wanless'theorem-fermat'sbigtheorem

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]