def gcd(a, b):
if a < b:
r = b % a
return gcd(a, r)
elif b == a:
return b
elif b == 0:
return a
else:
r = a % b
return gcd(b, r)
def f(n):
n2 = int(n**0.5)
s = g(n, 1, 1)
for i in range(1, n2+1):
for j in range(i, n2+1):
if gcd(i, j) != 1:
continue
s += g(n, i**2+j**2, (i+j))*(2 if i != j else 1)
return s
def a_b_n(k, D):
h1, k1 = 1, 0
a_i = int(D**0.5)
r = 1, 1, -a_i, 1
h2, k2 = a_i, 1
for i in range(k):
a, b, c, d = r
a, b, c, d = (a * b * d**2, a**2 * d**2 * D - c**2 * b**2,
-c * b**2 * d, a**2 * d**2 * D - c**2 * b**2)
d1 = gcd(abs(a), abs(b))
a //= d1
b //= d1
a_i = int(a * (D**0.5) / b + c / d)
c -= a_i * d
d2 = gcd(abs(c), abs(d))
c //= d2
d //= d2
r = a, b, c, d
h1, h2 = h2, a_i * h2 + h1
k1, k2 = k2, a_i * k2 + k1
return h2, k2
def fx(n, m):
d1 = gcd(n, m)
n0 = n
phin0 = phi2(n)
c = phi2(m) - phin0
if c % d1 != 0:
return 0
c //= d1
m //= d1
n //= d1
v, u = bezout(n, m)
k = (u*c) % m
return n0*k+phin0
def a_b(D):
h1, k1 = 1, 0
a_i = int(D ** 0.5)
r = 1, 1, -a_i, 1
h2, k2 = a_i, 1
while h2 ** 2 - D * k2 ** 2 != 1:
a, b, c, d = r
a2, b2 = a * b * d ** 2, a ** 2 * d ** 2 * D - c ** 2 * b ** 2
c2, d2 = -c * b ** 2 * d, a ** 2 * d ** 2 * D - c ** 2 * b ** 2
a, b, c, d = a2, b2, c2, d2
d1 = gcd(abs(a), abs(b))
a //= d1
b //= d1
a_i = int(a * (D ** 0.5) / b + c / d)
c -= a_i * d
d2 = gcd(abs(c), abs(d))
c //= d2
d //= d2
r = a, b, c, d
h1, h2 = h2, a_i * h2 + h1
k1, k2 = k2, a_i * k2 + k1
return h2, k2
def period(D):
h1, k1 = 1, 0
a_i = int(D ** 0.5)
r = 1, 1, -a_i, 1
h2, k2 = a_i, 1
L = []
while r not in L:
L.append(r)
a, b, c, d = r
a2, b2 = a * b * d ** 2, a ** 2 * d ** 2 * D - c ** 2 * b ** 2
c2, d2 = -c * b ** 2 * d, a ** 2 * d ** 2 * D - c ** 2 * b ** 2
a, b, c, d = a2, b2, c2, d2
d1 = gcd(abs(a), abs(b))
a //= d1
b //= d1
a_i = int(a * (D ** 0.5) / b + c / d)
c -= a_i * d
d2 = gcd(abs(c), abs(d))
c //= d2
d //= d2
r = a, b, c, d
h1, h2 = h2, a_i * h2 + h1
k1, k2 = k2, a_i * k2 + k1
return len(L) - L.index(r)
def problem_33():
L = []
for b in range(1, 9):
for c in range(b + 1, 10):
q, r = divmod(9 * b * c, 10 * b - c) # second case
if r == 0 and c < q < 10:
L.append((b, c))
q, r = divmod(9 * b * c, 10 * c - b) # first one
if r == 0 and 0 < q < b:
L.append((b, c))
num = 1
denominator = 1
for n, d in L:
num *= n
denominator *= d
delta = gcd(num, denominator)
print(denominator // delta) # sol = 100
# %% Problem 168
from Tools import gcd
s = 0
for d in range(2, 10):
for a in range(d, 10):
q = 10*d-1
qq = q // gcd(q, a)
ll = []
prod = 10
for n in range(2, 101):
if prod == d:
ll.append(n)
prod *= 10
prod %= qq
for n in ll:
b = ((10**(n-1)-d)//qq)*(a//gcd(q, a))
nb = int(str(b)+str(a))
s += int(nb)%10**5
s %= 10**5
s += 45*(11+111+1111+11111*96)
s %= 10**5
print(s) # 59206
def d(n):
k = k_max(n)
if terminating(k / gcd(k, n)):
return -n
return n
# %% Problem 329
from Tools import numpy_sieve, gcd
lp = numpy_sieve(500)
ok = "PPPPNNPPPNPPNPN"
L = [1] * 501
for k in range(15):
L2 = [0] * 501
L2[2] += L[1] * (2 if ok[k] == 'P' else 4)
for p in range(2, 500):
if p in lp:
pp, pn = 2, 1
else:
pp, pn = 1, 2
a = L[p] * (pp if ok[k] == 'P' else pn)
L2[p - 1] += a
L2[p + 1] += a
L2[499] += L[500] * (2 if ok[k] == 'P' else 4)
L = L2
s = 0
for a in L:
s += a
d = gcd(s, 500 * 6**15)
print(str(s // d) + "/" + str(
(500 * 6**15) // d)) # sol = 199740353/29386561536000
if a < b:
r = b % a
return gcd(a, r)
elif b == a:
return b
elif b == 0:
return a
else:
r = a % b
return gcd(b, r)
nb = 3
for d in range(9, 12001):
for n in range(d // 3 + 1, d // 2 + 1):
if gcd(d, n) == 1:
nb += 1
print(nb)
# %% Problem 74
l_fact = [1]
for i in range(1, 10):
l_fact.append(l_fact[-1] * i)
# print(l_fact)
def next_one(n):
s = 0
while n > 0:
s += l_fact[n % 10]