Find the greatest common divisor

$(1024^{2014}+1023,1024^{2014}-1)=(1024^{2014}+1023-1024^{2014}+1,1024^{2014}-1)=(1024,1024^{2014}-1)=(1024,1024^{2014}-1)mod(1024)= (0,-1)mod(1024)=(1024,1023)=1$

Let's say that gcd(a,b)=d. Then d|a and d|b. Therefore d|b-a=1024. So d|1024, then d can be 1, 2, 4, 8,..., 512, 1024. Of course 1 divides both a and b. For d>1 d can only be a even number, but a and b are odd numbers, so d>1 can't be a divisor for a and b. The only divisor is 1.