def test_is_square(self):
for k in [-2, -1, 2, 3, 5, 7, 11, 2545]:
self.assertEqual(is_square(k), False)
for (k, r) in [(9, 3), (16, 4), (289, 17), (130321, 361)]:
self.assertEqual(is_square(k), r)
self.assertEqual(is_square([(2, 4), (5, 2)]), 20)
self.assertEqual(is_square([(2, 4), (5, 3)]), False)
values = [
0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529,
576, 625, 676, 729, 784, 841, 900, 961
]
squares = [a for a in xrange(1000) if is_square(a)]
self.assertEqual(squares, values)
def test_is_square(self):
self.assertRaisesRegexp(ValueError, "is_square: Must have n >= 0.", is_square, -2)
for k in [2, 3, 5, 7, 11]:
self.assertEqual(is_square(k), False)
for (k, r) in [(9, 3), (16, 4), (289, 17), (130321, 361)]:
self.assertEqual(is_square(k), r)
self.assertEqual(is_square([(2, 4), (5, 2)]), 20)
self.assertEqual(is_square([(2, 4), (5, 3)]), False)
values = [
1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576,
625, 676, 729, 784, 841, 900, 961
]
squares = [a for a in xrange(1, 1000) if is_square(a)]
self.assertEqual(squares, values)
def pell_fundamental_solution(D, n=1):
"""Returns the fundamental solution to the Pell equation x**2 - D*y**2 == n,
where n in (-1, 1).
Given D > 0 not a square, and n in (-1, 1), this method returns the
fundamental solution to the Pell equation described above. The fundamental
solution (x, y) is the one with least positive value of x, and
correspondingly the least positive value of y.
Input:
* D: int (D > 0)
* n: int (n in (-1, 1)).
Returns:
* (x, y): tuple
Raises:
* ValueError: If D <= 0, D is perfect square, n not in (-1, 1), or if no
solutions exist.
Examples:
>>> pell_fundamental_solution(61)
(1766319049, 226153980)
>>> 1766319049**2 - 61*226153980**2
1
>>> pell_fundamental_solution(17, -1)
(4, 1)
>>> 4**2 - 17*1**2
-1
>>> pell_fundamental_solution(15, -1)
Traceback (most recent call last):
...
ValueError: pell_fundamental_solution: Solution nonexistent.
>>> pell_fundamental_solution(15, -2)
Traceback (most recent call last):
...
ValueError: pell_fundamental_solution: Must have D > 0 not a perfect square and n in (-1, 1).
Details:
For D > 0 not a perfect square, the equation x**2 - D*y**2 == 1 always
has solutions, while the equation x**2 - D*y**2 == -1 only has solutions
when the continued fraction expansion of sqrt(D) has odd period length.
See Corollary 5.7 of "Fundamental Number Theory with Applications" by
Mollin for details. See also the article "Simple Continued Fraction
Solutions for Diophantine Equations" by Mollin.
"""
if D <= 0 or is_square(D) or n not in (1, -1):
raise ValueError("pell_fundamental_solution: Must have D > 0 not a perfect square and n in (-1, 1).")
contfrac = QuadraticIrrational(D)
# No solution for n == -1 if the period length is even.
if n == -1 and contfrac.period_length % 2 == 0:
raise ValueError("pell_fundamental_solution: Solution nonexistent.")
# Otherwise, solutions always exist for D not a perfect square.
for (x, y) in contfrac.convergents():
if x*x - D*y*y == n:
return (x, y)
