def solve():
c,m = 0,0
while c < N:
m += 1
for a in xrange(2,2*m+1):
if is_square(a*a + m*m):
c += (m + m + 2 - a)//2 if a > m+1 else a//2
return m,c
def gen_integral_almost_equilateral_triangles(max_x, limit):
for x in range(2, max_x):
a = 4 * x * x * x * x - (2 * x * x - (x - 1) * (x - 1)) * \
(2 * x * x - (x - 1) * (x - 1))
if a % 16 == 0:
if is_square(a / 16):
perimeter = 3 * x - 1
if perimeter <= limit:
yield perimeter
a = 4 * x * x * x * x - (2 * x * x - (x + 1) * (x + 1)) * \
(2 * x * x - (x + 1) * (x + 1))
if a % 16 == 0:
if is_square(a / 16):
perimeter = 3 * x + 1
if perimeter <= limit:
yield perimeter
def rectilinear_shape(population):
try:
pos = population.positions
except Exception, e:
LOGGER.warning(("Could not retrieve units positions for population "
"%s; assuming square shape."), population.label)
if not is_square(population.size):
raise TypeError(("The shape population %s is not square and could "
"neither be retreived nor guessed."), population.label)
dim1 = dim2 = int(math.sqrt(population.size))
def main():
cnt = 0
for a in count(1):
for s in range(2, 2 * a + 1):
if is_square(a * a + s * s):
cnt += min(s, a + 1) - (s + 1) // 2
if cnt > 1_000_000:
return a
def conjecture_correct(current_odd):
for i in xrange(1, current_odd - 1):
if not is_prime_c(i):
continue
r = current_odd - i
if not is_even(r):
continue
r /= 2
if is_square(r):
#print "%d = %d + 2 * %d" % (current_odd, i, r)
return True
return False
def continued_fraction(n):
if is_square(n): return [int(n**0.5)]
num = 0
denom = 1
rep = []
while True:
m = int((n**0.5 + num) / denom)
rep.append(m)
if m == 2 * int(n**0.5):
return rep
num, denom = denom * m - num, (n - (num - denom * m)**2) / denom
def decrypt(self, criptotext):
message = ""
for c in criptotext:
punto, x = c
s = (pow(punto[0], 3) + self.curve.A*punto[0] + self.curve.B) % self.curve.p
while not is_square(s):
s += self.curve.p
raiz = int(math.sqrt(s))
if punto[1]==1:
raiz *= -1
pt_desc = ec.scalar_multiplication((punto[0], raiz), self.s, self.curve)
valorFinal = (x * modinv(pt_desc[0]%self.curve.p, self.curve.p)) % self.p
message += self.alphabet[valorFinal]
return message
def convergents(number):
if is_square(number):
return number ** 0.5
m, d, a = 0, 1, int(number ** 0.5)
a0 = a
history = []
yield a
while a != 2 * a0 or (m, d, a) not in history:
history.append((m, d, a))
m = int(d * a - m)
d = int((number - m * m) / d)
a = int((a0 + m) / d)
yield a
def convergents(number):
if is_square(number):
return number**0.5
m, d, a = 0, 1, int(number**0.5)
a0 = a
history = []
yield a
while a != 2 * a0 or (m, d, a) not in history:
history.append((m, d, a))
m = int(d * a - m)
d = int((number - m * m) / d)
a = int((a0 + m) / d)
yield a
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 cal_len(n):
if utils.is_square(n):
return 0
h = dict()
cnt = 0
s = utils.FracWithSqrt(0, 1, 1, n)
while True:
cnt += 1
rem = s.minus_int(int(s.get_exact()))
if rem.get_val() not in h:
h[rem.get_val()] = cnt
else:
return cnt - h[rem.get_val()]
s = rem.reciprocal()
return None
def find(n):
if utils.is_square(n):
return None
s = utils.FracWithSqrt(0, 1, 1, n)
a_i = list()
while True:
a_i.append(int(s.get_exact()))
cur = utils.FracWithSqrt(a_i[-1], 0, 1, 0)
for a in a_i[::-1][1:]:
cur = cur.reciprocal().minus_int(-a)
x, _, y = cur.get_val()
if x * x - n * y * y == 1:
return x
rem = s.minus_int(a_i[-1])
s = rem.reciprocal()
return None
def __init__(self, init_params, xform, num_frames=12):
random.seed(RANDOM_SEED)
self.gen_id = str(uuid4())[:18]
self.xform = xform
self.num_frames = num_frames
self.init_params = init_params
assert utils.is_square(self.xform)
assert len(self.init_params) == len(self.xform)
print("calc params")
self.params = self.gen_params()
print("calc seeds")
self.seeds = self.gen_seeds()
print(f"calc frames {self.gen_id}")
self.frames = self.gen_frames()
def run():
"""
Solution: We observe that since a triangle number is given by x = n(n+1)/2,
8*x + 1 = 4n(n+1) + 1 = 4n^2 + 4n + 1 = (2n + 1)^2 which is a perfect
square.
So to test for whether a number is a triangle number, we test if 8x + 1 is
a perfect square.
"""
words = read_file("data/p042 words.txt")
count = 0
for word in words:
n = sum(map(lambda s: alphabet[s], word.lower()))
if is_square(8 * n + 1):
count += 1
return count
squares = defaultdict(list)
for n in range(1, 1000):
x = n**2
squares[generalize(str(x))].append(x)
words = defaultdict(list)
with open("./p098_words.txt") as f:
for word in f.read().split(","):
word = word[1:-1]
words["".join(sorted(word))].append(word)
words = {k: l for k, l in words.items() if len(l) > 1}
res = 0
for word_chain in words.values():
for a in word_chain:
for b in word_chain:
if a == b: continue
if generalize(a) in squares:
for n in squares[generalize(a)]:
if len(a) != len(str(n)): continue
mapping = {c: d for c, d in zip(a, str(n))}
m = int("".join(mapping[c] for c in b))
if is_square(m) and len(str(m)) == len(b):
res = max(res, n, m)
print res
def refutes(i):
for p in PRIMES:
if is_square((i - p) / 2):
return False
return True
# Odd period square roots
from utils import is_square
def continued_fraction(n):
m = 0
d = 1
a = a0 = int(n**0.5)
expansion = [a0]
while a != 2 * a0:
m = d * a - m
d = (n - m**2) / d
a = (a0 + m) / d
expansion.append(a)
return expansion
max_N = 10000
n = 0
for i in range(max_N + 1):
if is_square(i):
continue
e = continued_fraction(i)
if len(e) % 2 == 0:
n += 1
print n
def continous_fraction(sequence):
if len(sequence) == 1:
return sequence.pop()
else:
return Fraction(sequence.pop(0) + Fraction(1, continous_fraction(sequence)))
# VARIABLES
largest_x = 0
d_answer = 0
timer.start()
for D in range(2, 1001):
# skip perfect squares
if is_square(D): continue
# init variables
it = convergents(D)
expansion = []
expansion.append(next(it))
# compute continous fraction and test it against Pell's equation.
# keep expanding until a solution is found
while True:
frac = continous_fraction(expansion.copy())
if frac.numerator ** 2 - D * frac.denominator ** 2 == 1:
if frac.numerator > largest_x:
largest_x = frac.numerator
d_answer = D
break
def test_is_square(self):
self.assertTrue(is_square(4))
self.assertFalse(is_square(8))
x = 12345678987654321234567**2
self.assertTrue(is_square(x))
self.assertFalse(is_square(x + 2))
from utils import is_square
from decimal import Decimal, getcontext
getcontext().prec = 120
res = 0
for n in range(1, 101):
if is_square(n): continue
root = Decimal(n)**Decimal(0.5)
res += sum(int(d) for d in str(root).replace(".", "")[:100])
print(res)
sqr_list = set()
i = 1
while i * i < RANGE:
sqr_list.add(i * i)
i += 1
ans = 0
for p in pairs:
for s in sqr_list:
if len(p[0]) == len(str(s)):
c2d = dict()
d2c = dict()
match = True
for c, d in zip(p[0], str(s)):
if c not in c2d:
c2d[c] = d
elif c2d[c] != d:
match = False
break
if d not in d2c:
d2c[d] = c
elif d2c[d] != c:
match = False
break
if match:
n2 = int(''.join([c2d[c] for c in p[1]]))
if len(str(s)) == len(str(n2)) and utils.is_square(n2):
ans = max(ans, max(s, n2))
print(ans)
from fractions import gcd
from utils import is_square
limit = 10**12
res = set()
for a in range(2, 10**4):
for b in range(1, min(a, limit / a**3)):
if gcd(a, b) > 1: continue
for c in range(1, int((limit // a**3 // b)**0.5)):
n = a**3 * b * c * c + c * b**2
if is_square(n):
res.add(n)
print sum(res)
from utils import is_square
limit = 10**6
cur = 0
m = 2
while cur <= limit:
m += 1
for ab in range(3, 2*m):
if is_square(ab*ab + m*m):
cur += min(ab, m+1) - (ab+1)//2
print m
"""
Created on Mon Apr 24 15:59:39 2018
@author: lamwa
"""
import utils
SUP = 10**12
def f(a, x, y):
return a * a * x * x * x * y + a * y * y
ans_set = set()
x = 2
while f(1, x, 1) < SUP:
y = 1
while y < x and f(1, x, y) < SUP:
a = 1
while f(a, x, y) < SUP:
ns = f(a, x, y)
if utils.is_square(ns):
ans_set.add(ns)
a += 1
y += 1
x += 1
ans = sum(ans_set)
print(ans)
from utils import is_square
def combs(x, y, z):
return (x + y, x - y, x + z, x - z, y + z, y - z)
l = 1000
for i in range(1, l):
e = i * i
for j in range(i + 1, l):
c = j * j
if not is_square(c - e): continue
off = 2 - i % 2
for k in range(j + off, int((c + e)**0.5) + 1, 2):
a = k * k
if all(is_square(x) for x in (a - c, a - e, c - e)):
print(a + c + e) / 2
if len(sequence) == 1:
return sequence.pop()
else:
return Fraction(
sequence.pop(0) + Fraction(1, continous_fraction(sequence)))
# VARIABLES
largest_x = 0
d_answer = 0
timer.start()
for D in range(2, 1001):
# skip perfect squares
if is_square(D): continue
# init variables
it = convergents(D)
expansion = []
expansion.append(next(it))
# compute continous fraction and test it against Pell's equation.
# keep expanding until a solution is found
while True:
frac = continous_fraction(expansion.copy())
if frac.numerator**2 - D * frac.denominator**2 == 1:
if frac.numerator > largest_x:
largest_x = frac.numerator
d_answer = D
break
def test_is_square(self):
self.assertTrue(is_square(4))
self.assertFalse(is_square(8))
x = 12345678987654321234567 ** 2
self.assertTrue(is_square(x))
self.assertFalse(is_square(x + 2))
def fracs(x):
first, l = get_period(x)
n = 0
d = 1
for a in period_to_generator(l):
n += a * d
g = gcd(d, n)
d, n = n / g, d / g
yield first * d + n, d
def find_solution(D):
for x, y in fracs(D):
if x ** 2 - D * y ** 2 == 1:
return x, y
m = 0
best = 0
for D in range(1001):
if not is_square(D):
x, y = find_solution(D)
print D, x, y
if x > m:
best = D
m = x
print m, best
#
# print find_solution(61)
from utils import is_square
pos = 15
res = []
for l in range(900000):
for add in (2, 4):
k = (l + (5 * l * l + add)**0.5) / 2.0
n = k * l
if n % 1 == 0: res.append(int(n))
if l > 0:
for sub in (1, 2):
k = 2 * l + (5 * l * l - sub)**0.5
n = k * l
if n % 1 == 0: res.append(int(n))
res = [n for n in sorted(set(res)) if is_square((n + 1)**2 + 4 * n**2)]
print res[pos]
# -*- coding: utf-8 -*-
"""
Created on Tue Mar 27 00:26:01 2018
@author: lamwa
"""
from decimal import getcontext, Decimal
import utils
RANGE = 100
getcontext().prec = 105
ans = 0
for n in range(2, RANGE):
if utils.is_square(n):
continue
sqrt = str(Decimal(n) ** Decimal('0.5'))
sqrt = sqrt.replace('.', '')
ans += sum(int(c) for c in sqrt[ : 100])
print(ans)
def test_is_square(self):
cases = {1:True, 2:False, 3:False, 4:True}
for case in cases:
result = utils.is_square(case)
if result != cases[case]:
self.fail("is_square failed for %s: it got %s" % (case, result))
# -*- coding: utf-8 -*-
"""
Created on Fri Apr 20 19:54:20 2018
@author: lamwa
"""
import utils
import math
a = [8, 136, 2448]
for i in range(4, 13):
a.append(17 * a[-1] + 17 * a[-2] - a[-3])
ans = 0
for k in a:
h1 = 2 * k - 1
h2 = 2 * k + 1
if utils.is_square(k**2 + h1**2):
ans += round(math.sqrt(k**2 + h1**2))
else:
ans += round(math.sqrt(k**2 + h2**2))
print(ans)
Created on Mon Apr 24 16:56:01 2018
@author: lamwa
"""
import utils
done = False
i = 4
while not done:
a = i * i
j = i - 1
while not done and j > 0:
c = j * j
f = a - c
if not utils.is_square(f):
j -= 1
continue
k = j - 1
while not done and k > 0:
e = k * k
d = a - e
b = c - e
if not utils.is_square(d) or not utils.is_square(b):
k -= 1
continue
if (a + c + e) % 2 == 0:
ans = (a + c + e) // 2
done = True
k -= 1
j -= 1
# -*- coding: utf-8 -*-
"""
Created on Wed Mar 28 14:07:27 2018
@author: lamwa
"""
import utils
n = 3
cnt = 0
while True:
for k in range(2, n * 2 + 1):
if utils.is_square(k * k + n * n):
l = max(k - n, 1)
r = k // 2
cnt += r - l + 1
#print(r - l + 1)
if cnt > 1000000:
print(n)
break
n += 1
def solve_eq_for(d):
""" Solve for minimum integer x: x**2 - d*y**2 = 1 """
x0, y0 = 0, 1
x1, y1 = 1, 0
n = d
num = 0
denom = 1
while True:
m = int((n**0.5 + num) / denom)
x, y = m * x1 + x0, m * y1 + y0
x0, y0 = x1, y1
x1, y1 = x, y
num, denom = denom * m - num, (n - (num - denom * m)**2) / denom
if x * x - d * y * y == 1:
return x
return -1
res = None
max_x = 0
for d in range(1, 1001):
if is_square(d): continue
x = solve_eq_for(d)
if x > max_x:
max_x = x
res = d
print(res)
l = len(expansion) - 1
for i in range(n - 1, -1, -1):
frac = 1 / (a1[i % l] + frac)
return a0 + frac
def solve_diophantine_equation(D):
n = 0
expansion = continued_fraction(D)
while True:
frac = approximation(expansion, n)
x = frac.numerator
y = frac.denominator
if x**2 - D * y**2 == 1:
return x, y
break
n += 1
x_max = 0
D_max = 0
for D in range(1000):
if is_square(D):
continue
x, y = solve_diophantine_equation(D)
if x > x_max:
x_max = x
D_max = D
print D_max
# -*- coding: utf-8 -*-
"""
Created on Wed Mar 28 17:06:55 2018
@author: lamwa
"""
import utils
RANGE = 1000000000
f1 = list()
for i in range(1, 1000):
if utils.is_square((3 * i + 1) * (i + 1)):
f1.append(i)
while True:
new = 14 * f1[-1] - f1[-2] + 8
if 6 * new + 2 <= RANGE:
f1.append(new)
else:
break
f2 = list()
for i in range(2, 1000):
if utils.is_square((3 * i - 1) * (i - 1)):
f2.append(i)
while True:
# -*- coding: utf-8 -*-
"""
Created on Mon Apr 25 10:56:39 2018
@author: lamwa
"""
import utils
import math
N = 120000
mapping = [set() for _ in range(N)]
for p in range(1, N):
for q in range(1, min(p, N - p) + 1):
b = p * p + q * q + p * q
if utils.is_square(b):
mapping[p].add(q)
ans_set = set()
for p in range(1, N):
for q in mapping[p]:
for r in mapping[q]:
if p + q + r <= N and r in mapping[p]:
ans_set.add(p + q + r)
ans = sum(ans_set)
print(ans)