def mirror_str_test(a, b):
ab = a + b
if len(ab) > 5:
if is_prime(int(ab)) and is_prime(int(b + a)):
return True
elif ab in primes and b + a in primes:
return True
return False
def prime_summations(n):
'''
problem 77
'''
prime_sums, primes = {0:1,1:0}, []
for i in range(2,n):
if is_prime(i):
primes.append(i)
return prime_sums
def pandigital_prime(num_digits = 4):
'''
find largest number that uses all digits 1 through num_digits and is prime
problem 41
'''
for num_d in range(num_digits,0,-1):
digits = [str(d) for d in range(1,num_d+1)]
perms = itertools.permutations(''.join(reversed(digits)))
for i in perms:
num = int(''.join(i))
if is_prime(num): return num
def truncatable_primes(maximum = 1000000):
'''
find all eleven 'truncatable' primes
problem 37
'''
poss_digits = set('1379')
all_primes = get_all_primes_under(maximum)
poss_primes = [p for p in [str(p) for p in all_primes] if set(p[1:]) <= poss_digits]
#primes can only be composed of the digits 1,3,7, and 9
#after the first digit
final_primes = []
for p in poss_primes:
found = True
length = len(p)
for i in range(0,length):
if not is_prime(int(p[i:])) or not is_prime(int(p[:i+1])):
found = False
break
if found: final_primes.append(int(p))
return final_primes
def is_circular_prime(n):
'''
return whether circle prime or not
problem 35
'''
if type(n) != int:
raise TypeError
if not is_prime(n):
return False
s = str(n)
length = len(s)
i = 0
while i < length:
i += 1
if not is_prime(int(rotate_string(s,i))):
return False
return True
def alternate_prime_sum():
ps=get_all_primes_under(3946)
mt,ms,mcps=len(ps),sum(ps),(0,0)
while mt>mcps[0]:
ls=ms if ms%2 else ms-1
while not is_prime(ls): ls-=2
k,ts=0,ms
while ts-ls>0: ts-=ps[k]; k+=1
if ts==ls and mt-k>mcps[0]: mcps=(mt-k,ts)
mt-=1
ms-=ps[mt]
print(mcps)
def quadratic_prime_with_a_and_b_under(maximum):
'''
returns a list of tuple (a,b) from 'n^2 + an + b' and an integer
representing how effective that quad is at predicting primes for
consecutive n values.
b must be a prime (0 + 0a + b) if at all effective, and a+b+1 must
be a prime to get 2
problem 27
'''
primes_under_max = get_all_primes_under(maximum)
products = itertools.product(range(-maximum,maximum+1),primes_under_max)
terms = {(a,b):2 for (a,b) in products if a+b+1 in primes_under_max or is_prime(a+b+1)}
for (a,b),num in terms.items():
n = 2
q = get_quadratic(n,a,b)
while q in primes_under_max if q < maximum else is_prime(q):
n += 1
q = get_quadratic(n,a,b)
terms[(a,b)] = n-1
return terms
def main():
start = time.time()
primes = list()
p = 2
limit = 2*10**6
while p <=limit:
if problem_7.is_prime(p):
primes.append(p)
p +=1
print sum(primes)
end = time.time()
print "time: %s" % (end-start)
def spiral_prime_ratio(i = 10):
'''
find the first ring of spiral prime at which the ratio of primes
is less than given %
problem 58
'''
percent = 100/i
primes = 0
corners = 1
for i in range(2,10**6,2):
side = i+1
square = side**2
primes += len([c for c in range(1,4) if is_prime(square-(c*i))])
corners += 4
if percent*primes < corners:
return (side,primes,corners)
def prime_summations(limit):
primes = []
sums = defaultdict(int)
n = 0
while True:
if is_prime(n):
sums[n] += 1
primes.append(n)
sums[n] += sum(sums[n-k] for k in primes if k <= n / 2)
print sums
if n == 7:
print [(n-k, sums[n-k]) for k in primes if k <= n / 2]
if sums[n] >= limit:
return n
n += 1
def get_circular_primes_under(maximum):
'''
find all circular primes under a given maximum
problem 35
'''
if type(maximum) != int:
raise TypeError
sieve = {d:0 for d in range(1,maximum+1)}
for (i,v) in sieve.items():
if v != 0:
continue
strung = str(i)
length = len(strung)
if '0' in strung:
possibilities = [i] + [int(rotate_string(strung,j)) for j in range(1,length) if rotate_string(strung,j)[0] != '0']
else:
possibilities = [i]+[int(rotate_string(strung,j)) for j in range(1,length)]
found = 1 if len([p for p in possibilities if is_prime(p)]) == length else -1
sieve.update({p:found for p in possibilities})
return [k for (k,v) in sieve.items() if v == 1]
