n**n + (n+1)**(n+1) = 0 mod p**2

then

n**n = - (n+1)**(n+1) mod p**2

Since for n > 0, n and n+1 are always relatively prime,

p cannot divide both n and (n+1).

Thus from the equation

n**n = - (n+1)**(n+1) mod p**2

p cannot divide either of n or (n+1).

Divide by (n+1)**n

(n/(n+1)) ** n = -(n+1) mod p**2

How will the order of (n/(n+1)) mod p**2 and the order of (-(n+1)) mod

p**2 relate to each other?

Kermit