<s>[INST] On representation of even integers as $a^x+b^y$ Does there exist some $k \in \mathbb N$ such that for every even integer $n \ge k $ we can find positive integers $a,b,x,y$ such that $n=a^x+b^y$ , where $\gcd (a,b)=1 ; a,b>1$ and at least one of $x,y$ is more than $1$ ? I know that for every even integer $n>6$ there are integers $a,b>1$ such that $\gcd(a,b)=1$ and $n=a+b$ , so here  $x=y=1$ for every $n$ , I want to see if at least one power can be bigger.
 [/INST] Let $n \ge 6$.  Choose a prime $p$ such $p^2 \le n-2$ and $p$ does not divide $n$.  Then $n=p^x+(n-p^2)^y$ with $x=2$ and $y=1$, and $p$ and $n-p^2$ are greater than $1$ and relatively prime.  This fails only if there is no such prime, which happens only if $n$ is divisible by every prime between $2$ and $\sqrt{n-2}$.  But in that case $n \ge \prod_{p \le \sqrt{n-2}}p$, which can't happen for very large $n$, since clearly the right-hand side grows much faster than the left.  Indeed,
$$
\prod_{p \le 7=\sqrt{51-2}} p= 2\cdot 3\cdot 5\cdot 7 = 210 \gg 51,
$$
so for $n\ge 51$ we can write $n=a^2+b$ where $\gcd(a,b)=1$.
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