def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A * Matrix([X, Y]) + B)[0]
v = (A * Matrix([X, Y]) + B)[1]
simplified = diop_simplify(eq.subs(zip((x, y), (u, v))))
coeff = dict(
[reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X * Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, 1]:
if term not in coeff.keys():
coeff[term] = 0
if coeff[X**2] != 0:
return divisible(coeff[Y**2], coeff[X**2]) and \
divisible(coeff[1], coeff[X**2])
return True
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
simplified = _mexpand(Subs(eq, (x, y), (u, v)).doit())
coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X*Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
if coeff[X**2] != 0:
return isinstance(S(coeff[Y**2])/coeff[X**2], Integer) and isinstance(S(coeff[Integer(1)])/coeff[X**2], Integer)
return True
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A*Matrix([X, Y]) + B)[0]
v = (A*Matrix([X, Y]) + B)[1]
simplified = diop_simplify(eq.subs(zip((x, y), (u, v))))
coeff = dict([reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X*Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, 1]:
if term not in coeff.keys():
coeff[term] = 0
if coeff[X**2] != 0:
return divisible(coeff[Y**2], coeff[X**2]) and \
divisible(coeff[1], coeff[X**2])
return True
def is_pell_transformation_ok(eq):
"""
Test whether X*Y, X, or Y terms are present in the equation
after transforming the equation using the transformation returned
by transformation_to_pell(). If they are not present we are good.
Moreover, coefficient of X**2 should be a divisor of coefficient of
Y**2 and the constant term.
"""
A, B = transformation_to_DN(eq)
u = (A * Matrix([X, Y]) + B)[0]
v = (A * Matrix([X, Y]) + B)[1]
simplified = _mexpand(Subs(eq, (x, y), (u, v)).doit())
coeff = dict(
[reversed(t.as_independent(*[X, Y])) for t in simplified.args])
for term in [X * Y, X, Y]:
if term in coeff.keys():
return False
for term in [X**2, Y**2, Integer(1)]:
if term not in coeff.keys():
coeff[term] = Integer(0)
if coeff[X**2] != 0:
return isinstance(S(coeff[Y**2]) / coeff[X**2],
Integer) and isinstance(
S(coeff[Integer(1)]) / coeff[X**2], Integer)
return True
# transform diophantine eq into a pell eq
# via find_DN
# find 2 initial solutions
# via brute force
# transform those two solutions into many solutions
# via brahmagupta
# transform pell solutions back to diophatine
# via A & B from transformation_to_DN
# find first solution pair that red+blue > 10**12
from sympy.solvers.diophantine import transformation_to_DN, diophantine, find_DN
from sympy import *
x, y = symbols("x, y", integer=True)
s = diophantine(x**2 - 2 * x * y - y**2 - x + y)
A, B = transformation_to_DN(s)
D, N = find_DN(s)
def pell(y, D, N):
return ((D * y**2 + N)**.5)
answers = []
y = 0
while len(answers) < 2:
y += 1
if pell(y).is_integer():
answers.append([pell(y, D, N), y])