def problem27():
"""
Considering the quadratic formula n^2 + an + b where |a| < 1000 and
|b| < 1000, find the product of the coefficients, a and b, for the
quadratic expression that produces the maximum number of primes for
consecutive values of n, starting with n = 0.
"""
limit = 1000
primes, primeset = util.primes(limit)
def quad(a, b):
return lambda n: n**2 + a*n + b
max_n = 0
max_coef = 0, 0
for a in xrange(-1000, 1000):
for b in primes:
q = quad(a, b)
n = 0
while q(n) in primeset:
n += 1
if n > max_n:
max_n = n
max_coef = a, b
return max_coef[0] * max_coef[1]
def problem46():
"""
It was proposed by Christian Goldbach that every odd composite number can
be written as the sum of a prime and twice a square.
9 = 7 + 2x1^2
15 = 7 + 2x2^2
21 = 3 + 2x3^2
25 = 7 + 2x3^2
27 = 19 + 2x2^2
33 = 31 + 2x1^2
It turns out that the conjecture was false.
What is the smallest odd composite that cannot be written as the sum of
a prime and twice a square?
"""
limit = 10000
dsquares = [2*i*i for i in xrange(1, limit)]
primes, primeset = util.primes(limit)
def is_goldbach(i):
for p in primes:
if p > i:
return False
for ds in dsquares:
if i == p + ds:
return True
elif i < p + ds:
break
return False
for i in xrange(3, limit, 2):
if i not in primeset and not is_goldbach(i):
return i
def problem49():
"""
The arithmetic sequence, 1487, 4817, 8147, in which each of the terms
increases by 3330, is unusual in two ways: (i) each of the three terms
are prime, and, (ii) each of the 4-digit numbers are permutations of
one another.
There are no arithmetic sequences made up of three 1-, 2-, or 3-digit
primes, exhibiting this property, but there is one other 4-digit
increasing sequence.
What 12-digit number do you form by concatenating the three terms in
this sequence?
"""
limit = 10000
primes, primeset = util.primes(limit)
for a in primes:
permset = set([int(x) for x in util.permutations(str(a))])
for b in util.permutations(str(a)):
b = int(b)
if b > a and b in primeset:
c = b + (b-a)
if c in primeset and c in permset and c != 8147:
return "{}{}{}".format(a, b, c)
def problem35():
"""
The number, 197, is called a circular prime because all rotations of the
digits: 197, 971, and 719, are themselves prime.
There are thirteen such primes below 100: 2, 3, 5, 7, 11, 13, 17, 31, 37,
71, 73, 79, and 97.
How many circular primes are there below one million?
"""
limit = 1000000
primes, primeset = util.primes(limit)
def rotations(digits):
for i in xrange(len(digits)):
yield digits[i:] + digits[:i]
count = 0
for i in xrange(2, limit):
if i in primeset:
for p in rotations(str(i)):
if int(p) not in primeset:
break
else:
count += 1
return count
def problem10():
"""
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
"""
limit = 2000000
primes, primeset = util.primes(limit)
return sum(primes)
def solve():
m = [0, 0, 0]
def max_prime(a, b):
c = count_prime(a, b)
if c > m[0]:
m[0] = c
m[1] = a
m[2] = b
pa = primes()
a = pa.next()
while a < 1000:
pb = primes()
b = pb.next()
while b < 1000:
max_prime(a, b)
max_prime(a, -b)
max_prime(-a, -b)
max_prime(-a, b)
b = pb.next()
a = pa.next()
return m
def solve():
print 'start'
s = 0
p = primes()
n = p.next()
cd = 11
while cd > 0:
if n < 11:
n = p.next()
continue
if is_removeable_prime(n):
print n
cd -= 1
s += n
n = p.next()
return s
def problem50():
"""
Which prime, below one-million, can be written as the sum of the most
consecutive primes?
"""
limit = 1000000
primes, primeset = util.primes(limit)
best = (1, 1)
for i in xrange(len(primes)):
total = 0
for j in xrange(i, len(primes)):
total += primes[j]
if total >= limit:
break
elif total in primeset and j-i > best[1]:
best = total, j-i
return best[0]
def problem47():
"""
The first two consecutive numbers to have two distinct prime factors are:
14 = 2 x 7
15 = 3 x 5
The first three consecutive numbers to have three distinct prime factors are:
644 = 2^2 x 7 x 23
645 = 3 x 5 x 43
646 = 2 x 17 x 19.
Find the first four consecutive integers to have four distinct prime factors.
What is the first of these numbers?
"""
limit = 1000000
fcount = 4
primes, primeset = util.primes(limit)
i = 3
count = 0
while i < limit:
j = i
factors = set()
while j not in primeset:
for p in primes:
if p > 1 and j % p == 0:
factors.add(p)
j /= p
break
factors.add(j)
if len(factors) != fcount:
count = 0
else:
count += 1
if count == fcount:
return i - fcount + 1
i += 1
def solve():
"""
The n/phi(n) will be large when phi(n) is small and n is large. To make
phi(n) small, we have to make n have as many divisors as possible. A good
way to do this is to include small primes as some of its divisors.
Multiples of these small primes will be ruled out from the totient, which
is a good thing. Not guaranteed to be the right answer, I suppose, but
turns out to be.
"""
primes = util.primes(1000000)
product = 1
phi = 0
for p in primes:
if product * p <= 1000000:
print "multiplying one more prime:", p
product *= p
phi += 1
else:
break
print "final n:", product
print "phi(n):", phi
print "n/phi(n):", product/float(phi)
def problem37():
"""
Find the sum of the only eleven primes that are both truncatable from
left to right and right to left.
"""
limit = 1000000
total = 0
primes, primeset = util.primes(limit)
def is_truncatable(i):
left = right = str(i)
while len(left) > 0 and len(right) > 0:
if int(left) not in primeset or int(right) not in primeset:
return False
else:
left, right = left[1:], right[:-1]
return True
for i in xrange(11, limit, 2):
if is_truncatable(i):
total += i
return total
def p50():
"""The prime 41, can be written as the sum of six consecutive primes:
41 = 2 + 3 + 5 + 7 + 11 + 13
This is the longest sum of consecutive primes that adds to a prime below one-hundred.
The longest sum of consecutive primes below one-thousand that adds to a prime, contains 21 terms, and is equal to 953.
Which prime, below one-million, can be written as the sum of the most consecutive primes?
"""
p = primes(999999)
work = p[:]
pSet = set(p)
bestPrime = 2
for i in range(1,len(work)):
for j in range(len(work)-i):
work[j]+=p[j+i]
if work[j] > 999999:
break
if work[j] in pSet:
bestPrime = work[j]
print bestPrime
print bestPrime
"""Which prime, below one-million, can be written as the sum of the most
consecutive primes?
"""
from util import primes
def IsPrime(n):
for d in xrange(2, int(n**0.5) + 1):
if n%d == 0: return False
return True
MAX = 1000000
memo = []
(pp, pl) = (0, 0)
for p in primes(MAX):
if p == 2:
memo.append(p)
continue
i = 0
while i < len(memo):
memo[i] += p
if memo[i] > MAX: break
else:
if IsPrime(memo[i]):
if i > pl:
pl = i
pp = memo[i]
i += 1
else:
memo.insert(0, p)
continue
break
from util import primes
product = 1
for p in primes():
next = product * p
if next > 10**6:
break
product = next
print(product)
from util import primes
for nth, prime in enumerate(primes()):
if nth == 10000:
print(prime)
break
#! /usr/bin/python
from util import primes
from util import inb
p = primes(1000000)
s = []
res = 21
for i in range(len(p)):
s.append(sum(p[i:i+res]))
if s[i] > 1000000:
break
while s != []:
res += 2
for i in range(len(s)):
s[i] += p[i+res-1] + p[i+res-2]
if s[i] > 1000000:
s = s[:i]
break
if inb(s[i],p):
print s[i],res
#! /usr/bin/python
from util import primes
from util import inb
p = primes(1000000)
s = []
res = 21
for i in range(len(p)):
s.append(sum(p[i:i + res]))
if s[i] > 1000000:
break
while s != []:
res += 2
for i in range(len(s)):
s[i] += p[i + res - 1] + p[i + res - 2]
if s[i] > 1000000:
s = s[:i]
break
if inb(s[i], p):
print s[i], res
import psyco
import util
primes = util.primes(9999)
primes_set = set(util.primes(99999999))
# pass 1: pairs
print "pairs"
conc_pairs = set()
for p1 in primes:
for p2 in reversed(primes):
if p1 < p2:
p1str = str(p1)
p2str = str(p2)
if int(p1str+p2str) in primes_set and\
int(p2str+p1str) in primes_set:
conc_pairs.add((p1,p2))
else:
break
print conc_pairs
# pass 2: triples
print "triples"
conc_triples = set()
for p1,p2 in conc_pairs:
for p3 in reversed(primes):
if p2 < p3:
if (p1,p3) in conc_pairs and\
(p2,p3) in conc_pairs:
conc_triples.add((p1,p2,p3))
else:
from util import primes
if __name__ == '__main__':
print(sum(primes(2000000)))
def first_prime_factor(n):
for prime in primes():
if n % prime == 0:
return prime
from util import is_prime, primes
pg = primes()
while next(pg) <= 56003:
pass
answer = None
for p in pg:
s = str(p)
for d in range(3):
if str(d) in s:
n = 1
for e in range(d+1, 10):
if is_prime(int(s.replace(str(d), str(e)))):
n += 1
if n >= 8:
answer = p
break
if answer:
break
print(answer)
from util import primes
if __name__ == '__main__':
size = 2 ** 16
while True:
ps = list(primes(size))
try:
print(ps[10000])
except IndexError:
size *= 2
else:
break
#! /usr/bin/python
from util import primes
product = 1
for i in primes(20):
product *= i
if product > 10 ** 6:
print product / i
import math
from util import primes
def g(a, x):
y_max = math.ceil(x / a - 1) + 1
mem = [1] * (y_max+1)
z_start = math.floor(x - a)
for z in range(z_start, -1, -1):
y_start = math.floor((x - z) / a - 1)
for y in range(y_start, -1, -1):
mem[y] += mem[y+1]
return mem[0]
def G(n):
return g(n ** 0.5, n)
if __name__ == '__main__':
all_primes = primes(start=10000000, end=10010000)
summed = sum([G(p) for p in all_primes])
print(summed % 1000000007)
"""What is the 10001st prime number? """ from util import primes print[p for p in primes(10001)][-1]
import psyco
import util
primes = set(util.primes(1000000))
def left_right(p):
s = str(p)
for i in range(len(s)):
yield int(s[i:])
def right_left(p):
s = str(p)
for i in range(1, len(s)+1):
yield int(s[:i])
def all_prime(l):
return all(p in primes for p in l)
if __name__ == "__main__":
sum = 0
num = 0
for p in primes:
if p not in [2,3,5,7]:
if all_prime(left_right(p)) and all_prime(right_left(p)):
sum += p
num += 1
print "found:", num
print "sum:", sum
def problem_7(): return last(primes(10001))
def primes_test():
p = 0
for prime in primes(max_step=6):
p = prime
assert_equal(p, 13)
from collections import deque
from itertools import takewhile, imap
from operator import itemgetter
from util import primes
print deque(
imap(itemgetter(1),
takewhile(lambda (i, v): i < 10001, enumerate(primes()))),
maxlen=1).pop()
def main():
p = 0
for prime in primes(max_step=10001):
p = prime
return p
If we list the set of reduced proper fractions for d <= 8 in ascending order of size, we get:
1/8, 1/7, 1/6, 1/5, 1/4, 2/7, 1/3, 3/8, 2/5, 3/7, 1/2, 4/7, 3/5, 5/8, 2/3, 5/7, 3/4, 4/5, 5/6, 6/7, 7/8
It can be seen that there are 21 elements in this set.
How many elements would be contained in the set of reduced proper fractions for d <= 1,000,000?
"""
# return int(n * reduce(lambda x,y: x*y, [1-(1./i) for i in pfact]))
from util import inb, pf, primes, totient
from math import sqrt
end = 10 ** 6
ps = primes(end)
ps_sq = primes(int(sqrt(end)))
pfacts = {}
# print ps, inb(2, ps)
for i in xrange(2, end + 1):
if inb(i, ps) != -1:
pfacts[i] = (set([i]), i - 1)
else:
for prime in ps_sq:
if i % prime == 0:
pf, totient = pfacts[i / prime]
if prime in pf:
pfacts[i] = (pf, totient * prime)
else:
pfacts[i] = (pf.union(set([prime])), totient * (prime - 1))
#! /usr/bin/python
from util import primes, is_prime, inb
MAX = 2000
LIST_LENGTH = 5
PRIMES_TO = 10**6
print "Computing first %d primes..." % PRIMES_TO
p = primes(PRIMES_TO)
print "Done\n"
pair_dict = {}
# Use a simple recursive backtracking algorithm
def solve(cur_list):
print cur_list
if len(cur_list) == LIST_LENGTH:
print cur_list, sum(cur_list)
exit()
start = 1 if cur_list == [] else inb(cur_list[-1], p) + 1
for i in p[start:MAX]:
recurse = True
for j in cur_list:
pairing = int(str(j) + str(i))
if pair_dict.get(pairing) != None:
recurse = False
break
if recurse:
solve(cur_list + [i])
