a)/2}-1\Big)\Big(2^{(s_x+a)/2}+1\Big)$$
Once again $\frac{s_x+a}2=1$ cannot be true for $a\geq 4$, so $s_x$ must be $0$. We then have
$$7^{a-t_k}=2^a-1,$$
where $a\geq 5$ is odd. By looking into the equaiton above modulo $4$, we find that $t_k$ is even. Then, with $a=2\beta+1$ and $t_k=2\sigma$ we may rewrite the equation as
$$7\cdot {\left(7^2\right)}^{\beta-\sigma}=2^{2\beta+1}-1$$
Finally, looking at the equation above modulo $16$, we find that the LHS is congruent to $7$ while the RHS is congruent to $-1\equiv 15$. This completes the proof that when $a\geq 4$, no solution exists. $\square$
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