public static bool Lucas(ulong n)
{
var random = new System.Random();
var factorisation = Factorise.Optimised(n - 1);
for (ulong a = 2; a < n;)
{
c:
if (GCD.Standard(a, n) > 1)
{
return(false);
}
if (Power.BinaryMod(a, n - 1, n) == 1)
{
foreach (uint prime in factorisation)
{
if (Power.BinaryMod(a, (n - 1) / prime, n) == 1)
{
a++;
goto c;
}
}
return(true);
}
else
{
return(false);
}
}
return(false);
}
public static bool Strong(ulong n, ulong a)
{
if (n == 2)
{
return(true);
}
if (n < 2 || (n & 1) == 0)
{
return(false);
}
ulong d = (n - 1) >> 1, s = 1;
for (; (d & 1) == 0; d >>= 1, s++)
{
;
}
a = Power.BinaryMod(a, d, n);
if (a == 1 || a == n - 1)
{
return(true);
}
for (ulong r = 1; r < s; r++)
{
a = (a * a) % n;
if (a == 1)
{
return(false);
}
if (a == n - 1)
{
return(true);
}
}
return(false);
}
/// <summary>
///
/// </summary>
/// <param name="n"></param>
/// <param name="iterations"></param>
/// <returns>true if n is prime, false if n is possibly composite</returns>
public static bool Lucas(ulong n, ulong iterations)
{
ulong[] factorisation = Factorise.Optimised(n - 1);
for (ulong i = 0; i < iterations; i++)
{
c:
ulong a = (ulong)random.Next(2, (int)n);
if (GCD.Standard(a, n) > 1)
{
return(false);
}
if (Power.BinaryMod(a, n - 1, n) == 1)
{
foreach (uint prime in factorisation)
{
if (Power.BinaryMod(a, (n - 1) / prime, n) == 1)
{
goto c;
}
}
return(true);
}
else
{
return(false);
}
}
return(false);
}
public static bool Fermat(ulong n, ulong a)
{
if (n == 2)
{
return(true);
}
if (n < 2 || (n & 1) == 0)
{
return(false);
}
return(Power.BinaryMod(a, n - 1, n) == 1);
}
public static bool Euler(ulong n, ulong a)
{
if (n == 2)
{
return(true);
}
if (n < 2 || (n & 1) == 0)
{
return(false);
}
a = Power.BinaryMod(a, (n - 1) >> 1, n);
return(a == 1 || a == n - 1);
}
public static short?Legendre(long a, long prime)
{
short s = (short)Power.BinaryMod((ulong)a, (ulong)((prime - 1) >> 1), (ulong)prime);
if (s == 0 || s == 1)
{
return(s);
}
if (s == a - 1)
{
return(-1);
}
return(null);
}
public static bool EulerJacobi(ulong n, ulong a)
{
if (n == 2)
{
return(true);
}
if (n < 2 || (n & 1) == 0)
{
return(false);
}
long jac = Symbol.Jacobi(a, n);
return((long)Power.BinaryMod(a, (n - 1) >> 1, n) == (jac < 0 ? (long)n - 1 : jac));
}
public static bool Pocklington(ulong n)
{
ulong[] primes = Sieve.Standard((ulong)System.Math.Log(n - 1)).Cast <ulong>().ToArray();
ulong f = n - 1, sqrt = (ulong)System.Math.Sqrt(n);
foreach (ulong prime in primes)
{
if (f / prime < sqrt)
{
goto g;
}
while (f % prime == 0)
{
if (f / prime < sqrt)
{
goto g;
}
f /= prime;
}
}
g:
var factors = Factorise.Standard(f, (z) => (ulong)Factoring.Special.TrialDivision((int)z));
for (uint i = 2; i < n; i++)
{
if (Power.BinaryMod(i, n - 1, n) == 1)
{
bool isPrime = true;
foreach (uint factor in factors)
{
if (GCD.Standard((uint)Power.Binary(i, (n - 1) / factor) - 1, n) != 1)
{
isPrime = false;
break;
}
}
if (isPrime)
{
return(true);
}
}
}
return(false);
}
public static bool MillerRabin(ulong n, ulong iterations)
{
if (n == 2)
{
return(true);
}
if (n < 2 || (n & 1) == 0)
{
return(false);
}
ulong d = (n - 1) >> 1, r = 1;
for (; (d & 1) == 0; d >>= 1, r++)
{
;
}
for (ulong i = 0; i < iterations; i++)
{
c:
ulong x = Power.BinaryMod((ulong)random.Next(2, (int)(n - 1)), d, n);
if (x == 1 || x == n - 1)
{
continue;
}
for (ulong j = 0; j < r - 1; j++)
{
x = (x * x) % n;
if (x == 1)
{
return(false);
}
if (x == n - 1)
{
goto c;
}
}
return(false);
}
return(true);
}
/// <summary>
/// Applies Miller test to n.
/// </summary>
/// <param name="n">number to be tested</param>
/// <returns>true if n is prime otherwise false</returns>
public static bool MillerTest(ulong n)
{
if (n == 2)
{
return(true);
}
if (n < 2 || (n & 1) == 0)
{
return(false);
}
ulong d = (n - 1) >> 1, s = 1;
for (; (d & 1) == 0; d >>= 1, s++)
{
;
}
ulong max = (ulong)System.Math.Min(n - 1, System.Math.Floor(2 * System.Math.Pow(System.Math.Log(n), 2)));
for (ulong a = 2; a <= max; a++)
{
ulong m = Power.BinaryMod(a, d, n);
if (m != 1 && m != n - 1)
{
for (ulong r = 1; r < s; r++)
{
if ((m = (m * m) % n) == n - 1)
{
goto c;
}
}
return(false);
}
c :;
}
return(true);
}
public static bool Pepin(ulong exponent)//fermat numbers
{
ulong n = (ulong)System.Math.Pow(2, System.Math.Pow(2, exponent));
return(Power.BinaryMod(3, n >> 1, n + 1) == n);
}
public static bool BinomialTheorem(int p, int a, int b) => (int)Power.BinaryMod((ulong)(a + b), (ulong)p, (ulong)p) == (int)(Power.BinaryMod((ulong)a, (ulong)p, (ulong)p) + Power.BinaryMod((ulong)b, (ulong)p, (ulong)p)) % p;