# 2017 UNCO Math Contest II Problems/Problem 10

## Problem

```Powerless Progressions
```

Find an infinite sequence of integers $a_1, a_2, a_3, \ldots$ that has all of these properties:

(1) $a_n = c + dn$ with c and d the same for all $n = 1, 2, 3, \ldots$

(2) $c$ and $d$ are positive integers, and

(3) no number in the sequence is the $r^{th}$ power of any integer, for any power $r = 2, 3, 4, \ldots$

Reminder: Justify answers. In particular, for maximum credit, make it clear in your presentation that your sequence possesses the third property.

## Solution

2, 6, 10, 14… (& other possibilities)

Assume $a_n=x^r.$ We now check modulo 4, seeing if any possible $a_n$ are congruent to 2 mod 4.

If $x$ is 0 mod 4, $x$ is a multiple of 4 and can never become 2 mod 4 when exponentiated.

If $x$ is 1 or 3 mod 4, $x$ is odd and cannot become even when exponentiated.

If $x$ is 2 mod 4, $x^r$ is a multiple of 4 for $r \ge 2,$ which is not equivalent to 2 mod 4.

Therefore, $a_n=2+4n$ can never be an $r^{th}$ power of an integer.