Solution 2 (Eigenvalue): Monkeys & Coconuts

Solution 2: Use Linear Algebra

Eigenvalue equation: A.X = λ.X

A =S(x)= $Latex \frac{4}{5}(x-1)$  where x = coconuts

S(x)=λx
Set λ = 1
$Latex \frac{4}{5}(x-1)=x$
x =-4
Also fifth monkey:
$Latex S^5 (x)$ = $Latex (\frac{4}{5})^5 (x-1)+(\frac{4}{5})^5+(\frac{4}{5})^4+\cdots+\frac{4}{5}$

$Latex 5^5$ divides (x-1)

Minimum positive x= -4 mod ($Latex 5^5$ )= $Latex 5^5$-4= 3121 [QED]

Note: λ=1 because "before" and "after" (transformation A)  is the same.