Carmichael Numbers

Certain cryptographic algorithms make use of big prime numbers. However, checking whether a big number is prime is not easy.

Randomized primality tests exist that offer a high degree of confidence of accurate determination at a low cost, such as the Fermat test. Let a be a random number between 2 and n - 1 where n is the number whose primality we are testing. Then, n is probably prime if the following equation holds:

aⁿ mod n = a

If a number passes the Fermat test several times, then it is prime with a high probability.

Unfortunately, there is bad news. Certain composite numbers (non-primes) still pass the Fermat test with every number smaller than themselves. These numbers are called Carmichael numbers.

Write a program to test whether a given integer is a Carmichael number.

Input: carmichael.in

The input will consist of a series of lines, each containing a small positive number n (2 ≤ n ≤ 65,000). A number n = 0 will mark the end of the input, and must not be processed.

Output: carmichael.out

For each number in the input, print whether it is a Carmichael number or not as shown in the sample output.

Sample Input

1729
17
561
1109
431
0

Sample Output

1729 is a Carmichael number.
17 is normal.
561 is a Carmichael number.
1109 is normal.
431 is normal.