Wanless' Lemma
a**(2**i)+b**(2**i)==c**(2**i) [mod 2**i]
=>a**(2**(i+1))+b**(2**(i+1))==c**(2**(i+1)) [mod 2**(i+1)] [i>1]
Proof:
a,b,c == 0 or 1 [mod 2]
=> a**(2**i), b**(2**i), c**(2**i)== 0 or 1 [mod 2**i] [Proof by induction]
=> WLOG a==0 [mod 2] [if i>1]
a**(2**i)+b**(2**i)==c**(2**i) [mod 2**i]
=> (a**(2**i)+b**(2**i))**2=(c**(2**i)+m2**i)**2
=> a**(2**(i+1))+2a**(2**i)b**(2**i)+b**(2**(i+1))=c**(2**(i+1))+2m2**i(c**(2**i))+m**2(2**2i)
=> a**(2**(i+1))+b**(2**(i+1))==c**(2**(i+1)) [mod 2**(i+1)] [if i>1]
Copyright 1997 James Wanless