def main():
N = 10**8
sq_coprime_sum = np.zeros((N + 1, ), dtype=np.uint32)
result_real = np.zeros((N + 1, ), dtype=np.uint32)
result_imag = np.zeros((N + 1, ), dtype=np.uint32)
print("successfully allocated")
sq_coprime_sum[2] = 2 # 1*1 + 1*1 = 2
for i in range(1, int(sqrt(N)) + 1):
for j in range(i + 1, int(sqrt(N)) + 1):
squared_sum = i * i + j * j
if squared_sum > N:
break
if gcd(i, j) == 1:
sq_coprime_sum[squared_sum] += 2 * (i + j)
# print(i, j, squared_sum)
# print(sq_coprime_sum)
print("sq_coprime_sum computed by", time() - starting_time)
for i in range(1, N + 1):
for j in range(i, N + 1, i):
result_real[j] += i
print("real part computed by", time() - starting_time)
for i in range(1, N + 1):
if sq_coprime_sum[i] == 0:
continue
for j in range(i, N + 1, i):
result_imag[j] += sq_coprime_sum[i] * result_real[j // i]
print("complex part computed by", time() - starting_time)
# print(list(zip(result_real, result_imag)))
print(sum(map(int, result_real)) + sum(map(int, result_imag)))
def main():
N = 10 ** 8
sq_coprime_sum = np.zeros((N + 1,), dtype=np.uint32)
result_real = np.zeros((N + 1,), dtype=np.uint32)
result_imag = np.zeros((N + 1,), dtype=np.uint32)
print("successfully allocated")
sq_coprime_sum[2] = 2 # 1*1 + 1*1 = 2
for i in range(1, int(sqrt(N)) + 1):
for j in range(i + 1, int(sqrt(N)) + 1):
squared_sum = i * i + j * j
if squared_sum > N:
break
if gcd(i, j) == 1:
sq_coprime_sum[squared_sum] += 2 * (i + j)
# print(i, j, squared_sum)
# print(sq_coprime_sum)
print("sq_coprime_sum computed by", time() - starting_time)
for i in range(1, N + 1):
for j in range(i, N + 1, i):
result_real[j] += i
print("real part computed by", time() - starting_time)
for i in range(1, N + 1):
if sq_coprime_sum[i] == 0:
continue
for j in range(i, N + 1, i):
result_imag[j] += sq_coprime_sum[i] * result_real[j // i]
print("complex part computed by", time() - starting_time)
# print(list(zip(result_real, result_imag)))
print(sum(map(int, result_real)) + sum(map(int, result_imag)))
def terminate(numerator, denominator):
left = denominator // gcd(numerator, denominator)
while left % 2 == 0:
left //= 2
while left % 5 == 0:
left //= 5
if left == 1: # terminate
return -1
else: # non-terminate
return 1
def parametrized():
prog_sq = []
for b in range(2, CBRT_N):
for a in range(1, b):
if a * a * a * b * b > N:
break
if gcd(a, b) != 1:
continue
for c in range(1, SQRT_N):
candidate = c * a * a + c * c * a * b * b * b
if candidate > N:
break
if is_square(candidate):
prog_sq.append(candidate)
print(candidate, c * a * b)
print(sum(prog_sq))
def parametrized():
prog_sq = []
for b in range(2, CBRT_N):
for a in range(1, b):
if a * a * a * b * b > N:
break
if gcd(a, b) != 1:
continue
for c in range(1, SQRT_N):
candidate = c * a * a + c * c * a * b * b * b
if candidate > N:
break
if is_square(candidate):
prog_sq.append(candidate)
print(candidate, c * a * b)
print(sum(prog_sq))
def _next_continued_fraction(x, y, z=1):
"""
(√x + y) reciprocal z * (√x - y) z * (√x - y)
-------- =============> --------------------- = ------------
z (√x + y) * (√x - y) x - y**2
"""
assert x > y * y
int_part = int(z * (sqrt(x) - y) / (x - y**2))
common_divisor = gcd(z, x - y**2)
denominator = (x - y**2) // common_divisor
y = -y - int_part * denominator
assert z == common_divisor, \
'x={}, y={}, z={}'.format(x, y, z)
return int_part, x, y, denominator
def _next_continued_fraction(x, y, z=1):
"""
(√x + y) reciprocal z * (√x - y) z * (√x - y)
-------- =============> --------------------- = ------------
z (√x + y) * (√x - y) x - y**2
"""
assert x > y * y
int_part = int(z * (sqrt(x) - y) / (x - y ** 2))
common_divisor = gcd(z, x - y ** 2)
denominator = (x - y ** 2) // common_divisor
y = -y - int_part * denominator
assert z == common_divisor, \
'x={}, y={}, z={}'.format(x, y, z)
return int_part, x, y, denominator
def least_common_multipler(S):
lcm = [1]
for s in range(2, S + 1):
lcm.append(lcm[-1] * s // gcd(lcm[-1], s))
return lcm
# project euler 141: Investigating progressive numbers,
# n, which are also square
"""
perfect_sq = d * q + r
d * d == q * r
perfect_sq = r + d ** 3 / r (r < d)
3
2 d
n = r + -----
r
parametrize
gcd(a, b) = 1, b > a
2
r = c * a
d = c * a * b
2 2 2 3
n = c * a + c * a * b
"""
from __future__ import division
from math import sqrt
from mathutil import gcd
N = 10**10 # not feasible for real problem size, see cpp version
SQRT_N = int(sqrt(N))
CBRT_N = int(N**(1 / 3))
# project euler 141: Investigating progressive numbers,
# n, which are also square
"""
perfect_sq = d * q + r
d * d == q * r
perfect_sq = r + d ** 3 / r (r < d)
3
2 d
n = r + -----
r
parametrize
gcd(a, b) = 1, b > a
2
r = c * a
d = c * a * b
2 2 2 3
n = c * a + c * a * b
"""
from __future__ import division
from math import sqrt
from mathutil import gcd
N = 10 ** 10 # not feasible for real problem size, see cpp version
