- 描述
- Given a prime P, 2 <= P < 2
^{31}, an integer B, 2 <= B < P, and an integer N, 2 <= N < P, compute the discrete logarithm of N, base B, modulo P. That is, find an integer L such that

B^{L} == N (mod P)

- 输入
- Read several lines of input, each containing P,B,N separated by a space.
- 输出
- For each line print the logarithm on a separate line. If there are several, print the smallest; if there is none, print "no solution".
- 样例输入
5 2 1
5 2 2
5 2 3
5 2 4
5 3 1
5 3 2
5 3 3
5 3 4
5 4 1
5 4 2
5 4 3
5 4 4
12345701 2 1111111
1111111121 65537 1111111111

- 样例输出
0
1
3
2
0
3
1
2
0
no solution
no solution
1
9584351
462803587

- 提示
- The solution to this problem requires a well known result in number theory that is probably expected of you for Putnam but not ACM competitions. It is Fermat's theorem that states

B^{(P-1)} == 1 (mod P)

for any prime P and some other (fairly rare) numbers known as base-B pseudoprimes. A rarer subset of the base-B pseudoprimes, known as Carmichael numbers, are pseudoprimes for every base between 2 and P-1. A corollary to Fermat's theorem is that for any m

B^{(-m)} == B^{(P-1-m)} (mod P) .

- 来源
- Waterloo Local 2002.01.26