catalan'sconjecture

Catalan's Conjecture

a**p=b**q +/- 1

a==0 [mod 2] or b==0 [mod 2]

=> WLOG b==0 [mod 2]

Therefore, by similar to FLT, p=2

a**2=b**q +/- 1

b==0 [mod 2] => b**q==0 [mod 4]

a==1 [mod 2] => a**2==1 [mod 4]

Therefore,

b**q=a**2 - 1=(a+1)(a-1)

hcf (a+1,a-1)=2

=> either

a+1=b1**q, a-1=b2**q (clearly not possible) or

a+1=2b1**q, a-1=2**(q-1)b2**q => 2**(q-2)b2**q=b1**q - 1 or

a+1=2**(q-1)b2**q, a-1=2b1**q => 2**(q-2)b2**q=b1**q + 1

Therefore, by similar to FLT, b1=1

=> b2=1, q=3

i.e.

a=3, p=2, b=2, q=3