Pick “a” and ‘c” odd, b even (for example a=c=3, b=2). Then trivially both sides will be even, and for all n you’ll have a GCD greater than 1.

gcd(3^3 – 1, 3^3 + 2^3 – 1) = gcd(26, 34) = 2.

I have this expression that’s beyond me, but relates to Fermat’s Last theorem ’s diophantine equation, a^n + b^n = c^n, a,b,c,n all positive integers and n > 2. We all knew that it is now a theorem. I seems to me that it is related to this equation,

gcd(c^n – 1, a^n + b^n – 1) = 1

can you show or give a counterexample that will give a common factor larger than 1 that is positive ODD integer?

Thanks