def getprimes(n=None, below=None):
primes = [2]
num = 3
if below is None:
while len(primes) < n:
if isprime(num):
primes.append(num)
num += 2
elif n is None:
while num < below:
if isprime(num):
primes.append(num)
num += 2
return primes
def largestprimefactor(n):
if type(n) is not int and type(n) is not long:
raise TypeError
allfactors = factors(n)
# Now we have a list of factor pairs (non-prime)
# Let's find which of these are prime numbers
pfactors = []
for factorpair in allfactors:
if isprime(factorpair[0]):
pfactors.append(factorpair[0])
if isprime(factorpair[1]):
pfactors.append(factorpair[1])
# print(pfactors)
return max(pfactors)
def digitreplace(n, checkprime=False):
# This function will return all possible digit replacements
# excluding replacing all digits
# Note that a preceeding zero would cause the number to be omitted
number = list(str(n))
length = len(number)
indices = list(range(length))
replacement_digits = "0123456789"
new_numbers = []
for num_digits_to_replace in range(1, length): # 1 to length-1 digits
for replacement_indices in combinations(indices,
num_digits_to_replace):
this_pattern = []
for digit in replacement_digits:
new_number = deepcopy(number)
for replacement_index in replacement_indices:
new_number[replacement_index] = digit
# Now we have a new number
# We want to omit numbers which start with a zero
if new_number[0] != "0":
# Convert to an int
new_number = int("".join(new_number))
else:
continue
if checkprime is False or isprime(new_number):
this_pattern.append(new_number)
if len(this_pattern) > 0:
new_numbers.append(this_pattern)
return new_numbers
def iscircularprime(n):
s = str(n)
for i in range(len(s)):
if isprime(int(s)) is False:
return False
s = rotate(s)
return True
def primeFactors(n, conv=False):
prime_factors = {}
for pair in factors(n):
for num in pair:
if isprime(num):
prime_factors[num] = 1
prod = 1
for prime_factor in prime_factors:
prod = prod * prime_factor
while prod != n:
quotient = n / prod # i.e. what we still need to represent with the prime factors
for prime_factor in prime_factors:
if quotient % prime_factor == 0:
prod = prod * prime_factor
prime_factors[prime_factor] += 1
if conv is False:
return prime_factors
else:
return primefactorconverter(prime_factors)
def findValidSets(input_set, member_length):
# Takes in an iterable of iterables (preferably set of frozensets)
# Gives back only those which have the prime pair set property
num_permutations = len(
list(permutations([0]*member_length, 2))) # A bit gross
print("Searching through " + str(len(input_set)) + " possible combinations...")
valid_sets = set()
for primes in input_set:
num_prime_pairs = 0
# This will give us the pairs both ways round
for x in permutations(primes, 2):
if isprime(concatenate(x)):
num_prime_pairs += 1
else:
break
if num_prime_pairs == num_permutations:
# All the pairs we checked were prime
valid_sets.add(frozenset(primes))
return valid_sets
#!/usr/bin/python3
# Getting all primes with < 10 digits takes way too much time
# Get all pandigital numbers < 10 digits then check for primeness
# Certainly quicker to generate pandigital numbers than to iterate and find them
from itertools import permutations
from lib import isprime
digits = "123456789"
if __name__ == "__main__":
primepandigitals = []
for n in range(2, 10):
for pandigital in permutations(digits[:n], n):
num = ""
for digit in pandigital:
num += digit
if isprime(int(num)):
primepandigitals.append(int(num))
print("Found " + str(len(primepandigitals)) + " pandigital prime numbers")
print(max(primepandigitals))
def truncateleft(n):
s = str(n)
return int(s[1:])
def truncateright(n):
s = str(n)
return int(s[:-1])
if __name__ == "__main__":
truncatable_primes = []
n = 11 # Single digit primes don't count
while len(truncatable_primes) < 11:
if isprime(n):
left = n
right = n
for i in range(len(str(n)) - 1):
left = truncateleft(left)
right = truncateright(right)
if isprime(left) and isprime(right):
if len(str(left)) == 1:
# That's the last time we will be truncating
truncatable_primes.append(n)
print(n)
else:
break
n += 2
print(sum(truncatable_primes))
#!/usr/bin/python3
from lib import isprime
# Let's just run through all the odd natural numbers starting at 3
primes = [2]
testnum = 3
while len(primes) != 10001:
if isprime(testnum):
primes.append(testnum)
testnum += 2
print(primes[-1])
primes.append(num)
num += 2
elif n is None:
while num < below:
if isprime(num):
primes.append(num)
num += 2
return primes
if __name__ == "__main__":
# First get a list of all prime numbers below 1,000,000
primes = getprimes(below=1000000)
print("Got " + str(len(primes)) + " primes")
# We will store sums like (sum, num_terms)
best = (0, 0)
for i, startpoint in enumerate(primes[:-1]):
best_for_this_start = (0, 0)
length = 1
summation = startpoint
for prime in primes[i+1:]:
summation += prime
length += 1
if summation > 1000000:
break
if isprime(summation):
best_for_this_start = (summation, length)
if best_for_this_start[1] > best[1]:
best = deepcopy(best_for_this_start)
print(best)
#!/usr/bin/python3
from lib import isprime
# let's see if we can just check all odd numbers below 2,000,000
s = 2 # Sum starts at 2 because we're missing 2 out of the range
for n in range(3, 2000000, 2):
if isprime(n):
s += n
print(s)
# Takes about 30sec or so. Fit for purpose.
#!/usr/bin/python3
from lib import isprime
if __name__ == "__main__":
spiral_side_length = 1
num_diagonal_primes = 0
num_diagonals = 1
prime_ratio = 1.0 # So as not to immidiately exit the loop
current_num = 1
while prime_ratio > 0.1:
num_diagonals += 4
spiral_side_length += 2
increment_this_spiral = spiral_side_length - 1
# Now look at all four corners
for i in range(4):
current_num += increment_this_spiral
if isprime(current_num):
num_diagonal_primes += 1
# Re-calculate the prime ratio
prime_ratio = float(num_diagonal_primes) / float(num_diagonals)
print(spiral_side_length)
#!/usr/bin/python3
# for n^2 + an + b to give consecutive primes starting at n = 0, b must always be prime
# So we need a list of prime numbers below or up to 1,000.
# Prime numbers are all positive so we can take that into account
from lib import isprime
primes = []
for i in range(2, 1001):
if isprime(i):
primes.append(i)
longest_sequence = (0, 0, 0) # (a, b, n)
for a in range(-999, 999):
for b in primes:
n = 0
ans = (n * n) + (a * n) + b
while ans > 0 and isprime(ans):
n += 1
ans = (n * n) + (a * n) + b
if n > longest_sequence[2]:
longest_sequence = (a, b, n)
print(longest_sequence)
a, b, n = longest_sequence
print(a * b)
def rotate(s):
rotated = ""
for i in range(len(s)):
rotated += s[i - 1]
return rotated
def iscircularprime(n):
s = str(n)
for i in range(len(s)):
if isprime(int(s)) is False:
return False
s = rotate(s)
return True
if __name__ == "__main__":
primes = [2]
for i in range(3, 1000000, 2):
if isprime(i) is True:
primes.append(i)
print("found " + str(len(primes)) + " primes")
circular_primes = []
for prime in primes:
if iscircularprime(prime) is True:
circular_primes.append(prime)
# print(circular_primes)
print(len(circular_primes))
import sys import lib primes_encountered = set() i = 1 while True: i += 1 isprime = lib.isprime(i) if isprime: primes_encountered.add(i) if not isprime and i % 2 == 1: found_hit = False for pr in primes_encountered: diff = i - pr if diff % 2 == 0: possible_sq = diff / 2 if possible_sq in list(x**2 for x in xrange(1, possible_sq+1)): found_hit = True break if not found_hit: print i break
# Start assuming the first number is in the equispaced sequence
# We need to find at least three numbers to make it a sequence
# If no seqeunce is found, then assume the second number is the
# start of an equispaced sequence
if len(nums) < 3:
return []
nums.sort()
for i, start in enumerate(nums[:-2]):
gaps = []
for potential_sequence_member in nums[i + 1:]:
gaps.append(potential_sequence_member - start)
for gap in gaps:
if gap * 2 in gaps:
# We have found a sequence with gap
return [start, start + gap, start + (2 * gap)]
return []
if __name__ == "__main__":
for i in range(1000, 10000):
primeperms = []
for perm in permutations(str(i)):
if perm[0] == "0":
continue # We need 4 digit numbers
n = int(perm[0] + perm[1] + perm[2] + perm[3])
if n not in primeperms and isprime(n):
primeperms.append(n)
seqeunce = findequispaced(primeperms)
if len(seqeunce) > 0:
print(seqeunce)