def main():
i = 1
pentagonals = []
while True:
penta = i * (3 * i - 1) // 2
for p in pentagonals:
if is_pentagonal(penta - p) and is_pentagonal(penta + p):
print(penta - p)
return
i += 1
pentagonals += [penta]
def euler45():
n = 2
while True:
x = n * (2 * n - 1)
if is_triangular(x) and is_pentagonal(x) and n != 143:
return x
n += 1
def main():
cnt = 0
for num in hexagonal_numbers():
if is_pentagonal(num): # Every hexagonal number is a triangular number
cnt += 1
if cnt == 3:
return num
def main():
# P(n+k)-P(n) = 3kn + P(k), P(n+k)-P(n) = P(m)
pm = 1
for m in count(1):
if not pm & 1: # both numbers must be even
pk = 1
for k in range(1, m):
if not (pm - pk) % (3 * k):
n = (pm - pk) // (3 * k)
if is_pentagonal(pm + n * (3 * n - 1)):
return pm
pk += 3 * k + 1
pm += 3 * m + 1
def main():
# P(n+k)-P(n) = 3kn + P(k), P(n+k)-P(n) = P(m)
pm = 1
for m in count(1):
if not pm & 1: # both numbers must be even
pk = 1
for k in range(1, m):
if not (pm - pk) % (3*k):
n = (pm - pk) // (3*k)
if is_pentagonal(pm + n*(3*n - 1)):
return pm
pk += 3*k + 1
pm += 3*m + 1
def num_type(n):
types = list()
if utils.is_triangle(n):
types.append('A')
if utils.is_square(n):
types.append('B')
if utils.is_pentagonal(n):
types.append('C')
if utils.is_hexagonal(n):
types.append('D')
if utils.is_heptagonal(n):
types.append('E')
if utils.is_octagonal(n):
types.append('F')
return types
def main():
i = 1
cnt = 0
while True:
triangular = i * (i + 1) // 2
if is_pentagonal(triangular) and is_hexagonal(triangular):
print(triangular)
cnt += 1
if cnt == 3:
return
i += 1
def test_is_pentagonal(self):
self.assertTrue(is_pentagonal(1))
self.assertFalse(is_pentagonal(2))
self.assertTrue(is_pentagonal(5))
self.assertFalse(is_pentagonal(15))
n = 10**15
p = nth_pentagonal(n)
self.assertTrue(is_pentagonal(p))
self.assertFalse(is_pentagonal(p + 2))
def test_is_pentagonal(self):
self.assertTrue(is_pentagonal(1))
self.assertFalse(is_pentagonal(2))
self.assertTrue(is_pentagonal(5))
self.assertFalse(is_pentagonal(15))
n = 10**15
p = nth_pentagonal(n)
self.assertTrue(is_pentagonal(p))
self.assertFalse(is_pentagonal(p + 2))
def criteria(n, m):
return (is_pentagonal(pentagonal(n) - pentagonal(m))
and is_pentagonal(pentagonal(n) + pentagonal(m)))
def criteria(n,m):
return (is_pentagonal(pentagonal(n)-pentagonal(m)) and
is_pentagonal(pentagonal(n)+pentagonal(m)))
import utils
def next_hexagonal(sequence):
n = len(sequence) + 1
return utils.nth_hexagonal(n)
hexagonals = utils.Sequence(next_hexagonal)
lower_bound = 40755
for num in hexagonals.iterate():
if utils.is_triangular(num) and utils.is_pentagonal(num):
print(num, 'is triangular, pentagonal, and hexagonal')
if num > lower_bound:
break