8 divides $\displaystyle 5^n-4n-1$ for all integers n greater than or equal to 1.

Proof:

Basis step:

n = 1

$\displaystyle 5^1-4(1)-1$ = 0

0/8 = 0

8 divides $\displaystyle 5^n-4n-1$ if n=1

Induction step:

Assume $\displaystyle 5^n-4n-1$ is divisible by 8 (induction hypothesis)

[show 5^(n+1) - 4(n+1) - 1 is divisible by 8]

- $\displaystyle 5^(n+1) - 4(n+1) - 1 = 5^n+1 - 4n + 4 - 1$
- $\displaystyle = 5^n+1 - 4n +3$
- $\displaystyle =5^n *5^1 - 4n +3$

...now im not sure where to go