bool IsLucasProbablePrime(const Integer &n)
{
if (n <= 1)
return false;
if (n.IsEven())
return n==2;
assert(n>2);
Integer b=3;
unsigned int i=0;
int j;
while ((j=Jacobi(b.Squared()-4, n)) == 1)
{
if (++i==64 && n.IsSquare()) // avoid infinite loop if n is a square
return false;
++b; ++b;
}
if (j==0)
return false;
else
return Lucas(n+1, b, n)==2;
}
bool IsStrongLucasProbablePrime(const Integer &n)
{
if (n <= 1)
return false;
if (n.IsEven())
return n==2;
assert(n>2);
Integer b=3;
unsigned int i=0;
int j;
while ((j=Jacobi(b.Squared()-4, n)) == 1)
{
if (++i==64 && n.IsSquare()) // avoid infinite loop if n is a square
return false;
++b; ++b;
}
if (j==0)
return false;
Integer n1 = n+1;
unsigned int a;
// calculate a = largest power of 2 that divides n1
for (a=0; ; a++)
if (n1.GetBit(a))
break;
Integer m = n1>>a;
Integer z = Lucas(m, b, n);
if (z==2 || z==n-2)
return true;
for (i=1; i<a; i++)
{
z = (z.Squared()-2)%n;
if (z==n-2)
return true;
if (z==2)
return false;
}
return false;
}