Wanless' Theorem - Fermat's Big Theorem
Inductive hypothesis:
a**(p2**n)==a**(2**n) [mod p2**n]
=> a**(p2**n)=a**(2**n)+mp2**n
a**(p2**(n+1))=(a**(p2**n))**2
=(a**(2**n)+mp2**n)**2
=a**(2**(n+1))+2mp(2**n)(a**(2**n))+(m**2)(p**2)2**2n
=a**(2**(n+1))+(p2**(n+1))(ma**(2**n)+(m**2)p2**(n-1))
==a**(2**(n+1)) [mod p2**(n+1)][if n>0]
Basis for induction:
(a**2)**p==a**2 [mod 2]
(a**p)**2==a**2 [mod p][By Fermat's Little Theorem]
=> (a**2p)==a**2 [mod 2p]
By mathematical induction:
a**(p2**n)==a**(2**n) [mod p2**n]
Copyright 1997 James Wanless