Pisano period: Difference between revisions
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=={{header|C++}}==
<syntaxhighlight lang="c++">#include <functional>
#include <iostream>
#include <iomanip>
#include <numeric>
#include <vector>
using namespace std;
template<typename T>
pair<unsigned, unsigned> floyd(function<T(T)> f, T x0) {
// Floyd's cycle detection algorithm.
auto tortoise = f(x0);
auto hare = f(f(x0));
while (tortoise != hare) {
tortoise = f(tortoise);
hare = f(f(hare));
}
// Find the position μ of first repetition.
unsigned mu = 0;
tortoise = x0;
while (tortoise != hare) {
tortoise = f(tortoise);
hare = f(hare);
mu += 1;
}
// Find the length of the shortest cycle starting from x_μ
unsigned lam = 1;
hare = f(tortoise);
while (tortoise != hare) {
hare = f(hare);
lam += 1;
}
return make_pair(lam, mu);
}
unsigned pisano_period(unsigned p) {
if (p < 2) return 1;
function<pair<unsigned, unsigned>(pair<unsigned, unsigned>)> f = [&](auto xy) {
return make_pair(xy.second, (xy.first + xy.second) % p);
};
return floyd(f, make_pair(0u, 1u)).first;
}
bool is_prime(unsigned p) {
if (p < 2) return false;
if (0 == p % 2) return 2 == p;
if (0 == p % 3) return 3 == p;
unsigned d = 5;
while (d * d <= p) {
if (0 == p % d) return false;
d += 2;
if (0 == p % d) return false;
d += 4;
}
return true;
}
vector<pair<unsigned, unsigned>> factor(unsigned n) {
vector<pair<unsigned, unsigned>> ans;
if (n < 2) return ans;
auto work = [&](unsigned p) {
if (0 == n % p) {
unsigned k = 1;
n /= p;
while (0 == n % p) {
k += 1;
n /= p;
}
ans.emplace_back(p, k);
}
};
work(2);
work(3);
unsigned d = 5;
while (d <= n) {
work(d);
d += 2;
work(d);
d += 4;
}
return ans;
}
long long ipow(long long p, unsigned k) {
long long ans = 1;
while (0 != k) {
if (0 != k % 2) ans *= p;
k /= 2;
p *= p;
}
return ans;
}
unsigned pisano_prime(unsigned p, unsigned k) {
if (!is_prime(p) || k == 0) {
return 0;
}
return ipow(p, k - 1) * pisano_period(p);
}
unsigned pisano(unsigned n) {
auto prime_powers{factor(n)};
unsigned ans = 1;
for (auto [p, k]: prime_powers) {
if (1 < ans) {
ans = lcm(ans, pisano_prime(p, k));
} else {
ans = pisano_prime(p, k);
}
}
return ans;
}
int main() {
for (unsigned p = 2; p < 15; ++p) {
auto pp = pisano_prime(p, 2);
if (0 < pp) {
cout << "pisanoPrime(" << setw(2) << p << ": 2) = " << pp << endl;
}
}
cout << endl;
for (unsigned p = 2; p < 180; ++p) {
auto pp = pisano_prime(p, 1);
if (0 < pp) {
cout << "pisanoPrime(" << setw(3) << p << ": 1) = " << pp << endl;
}
}
cout << endl;
cout << "pisano(n) for integers 'n' from 1 to 180 are:" << endl;
for (unsigned n = 1; n <= 180; ++n) {
cout << setw(3) << pisano(n) << " ";
if (0 == n % 15) {
cout << endl;
}
}
return 0;
}</syntaxhighlight>
{{out}}
<pre>pisanoPrime( 2: 2) = 6
pisanoPrime( 3: 2) = 24
pisanoPrime( 5: 2) = 100
pisanoPrime( 7: 2) = 112
pisanoPrime(11: 2) = 110
pisanoPrime(13: 2) = 364
pisanoPrime( 2: 1) = 3
pisanoPrime( 3: 1) = 8
pisanoPrime( 5: 1) = 20
pisanoPrime( 7: 1) = 16
pisanoPrime( 11: 1) = 10
pisanoPrime( 13: 1) = 28
pisanoPrime( 17: 1) = 36
pisanoPrime( 19: 1) = 18
pisanoPrime( 23: 1) = 48
pisanoPrime( 29: 1) = 14
pisanoPrime( 31: 1) = 30
pisanoPrime( 37: 1) = 76
pisanoPrime( 41: 1) = 40
pisanoPrime( 43: 1) = 88
pisanoPrime( 47: 1) = 32
pisanoPrime( 53: 1) = 108
pisanoPrime( 59: 1) = 58
pisanoPrime( 61: 1) = 60
pisanoPrime( 67: 1) = 136
pisanoPrime( 71: 1) = 70
pisanoPrime( 73: 1) = 148
pisanoPrime( 79: 1) = 78
pisanoPrime( 83: 1) = 168
pisanoPrime( 89: 1) = 44
pisanoPrime( 97: 1) = 196
pisanoPrime(101: 1) = 50
pisanoPrime(103: 1) = 208
pisanoPrime(107: 1) = 72
pisanoPrime(109: 1) = 108
pisanoPrime(113: 1) = 76
pisanoPrime(127: 1) = 256
pisanoPrime(131: 1) = 130
pisanoPrime(137: 1) = 276
pisanoPrime(139: 1) = 46
pisanoPrime(149: 1) = 148
pisanoPrime(151: 1) = 50
pisanoPrime(157: 1) = 316
pisanoPrime(163: 1) = 328
pisanoPrime(167: 1) = 336
pisanoPrime(173: 1) = 348
pisanoPrime(179: 1) = 178
pisano(n) for integers 'n' from 1 to 180 are:
1 3 8 6 20 24 16 12 24 60 10 24 28 48 40
24 36 24 18 60 16 30 48 24 100 84 72 48 14 120
30 48 40 36 80 24 76 18 56 60 40 48 88 30 120
48 32 24 112 300 72 84 108 72 20 48 72 42 58 120
60 30 48 96 140 120 136 36 48 240 70 24 148 228 200
18 80 168 78 120 216 120 168 48 180 264 56 60 44 120
112 48 120 96 180 48 196 336 120 300 50 72 208 84 80
108 72 72 108 60 152 48 76 72 240 42 168 174 144 120
110 60 40 30 500 48 256 192 88 420 130 120 144 408 360
36 276 48 46 240 32 210 140 24 140 444 112 228 148 600
50 36 72 240 60 168 316 78 216 240 48 216 328 120 40
168 336 48 364 180 72 264 348 168 400 120 232 132 178 120</pre>
=={{header|Factor}}==
{{works with|Factor|0.99 2020-01-23}}
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