def euler49():
for p in sieve.primerange(1488, 9999):
for i in range(1, 9999 - p):
if is_permutation(p, p + i) and is_permutation(
p, p + 2 * i) and isprime(p + i) and isprime(p + 2 * i):
print(str(p) + str(p + i) + str(p + 2 * i))
return
def prime_permutations():
"""Input: None.
Output: The concatenation of three 4-digit numbers
that are prime permutations."""
ps = primes(10000)
ps = set(ps[ps > 1487])
for p in ps:
if p + 3330 in ps and p + 6660 in ps:
if is_permutation(p, p + 3330) and is_permutation(p, p + 6660):
return cat([p, p + 3330, p + 6660])
def permuted_multiples():
"""Input: None.
Output: The smallest positive integer x such that 2x,
3x, 4x, 5x, and 6x are permutations of each other."""
n = 125874
while True:
n += 1
if is_permutation(n, 2 * n):
if is_permutation(n, 3 * n):
if is_permutation(n, 4 * n):
if is_permutation(n, 5 * n):
if is_permutation(n, 6 * n):
return n
def permuted_multiples():
"""Input: None.
Output: The smallest positive integer x such that 2x,
3x, 4x, 5x, and 6x are permutations of each other."""
n = 125874
while True:
n += 1
if is_permutation(n, 2 * n):
if is_permutation(n, 3 * n):
if is_permutation(n, 4 * n):
if is_permutation(n, 5 * n):
if is_permutation(n, 6 * n):
return n
def main():
n = 3
num_cube_permutations = 5
is_found = False
while not is_found:
N = euler.cubic_number(n)
num_digits = len(str(N))
# generate all the cubic numbers with num_digits digits
cubes = []
while len(str(N)) <= num_digits:
cubes.append(N)
n += 1
N = euler.cubic_number(n)
# make a list of lists of permutations
perms = []
while cubes:
c = cubes[0]
p = [c]
p.extend([d for d in cubes[1:] if euler.is_permutation(c, d)])
for x in p:
cubes.remove(x)
perms.append(p)
perms = [p for p in perms if len(p) == num_cube_permutations]
if perms:
is_found = True
print(perms)
print(min(perms[0]))
def euler62():
cubes = SortedSet(i**3 for i in range(10000))
for i1, c1 in enumerate(cubes):
count = 1
for c2 in cubes[i1 + 1:]:
if is_permutation(c1, c2): count += 1
if count == 5:
print(c1)
return
def euler62():
cubes = SortedSet(i**3 for i in range(10000))
for i1, c1 in enumerate(cubes):
count = 1
for c2 in cubes[i1+1:]:
if is_permutation(c1, c2): count += 1
if count == 5:
print(c1)
return
def find_cubic_permutations(all_cubes, number):
# goes through the cubes, and finds those that are are digit permutation of number.
# to avoid having to go through all of them, we stop once we reach a cube whose
# length is greater than our number. Assumes all_cubes is sorted
cubic_permutations = []
max_length = len(str(number))
for x in all_cubes:
if len(str(x)) > max_length:
return cubic_permutations
if is_permutation(x, number):
cubic_permutations.append(x)
return cubic_permutations
def slow_solve():
min_ratio = 999999999
for x in primes[200:LOWER_BOUND]:
for y in primes:
n = x*y
if n > UPPER_BOUND:
break
p = phi(n)
if is_permutation(n, p):
ratio = n/p
if ratio < min_ratio:
min_ratio = ratio
print("(x,y)=({},{}), n={}, phi(n)={}, ratio={}".format(x,y,n,p, n/p))
def main():
N = 10000000
min_ratio = N
min_n = 2
for n in range(2, N):
t = euler.totient(n)
if euler.is_permutation(n, t):
ratio = n / t
if ratio < min_ratio:
min_ratio = ratio
min_n = n
print(min_n)
def euler70():
result = 1
min_ratio = 2
limit = 10000000
lowerbound, upperbound = int(0.7 * (limit)**0.5), int(1.3 * (limit)**0.5)
for p in sieve.primerange(lowerbound, upperbound):
for q in sieve.primerange(p + 1, upperbound):
n = p * q
if n > limit:
break
phi = (p - 1) * (q - 1)
ratio = n / phi
if is_permutation(n, int(phi)) and min_ratio > ratio:
result, min_ratio = n, ratio
print(result)
def fast_solve():
# find out where to start
sqrt_index = 0
for i in range(len(primes)):
if primes[i] > SQRT_UPPER:
sqrt_index = i
break
x_index = sqrt_index
while x_index > 0:
print("Trying with x at index {} with value {}".format(x_index, primes[x_index]))
y_index = sqrt_index
while y_index < len(primes):
n = primes[x_index] * primes[y_index]
if n > UPPER_BOUND:
break
p = phi(n)
if is_permutation(n, p):
print("n={}, phi(n)={}, ratio={}".format(n,p, n/p))
return
y_index += 1
x_index -= 1
# coding=utf-8
'''
Problem 52
12 September 2003
It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.
Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits.
'''
import euler
for i in range(1, 10**6):
valid = True
for j in range(2, 7):
if not euler.is_permutation(i, i*j):
valid = False
break
if (valid):
print i
break
def euler52():
print(next(i for i in count(start=99999, step=9) if is_permutation(i, 2*i, 3*i, 4*i, 5*i, 6*i)))
def euler52():
print(
next(i for i in count(start=99999, step=9)
if is_permutation(i, 2 * i, 3 * i, 4 * i, 5 * i, 6 * i)))
# coding=utf-8
'''
Problem 52
12 September 2003
It can be seen that the number, 125874, and its double, 251748, contain exactly the same digits, but in a different order.
Find the smallest positive integer, x, such that 2x, 3x, 4x, 5x, and 6x, contain the same digits.
'''
import euler
for i in range(1, 10**6):
valid = True
for j in range(2, 7):
if not euler.is_permutation(i, i * j):
valid = False
break
if (valid):
print i
break
def euler49():
for p in sieve.primerange(1488, 9999):
for i in range(1, 9999-p):
if is_permutation(p, p+i) and is_permutation(p, p+2*i) and isprime(p+i) and isprime(p+2*i):
print(str(p)+str(p+i)+str(p+2*i))
return