Problem 4

Determine all positive integers relatively prime to all the terms of the infinite sequence

a_n=2^n+3^n+6^n -1,\ n\geq 1.

Step 1 of 5: Rule out

2

and

3

using

a_2

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In plain words

A candidate $k$ must be coprime to every term, in particular to the very first term computed, which already carries factors of $2$ and $3$ .

a_2=2^2+3^2+6^2-1=48=2^4\cdot3

Detailed analysis

Let $k$ be a positive integer coprime to every $a_n$ . Since $a_2=2^2+3^2+6^2-1=48=2^4\cdot3$ , $k$ must be coprime to $48$ , so $k$ is not divisible by $2$ or by $3$ .