"Congratulations. You've made it to level 2. Go to XXXXXXXXXX.org and enter XXXXXXXXXXXXX as the login and the answer to this equation as the password.

f(1)= xxxxxx
f(2)= xxxxxx
f(3)= xxxxxx
f(4)= xxxxxx
f(5)= __________"

Part 2:

bradfitz@bini:~/proj/google-prime$ ./part-2.pl | head -10
  0   1   2  1: 718xxxxxxx
  4   5   6  2: 8182845904
 22  23  24  3: 8747135266
 98  99 100  4: 7427466391
126 127 128  5: xxxxxxxxxx

Just out of interest (although the answers are all over the web) I decided to see how hard it would be to brute-force this. My dumb script just tries all possible sets of ten after a bit of sanity checking, and then uses LWP to try the request with a short timeout. After 20 attempts it found it. It would have been less effective if people owned some of the other answers!

The second part is harder to brute force, but I imagine a few people still tried it anyway. I kinda wonder whether Google are more interested in people who were able to figure out the nature of the sequence or people who find creative ways (completely dumb brute force doesn't count) to solve the problem without figuring out the pattern separately first…

Not quite. I did some quick checks on the numbers before doing the requests. For example, if it starts with zero it is out (less than ten digits) and also a quick loop to test dividing between everything between 1 and some constant (I forget what it was now, and I've destroyed the program to solve the second problem!) to see if any of them give a whole number. Only then does it make the HTTP request.

I think it was somewhere above the 30th position although I can't remember exactly where it was and I'm too lazy to write the one line of code it'd take to find out, too.

Aha… I knew there was some other prime rule I was forgetting. It's been a while since I did anything particularly mathematical. I should “practice” more.