def main(limit):
# Case 1: a, a, a + 1
# for integer area a must be odd, a = 2 * k1 + 1
# this leads to:
# p1 = 3*a + 1 = 6*k1 + 4
# (3*k1+1)^2 - 3*s^2 = 1 for some integer s
# Case 2: a, a, a - 1
# for integer area a must be odd, a = 2 * k2 + 1
# this leads to:
# p2 = 3*a - 1 = 6*k2 + 2
# (3*k2+2)^2 - 3*s^2 = 1 for some integer s
rv = 0
# get all t and s such that t^2 - 3*s^2 = 1
for x, y in convergents(3):
if x >= 4 and x * x - 3 * y * y == 1:
t = x
p1 = 2 * (t - 1) + 4
p2 = 2 * (t - 2) + 2
if p1 > limit and p2 > limit:
return rv
if p1 <= limit and (t - 1) % 3 == 0:
rv += p1
if p2 <= limit and (t - 2) % 3 == 0:
rv += p2
def main(limit):
# Case 1: a, a, a + 1
# for integer area a must be odd, a = 2 * k1 + 1
# this leads to:
# p1 = 3*a + 1 = 6*k1 + 4
# (3*k1+1)^2 - 3*s^2 = 1 for some integer s
# Case 2: a, a, a - 1
# for integer area a must be odd, a = 2 * k2 + 1
# this leads to:
# p2 = 3*a - 1 = 6*k2 + 2
# (3*k2+2)^2 - 3*s^2 = 1 for some integer s
rv = 0
# get all t and s such that t^2 - 3*s^2 = 1
for x, y in convergents(3):
if x >= 4 and x * x - 3 * y * y == 1:
t = x
p1 = 2 * (t - 1) + 4
p2 = 2 * (t - 2) + 2
if p1 > limit and p2 > limit:
return rv
if p1 <= limit and (t - 1) % 3 == 0:
rv += p1
if p2 <= limit and (t - 2) % 3 == 0:
rv += p2
def main(limit):
# B*(B-1)/T/(T-1) = 1/2 => 2*(2*B-1)^2 - (2*T-1)^2 = 1
# the solutions are the odd convergents of the Pell equation y^2 - 2*x^2 = -1
for x, y in convergents(2):
if x % 2 == 1 and y % 2 == 1 and x * x - 2 * y * y == -1:
blue = (y + 1) // 2
total = (x + 1) // 2
if total > limit:
return blue
def main(limit): # B*(B-1)/T/(T-1) = 1/2 => 2*(2*B-1)^2 - (2*T-1)^2 = 1 # the solutions are the odd convergents of the Pell equation y^2 - 2*x^2 = -1 for x, y in convergents(2): if x % 2 == 1 and y % 2 == 1 and x * x - 2 * y * y == -1: blue = (y + 1) // 2 total = (x + 1) // 2 if total > limit: return blue
def minX(D):
for x, y in convergents(D):
if x * x - D * y * y == 1:
return x
return float('-inf')
def goodConvergentsOf(D):
for x, y in convergents(D):
if x > 2 and x * x - D * y * y == -1:
yield x, y
def goodConvergentsOf(D): for x, y in convergents(D): if x > 2 and x * x - D * y * y == -1: yield x, y