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 5 of 5: No prime can divide

k

, so

k=1

Previous step Next step

k\ \text{has no prime factor}\ \Longrightarrow\ k=1

Detailed analysis

By Step 1, $2\nmid k$ and $3\nmid k$ ; by Step 4, for every prime $p>3$ , $p\mid a_{p-2}$ , so $p\nmid k$ either (else $k$ and $a_{p-2}$ would share the factor $p$ ). Hence $k$ has no prime factor at all, forcing $k=1$ . Conversely $k=1$ is coprime to every positive integer, so it does satisfy the condition. The unique answer is $k=1$ .