#3239. Discrete Logging

Discrete Logging

题目描述

Given a prime PP, an integer BB, and an integer NN, compute the discrete logarithm of NN, base BB, modulo PP. That is, find an integer LL such that

BL=N(modP)B^L=N\pmod{P}

输入格式

Read several lines of input, each containing P,B,NP,B,N separated by a space.

输出格式

For each line, print the logarithm on a separate line. If there are several, print the smallest; If there is none, print "no solution".

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

BP1=1(modP)B^{P-1}=1\pmod{P}

For any prime PP 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 mm

Bm=BP1m(modP)B^{-m} = B^{P-1-m}\pmod{P}

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
0
1
3
2
0
3
1
2
0
no solution
no solution
1
9584351
462803587

数据规模与约定

  • For 100%100\% of Data, 2P<2312 \leq P < 2^{31}, 2B<P2 \leq B < P, 2N<P2 \leq N < P.