Should I use some test of simplicity (from Cormen or somewhere else) or is there any better way for such numbers?

If someone knows how to solve this proble without
cheatin' (if carefull reading of the definition is a cheat) then he can get a lot of money... :)

Sorry, I understood after all, as for me I just found all the numbers in the Internet:)

I don't think it is possible to solve it without cheating,
otherwise, why now the largest one is only 43th?

Any way, you can get most of the ans in the text, except 3~8...

I agree

Interesting

Of course, you cannot solve it without "cheating".
The most powerful tests work for months to test one number!

You already have all numbers written in the text - why do you think it is cheating?
And it's easy to find the rest - up to 7-th. It's enough to test all numbers 2^p-1 up to 2^30-1 (only 30 numbers!) using bruteforce.

its just impossible to calculate it in 1 second.

If using Wikipedia and OEIS aren't cheating I know how to do it)

I wonder , who can?

Using Wikipedia and OEIS may be cheating, but using the answers already in the problem definition definitely isn't.