Last week I was engaging in my hobby of typing numbers into my calculator at random when I noticed the following:

2⁹ + 8⁹ is divisible by 5

5⁷ + 19⁷ is divisible by 12

3¹¹ + 11¹¹ is divisible by 7

6³ + 20³ is divisible by 13

What conjecture do you think I came up with?

Can you prove it? (And can you prove a slightly stronger conjecture?)

What happens with even powers?