<s>[INST] Prove that there exist no natural number k such that $3^k+5^k$ is a square of an integer number? How to prove that there exist no natural number $k$ such that $3^k+5^k$ is a square of an integer number
 [/INST] An elementary solution without modular arithmetic. :)
If $k$ is odd, then write
$$a^2 = 3^k+5^k= 8(3^{k-1}-3^{k-2}5+...+5^{k-1})$$
Since expression in bracket is odd (we have odd odd numbers) we have $8|a^2$ but $16\not|a^2$. A contradiction.
Say $k$ is even. Then $k=2n$ so $(a-3^n)(a+3^n)= 5^{2n}$, so $a-3^n= 5^{x}$ and $a+3^n= 5^{y}$ for some non negative integer $x$ and $y$ (we see that $y>x$ and $x+y=2n$) thus:
$$ 2\cdot 3^n = 5^y-5^x= 5^x(5^{y-x}-1)$$
so $x=0$. So $2\cdot 3^n = 5^{2n}-1$ which is impossible since that left side is smaller then right side for each $n$ (easy to prove with induction).
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