[fdej]

Just check for the existence of at least one odd digit mod 10^B for some well chosen B.

Here is a C program that does the verification up to 2^(10^10) in 30 seconds: https://gist.github.com/fredrik-johansson/8924e10e5d74e39109...

Edit: made it multithreaded, goes up to 2^(10^12) in nine minutes on 8 cores.