special type of representation

Taking
8^m = 31 (( 2 n - 1 )^2) + (( 2 w - 1 )^2 )
making it
2^(3m) = 31 (( 2 n - 1 )^2) + (( 2 w - 1 )^2 )
using
k=( 2 n - 1 )^2) + ( 2 w - 1 )^2
this becomes
2^(3m)=31k
k=2^(3m)/31 where k is an integer
as 31 is prime and not divisible by 2
I don't think there is an integral solution.

Rap

There is no 31*(2w-1)^2 term in the original equation.

Oops--miscounted the brackets