def is_circular(n):
if is_prime(n):
r = rotate(n)
while (r != n):
if not is_prime(r):
return False
r = rotate(r)
return True
else:
return False
def find_nth_prime(n):
cur = 2 # init: 第一个prime是2
cnt = 1 # init
while cnt <= n:
if is_prime(cur) and cnt == n: # 退出条件先写
return cur
if is_prime(cur):
cnt += 1
cur += 1
return -1
def is_trunc(p): single_primes = [2,3,5,7] if not ((p % 10) in single_primes): return False if not ((p / 10 ** (len(str(p))-1)) in single_primes): return False left = p / 10 while left > 0: if not euler_util.is_prime(left): return False left /= 10 right = p % (10 ** (len(str(p))-1)) while right > 0: if not euler_util.is_prime(right): return False right = right % (10 ** (len(str(right))-1)) return True
def is_trunc(p):
single_primes = [2, 3, 5, 7]
if not ((p % 10) in single_primes): return False
if not ((p / 10**(len(str(p)) - 1)) in single_primes): return False
left = p / 10
while left > 0:
if not euler_util.is_prime(left): return False
left /= 10
right = p % (10**(len(str(p)) - 1))
while right > 0:
if not euler_util.is_prime(right): return False
right = right % (10**(len(str(right)) - 1))
return True
def find_prime_under_ceiling_sum(ceiling):
cur = 0
res = 0
while cur < ceiling:
if is_prime(cur):
res += cur
cur += 1
return res
def is_truncatable(n, leftToRight): if is_prime(n): if len(str(n)) == 1: return True if leftToRight: return is_truncatable(int(str(n)[1:]), True) else: return is_truncatable(int(str(n)[:-1]), False) else: return False
def euler41():
"""http://projecteuler.net/index.php?section=problems&id=41
What is the largest n-digit pandigital prime?"""
primes = []
# note: sum(1..8) % 3 == sum(1..9) % 3 == 0, so 7 digits is the max
for perm in permutations("1234567"):
val = int(''.join(perm))
if is_prime(val):
primes.append(val)
return max(primes)
def euler58():
"""http://projecteuler.net/problem=58
Investigate the number of primes that lie on the diagonals of the spiral
grid."""
value, side, count = 1, 2, 0
primes, total = 0, 0
while True:
primes, total = primes + is_prime(value), total + 1
if count == 0:
if primes > 0 and float(primes)/total < 0.1:
return side - 1
count, value = count + 1, value + side
if count == 4:
side, count = side + 2, 0
def euler41():
"""
We shall say that an n-digit number is pandigital if it makes use of all the
digits 1 to n exactly once. For example, 2143 is a 4-digit pandigital and is
also prime.
What is the largest n-digit pandigital prime that exists?
"""
# last digit can't be 5 or even
last_digits = [1,3,7,9]
# all pandigitals of length 8 and 9 are divisible by 3 since the sum of their
# digits (36 and 45) are divisible by 3 so start with length 7
cands = ([int("".join(p)) for p in permutations("1234567")])[::-1]
for cand in cands:
if not cand % 10 in last_digits: continue
if euler_util.is_prime(cand): return cand
def euler58():
"""
Starting with 1 and spiralling anticlockwise in the following way, a square
spiral with side length 7 is formed.
37 36 35 34 33 32 31
38 17 16 15 14 13 30
39 18 5 4 3 12 29
40 19 6 1 2 11 28
41 20 7 8 9 10 27
42 21 22 23 24 25 26
43 44 45 46 47 48 49
It is interesting to note that the odd squares lie along the bottom right
diagonal, but what is more interesting is that 8 out of the 13 numbers lying
along both diagonals are prime; that is, a ratio of 8/13 ~= 62%.
If one complete new layer is wrapped around the spiral above, a square spiral
with side length 9 will be formed. If this process is continued, what is the
side length of the square spiral for which the ratio of primes along both
diagonals first falls below 10%?
"""
seed = 1
side_length = 1
ratio = 1.0
primes = 0
total = 1.0
while ratio >= 0.1:
side_length += 2
inc = side_length - 1
for i in range(1, 5):
cand = seed + (i * inc)
if euler_util.is_prime(cand): primes += 1
total += 4
seed += inc * 4
ratio = primes / total
return side_length
def euler58(): """ Starting with 1 and spiralling anticlockwise in the following way, a square spiral with side length 7 is formed. 37 36 35 34 33 32 31 38 17 16 15 14 13 30 39 18 5 4 3 12 29 40 19 6 1 2 11 28 41 20 7 8 9 10 27 42 21 22 23 24 25 26 43 44 45 46 47 48 49 It is interesting to note that the odd squares lie along the bottom right diagonal, but what is more interesting is that 8 out of the 13 numbers lying along both diagonals are prime; that is, a ratio of 8/13 ~= 62%. If one complete new layer is wrapped around the spiral above, a square spiral with side length 9 will be formed. If this process is continued, what is the side length of the square spiral for which the ratio of primes along both diagonals first falls below 10%? """ seed = 1 side_length = 1 ratio = 1.0 primes = 0 total = 1.0 while ratio >= 0.1: side_length += 2 inc = side_length - 1 for i in range(1, 5): cand = seed + (i * inc) if euler_util.is_prime(cand): primes += 1 total += 4 seed += inc * 4 ratio = primes / total return side_length
def euler131(upper_bound=10**6):
"""http://projecteuler.net/problem=131
There are some prime values, p, for which there exists a positive integer,
n, such that the expression n**3 + pn**2 is a perfect cube.
For example, when p = 19, 83 + 82*19 = 123.
What is perhaps most surprising is that for each prime with this property
the value of n is unique, and there are only four such primes below
one-hundred.
How many primes below one million have this remarkable property?
"""
# Since n**3 + p*n**2 = (n+p)*n**2 with p prime, n and n+p are both
# perfect cubes. So, this is just an alternative formulation for Cuban
# primes: http://oeis.org/A002407
#
# These are also identical to the centered hexagonal numbers that are
# also prime.
hexes = [3*n*(n+1)+1 for n in xrange(int(upper_bound**0.5))]
hexes = [n for n in hexes if n < upper_bound]
return len([1 for n in hexes if is_prime(n)])
from euler_util import primes_sieve, is_prime, is_odd
from math import sqrt
def is_sum_of_a_prime_and_twice_a_square(number):
return len([i for i in primes_sieve(number - 2) if sqrt((number - i)/2.0).is_integer()]) > 0
print [i for i in range(1, 10000) if is_odd(i) and not is_prime(i) and not is_sum_of_a_prime_and_twice_a_square(i)]
def concat_prime(p1, p2):
if not is_prime(int(str(p1)+str(p2))): return False
if not is_prime(int(str(p2)+str(p1))): return False
return True
def primos_entre(i, f):
return [i for i in range(i, f + 1) if is_prime(i)]
from euler_util import is_prime def is_truncatable(n, leftToRight): if is_prime(n): if len(str(n)) == 1: return True if leftToRight: return is_truncatable(int(str(n)[1:]), True) else: return is_truncatable(int(str(n)[:-1]), False) else: return False if __name__ == "__main__": p = 7 s = 0 n = 0 target = 11 while n < target: p += 1 while not is_prime(p): p += 1 if is_truncatable(p, True) and is_truncatable(p, False): s += p n += 1 print s
def divisores_primos(numero):
return [i for i in divisores(numero) if is_prime(i)]
from euler_util import is_prime
def completar_ceros(numero):
temp = '0000'+str(numero)
return temp[len(temp)-4:]
def ordernar_caracteres(string):
return ''.join(sorted(string))
p = [i for i in range(3340)][1:]
s = map(lambda x : x+3330,p)
t = map(lambda x : x+3330,s)
sp = map(lambda x : ordernar_caracteres(completar_ceros(x)), p)
ss = map(lambda x : ordernar_caracteres(completar_ceros(x)), s)
st = map(lambda x : ordernar_caracteres(completar_ceros(x)), t)
for i in range(3339):
if sp[i]==ss[i] and ss[i]==st[i] and is_prime(i+1)and is_prime(i+1+3330) and is_prime(i+1+3330*2):
print i+1, i+1+3330, i+1+3330*2, sp[i], ss[i], st[i]
from euler_util import is_prime
def completar_ceros(numero):
temp = '0000' + str(numero)
return temp[len(temp) - 4:]
def ordernar_caracteres(string):
return ''.join(sorted(string))
p = [i for i in range(3340)][1:]
s = map(lambda x: x + 3330, p)
t = map(lambda x: x + 3330, s)
sp = map(lambda x: ordernar_caracteres(completar_ceros(x)), p)
ss = map(lambda x: ordernar_caracteres(completar_ceros(x)), s)
st = map(lambda x: ordernar_caracteres(completar_ceros(x)), t)
for i in range(3339):
if sp[i] == ss[i] and ss[i] == st[i] and is_prime(i + 1) and is_prime(
i + 1 + 3330) and is_prime(i + 1 + 3330 * 2):
print i + 1, i + 1 + 3330, i + 1 + 3330 * 2, sp[i], ss[i], st[i]
from euler_util import is_prime
def primos_entre(i,f):
return [i for i in range(i,f+1) if is_prime(i)]
#b = primos_entre(2, 1000)
#c = map(lambda x : x*-1,b)
if __name__ == '__main__':
max = 0
for a in range(-999,999+1):
for b in primos_entre(-1000,1000):
n = 1
while is_prime(n**2+a*n+b):
n= n +1
if n > max:
maxa =a
maxb=b
max = n
print maxa, maxb, maxa * maxb
from euler_util import is_prime
def primos_entre(i, f):
return [i for i in range(i, f + 1) if is_prime(i)]
#b = primos_entre(2, 1000)
#c = map(lambda x : x*-1,b)
if __name__ == '__main__':
max = 0
for a in range(-999, 999 + 1):
for b in primos_entre(-1000, 1000):
n = 1
while is_prime(n**2 + a * n + b):
n = n + 1
if n > max:
maxa = a
maxb = b
max = n
print maxa, maxb, maxa * maxb
from euler_util import primes_sieve, is_prime, is_odd
from math import sqrt
def is_sum_of_a_prime_and_twice_a_square(number):
return len([
i for i in primes_sieve(number - 2)
if sqrt((number - i) / 2.0).is_integer()
]) > 0
print[
i for i in range(1, 10000) if is_odd(i) and not is_prime(i)
and not is_sum_of_a_prime_and_twice_a_square(i)
]
from euler_util import is_prime print 2+sum([2*x+1 for x in range(1, 1000000) if is_prime(2*x+1)])
def primos_entre(i,f):
return [i for i in range(i,f+1) if is_prime(i)]
