for i in range(1, 10**10):
for k in range(1, 5):
s = str(pow(2, i, 10**(10**k)))
if '1' in s or '3' in s or '5' in s or '7' in s or '9' in s:
break
else:
print(2**i)
It's really easily to parallelize, I was able to run it up to 10**8 in about 15min, so you would be able to run it up to 10**10 in a few hours with parallelization.