# When p is a prime number and a is coprime to p,
#
# a^(p-1)≡1 (mod p)
#
# Check if it works when p is lower than N.

N=100
(p100=(1:N)[!!gmp::isprime(1:N)] ) # primes < 100
m=length(p100)
cop=list()
for(i in 1:m){
p=p100[i]
cop[[i]]=(1:p)[gmp::gcd(1:p,p)==1] # disjoint,coprime to p100[i]
}
names(cop)=p100
cop
re=cop
for(i in 1:m){
p=p100[i]
for(j in 1:length(cop[[i]])){
re[[i]][j] = beki2(cop[[i]][j],p-1,p) # coprime^(p-1)%%p
}
}
re
> str(re)
List of 25
$ 2 : int 1
$ 3 : int [1:2] 1 1
...
$ 89: int [1:88] 1 1 1 1 1 1 1 1 1 1 ...
$ 97: int [1:96] 1 1 1 1 1 1 1 1 1 1 ...
フェルマーの定理