<s>[INST] Find $n\in\mathbb N$ so that: $\sqrt{1+5^n+6^n+11^n}\in\mathbb N$ Find $n\in\mathbb N$ so that: $\sqrt{1+5^n+6^n+11^n}\in\mathbb N$
My attempt led me to have $\quad n=2k+1:\quad k\in\mathbb N$
The expression under square root is odd, so the square root's value is also odd.
I assumed $\sqrt{1+5^n+6^n+11^n}=2a+1:\quad a\in\mathbb N \Rightarrow$
$5^n+6^n+11^n=(2a+1)^2-1 \Rightarrow 5^n+6^n+11^n=4a(a+1)$
Then the expression on the left must be divisible by 8
$5^n\equiv 5\pmod 8$ if $n$ is odd, $5^n\equiv 1\pmod 8$ if $n$ is even.
$6^n\equiv 0\pmod 8$ if $n\ge3$.
$11^n\equiv 3\pmod 8$ if $n$ is odd, $11^n\equiv 1\pmod 8$ if $n$ is even.
Then $n$ must be odd and $\ge3$ for the expression to be divisible by 8.
I'm stuck here, would anyone give a hand please?
 [/INST] $1+5^n+6^n+11^n$ has the unit digit = $3$ for any $n$ and therefore cannot be a square
Indeed $11^n$ ends always with $1$, 
$6^n$ ends with $6$, 
$5^n$ ends with $5$ adding $1$ makes $13$ so we get the unit digit to be $3$ in any case
Hope this can be useful
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