def test_sqrts_mod_n(self):
self.assertEqual(sqrts_mod_n(3, 178137047), [16152263, 28026114, 150110933, 161984784])
self.assertEqual(sqrts_mod_n(49, 53**3*61**4), [7, 721465236980, 1339862033577, 2061327270550])
self.assertEqual(sqrts_mod_n(-1, 5**3*7**2), [])
for n in xrange(2, 200):
values = defaultdict(list)
for a in xrange(n):
values[a*a % n].append(a)
for a in xrange(n):
self.assertEqual(values[a], sqrts_mod_n(a, n))
def _pell_general(D, N, one_solution=False):
"""
test with D = 1121311213, N = 11889485036288588
"""
# Find fundamental solution to Pell
for (t, u) in QuadraticIrrational(D).convergents():
if t*t - D*u*u == 1:
break
# Now find fundamental solution to general Pell
fundamental_solutions = []
residues = sqrts_mod_n(N, D)
L = N*(t - 1)
print residues
for r in residues:
for x in itertools.count(r, D):
w = (x*x - N)//D
if w < 0:
continue
elif 2*D*w > L:
break
y = integer_sqrt(w)
if y*y == w:
if one_solution:
yield (x, y)
return
fundamental_solutions.append((x, y))
(r, s) = (-x, y)
if (x*r - D*y*s) % N != 0 or (x*s - y*r) % N != 0:
fundamental_solutions.append((r, s))
minimal_positive_solutions = []
for (x, y) in fundamental_solutions:
if x < 0:
(r, s) = (-x, -y)
p = r*t + s*u*D
q = r*u + s*t
minimal_positive_solutions.append((p, q))
else:
minimal_positive_solutions.append((x, y))
minimal_positive_solutions.sort()
if not minimal_positive_solutions:
return
print minimal_positive_solutions
while True:
for i in xrange(len(minimal_positive_solutions)):
x, y = minimal_positive_solutions[i]
yield (x, y)
minimal_positive_solutions[i] = (x*t + y*u*D, x*u + y*t)
def _mollin(D, N):
positive_roots = sqrts_mod_n(D, abs(N))
roots = []
print positive_roots
for r in positive_roots:
for e in [-r, r]:
if -abs(N) < 2*e <= abs(N):
roots.append(e)
return roots
def pell_minimal_positive_solutions(D, N):
r"""Returns the minimal positive solutions to the Pell equation
x**2 - D*y**2 == N.
Given D > 0 not a square, and N != 0, this method returns a list of the
minimal positive solutions in each class for the Pell equation
described above.
Parameters
----------
D : int (D > 0)
N : int (N != 0)
Returns
-------
L : list
List of the minimal positive solutions of the equation. The list is
empty if there are no solutions.
Notes
-----
References
----------
.. [1] T. Andreescu, D. Andrica, "Quadratic Diophantine Equations",
Springer-Verlag, New York, 2015.
.. [2] R.A. Mollin, "Fundamental Number Theory with Applications",
Chapman & Hall/CRC, 2008.
Examples
--------
>>> pell_minimal_positive_solutions(13, 27)
[(12, 3), (40, 11), (220, 61), (768, 213)]
>>> pell_minimal_positive_solutions(
"""
(u, v) = pell_fundamental_solution(D, 1)
# If N == 1, there is only one class of solutions.
if N == 1:
return [(u, v)]
if N > 0:
# See Theorem 7.1 in [2]
B = int(((u + 1) * N / 2.0)**(0.5)) + 1
else:
# See Theorem 7.2 in [2]
B = int(((u - 1) * (-N) / 2.0)**(0.5)) + 1
# We can proceed in two different ways at this point:
# 1) We can check all values of y in the range described in Theorems
# 7.1 and 7.2, finding the corresponding valid values of x for each.
#
# 2) By solving a quadratic congruence, we get congruences that x must
# satisfy. By iterating through these progressions, we are able to
# check the entire range faster than in 1.
#
# We use the second method here.
residues = sqrts_mod_n(N, D)
fundamental_solutions = []
# Find the fundamental solutions for each class.
for r in residues:
for x in xrange(r, B, D):
y = is_square((x * x - N) // D)
if y > 0 and y is not False:
if (-x * x - D * y * y) % N == 0 and 2 * x * y % N == 0:
fundamental_solutions.append((x, y))
else:
fundamental_solutions.append((x, y))
fundamental_solutions.append((-x, y))
# Find the minimal positive solutions for each class.
minimal_positive_solutions = []
for (x, y) in fundamental_solutions:
if x < 0:
if N > 0:
(p, q) = (-x, -y)
else:
(p, q) = (x, y)
p, q = p * u + q * v * D, p * v + q * u
else:
(p, q) = (x, y)
minimal_positive_solutions.append((p, q))
minimal_positive_solutions.sort()
return minimal_positive_solutions
