<s>[INST] How do I show that the sum $(a+\frac12)^n+(b+\frac12)^n$ is an integer for only finitely many $n$? 
Show that if $a$ and $b$ are positive integers, then $$\left(a +\frac12\right)^n + \left(b+\frac{1}{2}\right)^n$$is an integer for only finitely many positive integers $n$.

I tried hard but nothing seems to work. :( 
 [/INST] Note that
$$\left(a+\frac{1}{2}\right)^n + \left(b+\frac{1}{2}\right)^n=\frac{1}{2^n}\left(c^n+d^n\right),$$
where $c=2a+1$ and $d=2b+1$. 
Let $2^e$ be the highest power of $2$ that divides $c+d$. 
If $n$ is even, the highest power of $2$ that divides $c^n+d^n$ is $2$.
For odd $n$, note that $u^n+v^n=(u+v)(u^{n-1}+u^{n-2}v+\cdots +v^{n-1})$. The second term in this product is odd. It follows that the highest power of $2$ that divides $c^n+d^n$ is $2^e$. 
Thus if $n\gt e$, then $\left(a+\frac{1}{2}\right)^n + \left(b+\frac{1}{2}\right)^n$ is not an integer. 
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