def isGoldbach(n):
if n % 2 == 0 or isPrime(n): # Not a composite
return True
for i in range(1, n + 1):
remainder = n - 2 * i**2
if remainder <= 0:
return False
elif isPrime(remainder):
return True
def compute():
timer = time.time()
for i in range(1000, 3341):
if isPrime(i) and isPrime(i + 3330) and isPrime(i + 6660) and set(
[d for d in str(i)]) == set([d for d in str(i + 3330)]) and set([
d for d in str(i)
]) == set([d for d in str(i + 6660)]) and i != 1487:
ans = int(str(i) + str(i + 3330) + str(i + 6660))
break
print("Answer is " + str(ans) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return ans
def p3(x=600851475143):
below = []
for i in xrange(2, int(x**0.5)):
if x % i == 0:
if isPrime(x / i):
print x / i
return
below.append(i)
for i in below[::-1]:
if isPrime(i):
print i
return
return x
def p3(x=600851475143):
below=[]
for i in xrange(2,int(x**0.5)):
if x%i==0:
if isPrime(x/i):
print x/i
return
below.append(i)
for i in below[::-1]:
if isPrime(i):
print i
return
return x
def compute():
timer = time.time()
tp = []
i = 11
while len(tp) < 11:
if isPrime(i):
s = str(i)
if all(
isPrime(int(s[:j])) and isPrime(int(s[j:]))
for j in range(1, len(s))):
tp.append(i)
i += 2
print("Answer is " + str(sum(tp)) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return sum(tp)
def genPrimes():
yield 2
i = 3
while True:
if util.isPrime(i):
yield i
i += 2
def test1B(args={'dbg': 1}):
'''(3, N, 2) util.isPrime() & list primes'''
test = 'test1B'
dbg = 1
# print('{} args={}'.format(test, args))
if 'dbg' in args: dbg = args['dbg']
if 'N' in args: N = args['N']
if dbg == 0: print('.', end='')
pList = []
m = 2
while m < N / 2:
m *= 2
pList.append(m)
# if dbg: print('m = {}'.format(m))
for n in range(3, N, 2):
if util.isPrime(n):
m = n
while m < N / n:
m *= n
pList.append(m)
if dbg:
# print('{}'.format(pList))
M = 1
for p in pList:
M *= p
print('\n{} {:,}'.format(test, M), end=' ')
def _keyGen(lamda=64, down=100, up=10000):
'''
lamda: 参数
'''
p = lamda
prikey_len = lamda * 2 + random.randint(1, lamda)
pubkey_len = lamda * 3 + random.randint(1, lamda)
prikey = random.randrange(2**(prikey_len-1) + 1, 2**prikey_len)
while prikey % 2 == 0:
prikey = random.randrange(2**(prikey_len-1) + 1, 2**prikey_len)
pubkeynum = pubkey_len + lamda
pubkey = []
c = 2**pubkey_len
d = 2**lamda
n = random.randrange(down, up)
while util.isPrime(n) == False:
n = random.randrange(down, up)
pubkey = []
for i in range(0, pubkeynum):
q = random.randrange(0, c // prikey)
r = random.randrange(0, d)
pubkey.append(prikey * q + n * r)
return (prikey, pubkey, n)
def sumPrimesBelow(goal):
primes = []
for n in [i for i in range(2, goal)]:
if isPrime(n):
primes.append(n)
print ("Number of primes less than %d: %d" % (goal, len(primes)))
print ("Sum of primes less than %d: %d" % (goal, sum(primes)))
def compute():
timer = time.time()
primes = []
for i in range(1000000):
s = str(i)
l = [int(s[j:] + s[:j]) for j in range(len(s))]
if all(isPrime(val) for val in l):
primes = primes + l
ans = len(set(primes))
print("Answer is " + str(ans) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return ans
def compute():
timer = time.time()
i = 644
consecutive = []
while len(consecutive) < 4:
if len([x for x in getFactors(i) if isPrime(x)]) == 4:
consecutive.append(i)
else:
consecutive = []
i += 1
print("Answer is " + str(consecutive[0]) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return consecutive[0]
def p58(n=0.10):
corners = [3, 5, 7]
#upper right, upper left, lower left, lower right is never prime
differences = [10, 12, 14]
#Notice that as the grid expands one rotation, all these differences increase by 8 every time
statistics = [3, 3, 5]
#current side length, currrent number of primes, total number of corners
while statistics[1] / (statistics[2] + 0.0) > n:
corners = [corners[i] + differences[i] for i in xrange(3)]
differences = [i + 8 for i in differences]
update = [2, sum([int(isPrime(i)) for i in corners]), 4]
statistics = [statistics[i] + update[i] for i in xrange(3)]
print statistics[0]
def countPrimes(n, exc):
primes = []
if 0 in exc:
a = 1
else:
a = 0
for x in range(a, 10):
s = str(n)
for val in exc:
s = s[:val] + str(x) + s[val + 1:]
if s[0] != '0' and isPrime(int(s)):
primes.append(int(s))
return primes
def p41():
#enumerate 7 digit pandigitals in decreasing order
for i in xrange(5039,-1,-1):
factoradic=toFactoradic(i,6)
array=[str(i) for i in xrange(1,8)]
permutation=''
for j in factoradic:
permutation+=array[int(j)]
del array[int(j)]
if isPrime(int(permutation)):
print permutation
return
#enumerate four digit pandigitals in decreasing order
for i in xrange(23,-1,-1):
factoradic=toFactoradic(i,3)
array=[str(i) for i in xrange(1,5)]
permutation=''
for j in factoradic:
permutation+=array[int(j)]
del array[int(j)]
if isPrime(int(permutation)):
print permutation
return
def p41():
#enumerate 7 digit pandigitals in decreasing order
for i in xrange(5039, -1, -1):
factoradic = toFactoradic(i, 6)
array = [str(i) for i in xrange(1, 8)]
permutation = ''
for j in factoradic:
permutation += array[int(j)]
del array[int(j)]
if isPrime(int(permutation)):
print permutation
return
#enumerate four digit pandigitals in decreasing order
for i in xrange(23, -1, -1):
factoradic = toFactoradic(i, 3)
array = [str(i) for i in xrange(1, 5)]
permutation = ''
for j in factoradic:
permutation += array[int(j)]
del array[int(j)]
if isPrime(int(permutation)):
print permutation
return
def compute():
timer = time.time()
ans = 0
maxprimes = 0
for b in range(1000, 0, -1):
if isPrime(
b
): # b must be positive and prime as when n=0, ans = b which must be prime
for a in range(-999, 1000):
primes = 1
for n in range(1, b):
if isPrime(n**2 + a * n + b):
primes += 1
else:
break
if primes > maxprimes:
maxprimes = primes
ans = a * b
if maxprimes == b: # Break when we find the largest value b where for all x, f(x) < b is prime
break
print("Answer is " + str(ans) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return ans
def compute():
timer = time.time()
i = 56003
ans = 0
while ans == 0:
if isPrime(i):
for j, combo in enumerate(powerset(range(len(str(i)))), 1):
primes = countPrimes(i, combo)
# print(i, combo, primes)
if len(primes) > 7:
ans = min(primes)
i += 1
print("Answer is " + str(ans) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return ans
def test1C(args={'dbg': 1}):
'''uses util.isPrime() for odd #s >= 3'''
test = 'test3'
dbg = 1
# print('{} args={}'.format(test, args))
if 'dbg' in args: dbg = args['dbg']
if 'N' in args: N = args['N']
if dbg == 0: print('.', end='')
n, p = 2, 3
while 1:
if util.isPrime(p):
# print('{} {}'.format(n, p))
n += 1
if n == N+1: break
p += 2
if dbg: print('\n{} {:,}th prime = {:,}'.format(test, N, p), end=' ')
def compute():
timer = time.time()
primes = getPrimes(1000000)
longest = 0
for i in range(len(primes) // 4):
chain = 1
total = primes[i]
while total + primes[i + chain] < 1000000:
total += primes[i + chain]
chain += 1
if chain > longest and isPrime(total):
longest = chain
ans = total
print("Answer is " + str(ans) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return ans
def p27():
max_consecutive_primes=40
pair=(1,41) #store the a and b values
for b in primes:
for a in xrange(-999,1000,2):
consecutive_primes=1
for n in xrange(1,b):
num=n*n+a*n+b
if num < 0:
break
if isPrime(num):
consecutive_primes+=1
else:
break
if consecutive_primes > max_consecutive_primes:
pair=(a,b)
max_consecutive_primes=consecutive_primes
print pair, pair[0]*pair[1]
def compute():
timer = time.time()
ans = 0
for length in range(9, 0, -1):
if ans == 0:
perms = sorted(list(
permutations([str(i) for i in range(1, length + 1)])),
reverse=True)
for p in perms:
s = ''
for d in p:
s = s + d
if isPrime(int(s)):
ans = int(s)
break
print("Answer is " + str(ans) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return ans
def p27():
max_consecutive_primes = 40
pair = (1, 41) #store the a and b values
for b in primes:
for a in xrange(-999, 1000, 2):
consecutive_primes = 1
for n in xrange(1, b):
num = n * n + a * n + b
if num < 0:
break
if isPrime(num):
consecutive_primes += 1
else:
break
if consecutive_primes > max_consecutive_primes:
pair = (a, b)
max_consecutive_primes = consecutive_primes
print pair, pair[0] * pair[1]
def test2(args={'dbg': 1}):
'''range(3, N) isPrime(n)'''
test = 'test2'
dbg = 1
# print('{} args={}'.format(test, args))
if 'dbg' in args: dbg = args['dbg']
if 'N' in args: N = args['N']
if dbg == 0: print('.', end='')
p, s = [2], 2
for n in range(3, N, 2):
if util.isPrime(n):
p.append(n)
s += n
# print('{}'.format(p))
if dbg:
print('\n{} the sum of primes below {:,} = {:,}'.format(test, N, s),
end=' ')
def test1(args={'dbg': 1}):
'''uses util.isPrime() (3, sqrt(N)+1, 2)'''
test = 'test1'
dbg = 1
# print('{} args={}'.format(test, args))
if 'dbg' in args: dbg = args['dbg']
if 'N' in args: N = args['N']
if dbg == 0: print('.', end='')
p = []
M = int(math.sqrt(N))
for i in (range(3, M + 1, 2)):
if N % i == 0:
if util.isPrime(i):
p.append(i)
# if dbg: print(p)
if dbg:
print('\n{} largest prime factor = {:,}'.format(test, p[-1]), end=' ')
def test1(args={'dbg': 1}):
'''(3, N, 2) util.isPrime()'''
test = 'test1'
dbg = 1
# print('{} args={}'.format(test, args))
if 'dbg' in args: dbg = args['dbg']
if 'N' in args: N = args['N']
if dbg == 0: print('.', end='')
m = 2
while m < N / 2:
m *= 2
M = m
# if dbg: print('m = {}'.format(m))
for n in range(3, N, 2):
if util.isPrime(n):
m = n
while m < N / n:
m *= n
M *= m
if dbg: print('\n{} {:,}'.format(test, M), end=' ')
def compute():
timer = time.time()
value = 1
increment = 2
primes = 0
diagonals = 1
found = False
while not found:
for i in range(4):
value += increment
if isPrime(value):
primes += 1
diagonals += 1
if primes / diagonals < 0.1:
found = True
ans = (diagonals + 1) / 2
increment += 2
print("Answer is " + str(ans) + ". Completed in " +
str(time.time() - timer) + " seconds.")
return ans
def test2(args={'dbg': 1}):
'''uses util.isPrime() i=odd while(i*i <= N)'''
test = 'test2'
dbg = 1
# print('{} args={}'.format(test, args))
if 'dbg' in args: dbg = args['dbg']
if 'N' in args: N = args['N']
if dbg == 0: print('.', end='')
p = []
i = 3
while i * i <= N:
if N % i == 0:
# print('{} is a factor of N ({})'.format(i, N/i), end=' ')
if util.isPrime(i):
p.append(i)
i += 2
# if dbg: print('{} {} {} {}'.format(N, i, p, p[-1]))
if dbg:
print('\n{} Largest prime factor = {:,}'.format(test, p[-1]), end=' ')
def test3(args={'dbg': 1}):
'''uses util.isPrime() (i <= M) (M % i)'''
test = 'test3'
dbg = 1
# print('{} args={}'.format(test, args))
if 'dbg' in args: dbg = args['dbg']
if 'N' in args: N = args['N']
if dbg == 0: print('.', end='')
pf = []
i = 3
M = N
while i <= M:
while M % i == 0:
if util.isPrime(i):
# if dbg: print('M={}, i={}'.format(M, i))
pf.append(i)
M = int(M / i)
i += 1
# if dbg: print('prime factors = {}'.format(pf))
if dbg:
print('\n{} largest prime factor = {:,}'.format(test, pf[-1]), end=' ')
'''
The sum of the primes below 10 is 2 + 3 + 5 + 7 = 17.
Find the sum of all the primes below two million.
'''
import util
MAX_NUM = 2000000
sum_val = 1
for i in range(0, MAX_NUM):
if util.isPrime(i):
sum_val += i
print sum_val
'''
The prime factors of 13195 are 5, 7, 13 and 29.
What is the largest prime factor of the number 600851475143 ?
'''
import math
import util
VALUE = 600851475143
maxPrime = 0
for i in range(1, int(math.sqrt(VALUE)), 2):
if VALUE % i == 0 and util.isPrime(i):
maxPrime = i
print maxPrime
'''
By listing the first six prime numbers: 2, 3, 5, 7, 11, and 13, we can see that the 6th prime is 13.
What is the 10,001st prime number?
'''
import util
MAX_NUM = 10001
current_number = 1
prime_count = 1
while prime_count < MAX_NUM:
current_number += 1
if util.isPrime(current_number):
prime_count += 1
print current_number
