def find_well_distr(s):
def find_distrib(s):
if s == "":
return [dict()]
l = []
for dictio in find_distrib(s[1:]):
if s[0] in dictio:
l.append(dictio)
else:
for i in range(10):
if i not in dictio.values():
d = dictio.copy()
d[s[0]] = i
l.append(d)
return l
ll = find_distrib(s)
l_well = []
for dictio in ll:
if dictio[s[0]] == 0:
continue
l = [str(dictio[lettre]) for lettre in s]
nb = int("".join(l))
if is_square(nb):
l_well.append(dictio)
return l_well
def c_b(k, b0=0):
b = b0 + 1
while not is_square(4 * k + 5 * b**2):
b += 1
if b >= (k / 5)**0.5:
return None
return b, int((4 * k + 5 * b**2)**0.5)
def problem_44():
def pent(n):
return (n * (3 * n - 1)) // 2
a = 1
while True:
d = pent(a)
a += 1
dd = 2 * d
i = 1
ok = False
while i * (3 * (i + 2) - 1) <= dd:
if dd % i == 0:
j = dd // i
if j % 3 != 2:
i += 1
continue
i2 = (j + 1) // 3
if i2 < i:
i += 1
continue
if (i + i2) % 2 != 0:
i += 1
continue
k = (i + i2) // 2
s = 3 * k**2 - k - d
delta = 1 + 24 * s
if is_square(delta) and int(delta**0.5 + 1) % 3 == 0:
ok = True
print(d)
break
i += 1
if ok:
break
def problem_66():
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
x_max = 9
D_max = 5
for D in range(10, 1001):
if not is_square(D):
x = a_b(D)[0]
if x > x_max:
D_max = D
x_max = x
print(D_max)
def problem_64():
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)
c = 0
for N in range(2, 10001):
if not is_square(N) and period(N) % 2 == 1:
c += 1
print(c)
def f(n):
nb = 0
for x in range(n // 2 + 1, int((n/2)*(1+2**0.5))+1):
c_2 = n ** 2 - 4 * x ** 2 + 4 * n * x
if is_square(c_2):
nb += 1
if nb > 105:
return 500
return nb * 4
def next1(b):
from Tools import is_square
while not is_square(1 + 5 * b**2):
b += 1
return int(sqrt(1 + 5 * b**2) + 0.05), b
def next_x(n, x=0):
x += 1
while not is_square(2 * n**2 - (2 * x - n)**2) and x <= n:
x += 1
return x if x <= n else 0
def nb(a, b, c, d):
s = 0
s += dic[(a, b)]
s += dic[(b, c)]
s += dic[(c, d)]
s += dic[(d, a)]
s += a + b + c + d - 3
return s
ll = []
for aa in range(1, m + 1):
for bb in range(aa + 1, m + 1):
for cc in range(aa + 1, m + 1):
for dd in range(bb, m + 1):
if is_square(nb(aa, bb, cc, dd)):
ll.append((aa, bb, cc, dd))
n_tot = 0
for aa, bb, cc, dd in ll:
if bb == dd:
n_tot += 1
# print(aa, bb, cc, dd, "ll", 1)
else:
n_tot += 2
# print(aa, bb, cc, dd, "ll", 2)
n_tot *= 4
ll2 = []
for aa in range(1, m + 1):
bb = aa
# %% Problem 141
from Tools import is_square
s = set()
for b in range(1, 10**6):
for p in range(1, 10**6):
if p**2 * b >= 10**6:
break
a = p + 1
c = a**3 * b**2 * p + p**2 * b
while c <= 10**12:
if is_square(c):
s.add(c)
a += 1
c = a**3 * b**2 * p + p**2 * b
print(sum(s)) # sol = 878454337159
# Todo: Too long ?
if abs(nb_ - nn) < mm:
mini = m, n
mm = abs(nb_ - nn)
print(mini[0] * mini[1]) # sol = 2772
# %% Problem 86
from math import sqrt
from Tools import is_square
c = 0
n = 0
while n < 10**6:
c += 1
for d in range(c + 1, int(c * 5**0.5 + 0.1) + 1):
if is_square(d**2 - c**2):
m = int((d**2 - c**2)**0.5 + 0.1)
if m <= c + 1:
n += m // 2
else:
n += (2 * c - m) // 2 + 1
# %% Problem 87
from Tools import numpy_sieve
l = [False for i in range(50000000)]
l_pp = list(numpy_sieve(7070))
for i in l_pp:
for j in l_pp: