De-RNG-ed (Large)

I want to build an online poker site. The random number generator is an important part of such a system, and it has to be fast and random enough. Here is the compromise I came up with.

I need random numbers of at most $D$ digits. First I pick a prime $P$ with $P \le 10^D$. I also pick non-negative integers $A$ and $B$. Finally I pick an integer seed $S$ with $0 \le S \le P-1$.

The sequence comes out like this. First I output $S$, then I update $S$ with

$S := (A \times S + B) \bmod P$

I output the updated $S$ as the next term of the sequence and update $S$ again with the same formula. I repeat this as many times as I want.

Do you think this is a good random number generator? Write a program that reads $K$ consecutive terms of a sequence made this way and prints the term that comes right after them.

The first line has the number of test cases $T$. Each test case then takes two lines. The first line has $D$ and $K$. The second line has $K$ consecutive terms of a sequence made by the generator described above.

Limits

• $1 \le T \le 100$
• $1 \le D \le 6$
• $1 \le K \le 10$
• The $K$ integers are consecutive terms of a sequence made by a generator of the kind above, so at least one $(P, A, B)$ produces exactly these $K$ terms.

For each test case, print one line in the form Case #x: y, where $x$ is the test case number and starts from 1.

Consider every combination of a prime $P \le 10^D$ and non-negative integers $A$ and $B$ that produces exactly the given $K$ terms. If all of those combinations give the same next term, print that value as $y$. If two of them give different next terms, print I don't know. as $y$, period included.