## Find the 200th prime-proof sqube containing the contiguous sub-string "200"

### Problem 200

Published on Saturday, 28th June 2008, 10:00 am; Solved by 1418; Difficulty rating: 65%We shall define a sqube to be a number of the form, `p`^{2}`q`^{3}, where `p` and `q` are distinct primes.

For example, 200 = 5^{2}2^{3} or 120072949 = 23^{2}61^{3}.

The first five squbes are 72, 108, 200, 392, and 500.

Interestingly, 200 is also the first number for which you cannot change any single digit to make a prime; we shall call such numbers, prime-proof. The next prime-proof sqube which contains the contiguous sub-string "200" is 1992008.

Find the 200th prime-proof sqube containing the contiguous sub-string "200".