def isConcatPrimePair(p1, p2):
pair = getPair(p1, p2)
if pair[0] in primePairsSet:
if pair[1] in primePairsSet:
return True
else:
if euler.isPrime(pair[0]) and euler.isPrime(pair[1]):
primePairsSet.update(pair)
return True
else:
return False
def check(a, b, c, d, e):
nums = map(str, [a, b, c, d, e])
for perm in perms:
new = nums[perm[0]] + nums[perm[1]]
if not (isPrime(int(new))):
return False
return True
def isTestedPrime( n ):
if n <= MAX_PRIME:
return n in PRIME_SET
elif n <= MAX_TEST:
return isPrime( n, PRIMES )
else:
raise Exception( 'n is too large to test.' )
def primeWithRepeat_is(numberOfDigits_i, digit_i):
digitOther_is = list(DIGIT_S)
digitOther_is.remove(digit_i)
repeat_is = [digit_i for i in range(0, numberOfDigits_i)]
for nonRepeats_i in range(0, numberOfDigits_i):
if nonRepeats_i == 0 and numberOfDigits_i % 2 == 0:
continue
candidate_is = []
for c in combinations(range(0, numberOfDigits_i), nonRepeats_i):
if c[0] != 0 and digit_i == 0:
continue
for c1 in product(digitOther_is, repeat=nonRepeats_i):
if c[0] == 0 and c1[0] == 0:
continue
c0 = list(deepcopy(repeat_is))
assert (len(c0) == numberOfDigits_i)
assert (len(c) == nonRepeats_i)
for j in range(0, len(list(c))):
c0[c[j]] = c1[j]
primeMaybe = int(''.join(map(str, c0)))
candidate_is.append(primeMaybe)
#print( c, i, c0, j, primeMaybe)
primeWithRepeat_is = []
for candidate_i in candidate_is:
if numberOfDigits_i < DIGIT_BIG and candidate_i in primea:
primeWithRepeat_is.append(candidate_i)
elif DIGIT_BIG <= numberOfDigits_i and isPrime(
candidate_i, primes):
primeWithRepeat_is.append(candidate_i)
if len(primeWithRepeat_is) != 0:
break
return digit_i, numberOfDigits_i - nonRepeats_i, len(
primeWithRepeat_is), sum(primeWithRepeat_is)
def truncatableLeft(a):
num = str(a)
while len(num) > 1:
num = num[1:]
if not isPrime(int(num)):
return False
return True
def smallestPrimeOfEightMemberFamily(numString):
DIGITS = "***"
# Generate digit pattern and all permutations.
pattern = ''.join([
''.join(['x' for x in range(len(numString[0:-1:]) - len(DIGITS))]),
DIGITS
])
patterns = list(map(lambda x: ''.join(x), set(permutations(pattern))))
# Check the digit patterns to determine if a number is an eight-member family.
for pat in patterns:
patMarker = [i for i, x in enumerate(pat) if x == '*']
candidatePattern = ''.join(
['*' if i in patMarker else x for i, x in enumerate(numString)])
tally = 0
smallestInFamily = 0
for i in range(10):
# If no prime has been found by now (0, 1, 2), then it won't be an eight-member family.
if i == 3 and tally == 0:
break
number = candidatePattern.replace('*', str(i))
if number[0] == '0':
continue
if euler.isPrime(int(number)):
tally += 1
if tally == 1:
smallest = int(candidatePattern.replace('*', str(i)))
if tally == 8:
return smallest
return 0
def isPrimeString( s ):
if s in PRIMECA:
return True
n = int( s )
if PRIMES[ -1 ] * PRIMES[ -1 ] < n:
raise Exception( 'isPrimeString: PRIMES[ -1 ] * PRIMES[ -1 ] < n' )
else:
return isPrime( n, PRIMES )
def sumFactors(n):
if isPrime(n, primes):
return 1
ps = primeFactors(n, primes)
pPowers = primePowerFactorization(ps)
sum = 1
for p in pPowers.keys():
sum *= (p**(pPowers[p] + 1) - 1) // (p - 1)
return sum - n
def evaluatePair(a,b):
"""determine how many consecutive primes are generated"""
n = 0
while True:
fx = n**2 + a*n + b
if isPrime(fx):
n += 1
else:
break
return n-1
def count_sets(permut, prev):
global count
if not permut:
prod = product(prev)
if prod not in uniq_prods:
uniq_prods.add(prod)
count += 1
for end in range(len(permut)):
candidate = get_num(permut, end)
if isPrime(candidate):
count_sets(permut[end+1:], prev + [candidate])
def testSuccess(self):
self.assertTrue(euler.isPrime(2))
self.assertTrue(euler.isPrime(3))
self.assertTrue(euler.isPrime(5))
self.assertTrue(euler.isPrime(7))
self.assertTrue(euler.isPrime(17))
self.assertTrue(euler.isPrime(617))
def eu7():
LIMIT = 10001
candidate = 1
numOfPrimes = 1
while (numOfPrimes != LIMIT):
candidate += 2
if isPrime(candidate):
numOfPrimes += 1
return candidate
def run():
isPrime = {}
for i in itertools.count(3, 2):
isPrime[i] = euler.isPrime(i)
if isPrime[i]:
continue
conjecture = False
for j in range(i-2, 1, -2):
if not isPrime[j]:
continue
n = (i-j)//2
if isPerfectSquare(n):
conjecture = True
break
if not conjecture:
return i
import math
import euler
i = 1
target = 600851475143
max = math.sqrt(600851475143)
value = 0
while i < max:
if target % i == 0 and euler.isPrime(i):
if i > value:
value = i
i += 1
print(value)
import math import euler highestPrimeCount = 0 for a in range(-999, 1000): for b in range(-999, 1000): n = -1 primeCount = 0 while True: n += 1 prime = int(math.pow(n, 2) + a*n + b) if not euler.isPrime(prime): break primeCount += 1 if primeCount > highestPrimeCount: highestPrimeCount = primeCount highestA = a highestB = b print highestA, highestB, highestPrimeCount
def run():
for ndigit in ["1234567"[:i] for i in range(1, 8)][::-1]:
for num in list(map("".join, itertools.permutations(ndigit)))[::-1]:
if euler.isPrime(int(num)):
return int(num)
return 0
#!/usr/bin/python
from euler import appendDigits, appendToPrimes, isPrime
PRIMES = []
LOWER = 10**3
UPPER = 10**4
appendToPrimes(UPPER, PRIMES)
fourDigitPrimes = [p for p in PRIMES if LOWER <= p < UPPER]
assert(fourDigitPrimes[0] == 1009)
assert(isPrime(1009, PRIMES))
assert(fourDigitPrimes[-1] == 9973)
assert(isPrime(9973, PRIMES))
for p1 in fourDigitPrimes:
digits1 = []
appendDigits(p1, digits1)
for p0 in fourDigitPrimes:
if p1 <= p0:
break
assert(p0 < p1)
digits0 = []
appendDigits(p0, digits0)
if digits0 != digits1:
continue
else:
p2 = p1 + (p1 - p0) # next in the arithmetic series
assert(p1 < p2)
if p2 not in fourDigitPrimes:
continue
from euler import isPrime
from itertools import permutations
N = 10
ans = 0
for d in range(0, 10):
for M in range(N-1, 1, -1):
curr = set() # make sure we don't repeat answers
for extra in range(0, 10**(N-M)):
for locs in permutations(range(0, N), N-M):
# build string defaulting to digit d, and filling in
# with permutations of other digits
num = [str(d)] * N
for loc,digit in zip(locs,str(extra)):
num[loc] = digit
num = int("".join(num))
if num < 10**(N-1):
continue
if num in curr:
continue
elif isPrime(num):
curr.add(num)
if len(curr) > 0:
ans += sum(curr)
break
print(ans)
from itertools import permutations
from euler import isPrime
for i in range(1000, 9999):
if not (isPrime(i)):
continue
perm = permutations(str(i), 4)
perms = []
for p in perm:
num = int("".join(p))
if isPrime(num) and num not in perms:
perms.append(num)
for j in perms:
if j <= i:
continue
for k in perms:
if k <= j:
continue
if (i + k) / 2 == j:
print i, j, k
import euler
maxi = 0
for i in range(4, 10):
perms = euler.getPerms(''.join(map(str, range(1,i))))
newperms = []
for perm in perms:
newperms.append(int(perm))
perms = newperms
perms.sort()
for perm in reversed(perms):
if euler.isPrime(perm):
print perm
break
print maxi
def test_isPrime(self): self.assertEqual(euler.isPrime(2), True) self.assertEqual(euler.isPrime(3), True) self.assertEqual(euler.isPrime(4), False) self.assertEqual(euler.isPrime(5), True) self.assertEqual(euler.isPrime(6), False)
from euler import primes, isPrime
from itertools import permutations
def get_permutations(num):
aux=str(num)
for i in permutations(aux):
yield ''.join(i)
primes=[x for x in primes(10000) if x>=1000]
for p in primes:
current=[]
for permut in get_permutations(p):
perm=int(permut)
if isPrime(perm) and perm not in current:
current.append(perm)
if len(current)==3:
current=sorted(current)
if current[1]-current[0]==current[2]-current[1]:
print str(current[0])+str(current[1])+str(current[2])
#296962999629
def isStrongHarshad(x):
dsum = digisum(x)
return x%dsum == 0 and isPrime(x//dsum)
from euler import isPrime, digisum
def isHarshad(x):
return x%digisum(x) == 0
def isStrongHarshad(x):
dsum = digisum(x)
return x%dsum == 0 and isPrime(x//dsum)
def isRTHarshad(x):
return x < 10 or (isHarshad(x) and isRTHarshad(x//10))
N = 14
RTHarshads = [[] for n in range(0,N)]
RTHarshads[1] = [i for i in range(1,10)]
ans = 0
for n in range(2,N):
for h in RTHarshads[n-1]:
for k in range(0,10):
x = h*10+k
if isHarshad(x):
RTHarshads[n].append(x)
for h in RTHarshads[n]:
if isStrongHarshad(h):
for k in range(1,10,2):
x = h*10+k
if isPrime(x):
ans += x
print(ans)
# Project Euler
# Problem 58
# Spiral primes
import sys
sys.path.append('../')
from euler import isPrime
n = 1
fracPrime = 1
numPrime = 0
while fracPrime >= 0.1:
n += 2
numOnDiags = 2 * n + 1
num = n**2 - 3 * n + 3
for _ in range(3):
numPrime += 1 if isPrime(num) else 0
num += (n - 1)
fracPrime = numPrime / numOnDiags
print(n)
#!/usr/bin/python
from itertools import permutations
from euler import appendToPrimes
from euler import isPrime
BASE = 10
D = 4 # 2143 is a pandigital prime
PRIMES = []
appendToPrimes(10**5, PRIMES)
maximum = 0
for n in range(D, BASE):
for p in list(permutations(range(1, n + 1))):
i = int(''.join(map(str, p)))
if isPrime(i, PRIMES):
if maximum < i:
maximum = i
print(maximum)
from euler import range_infinite, isPrime
total = 1
primes = 0
for n in range_infinite(3, 2):
total += 4
if isPrime((n - 2) * (n - 2) + n - 1):
primes += 1
if isPrime((n - 1) * (n - 1) + 1):
primes += 1
if isPrime((n - 1) * (n - 1) + n):
primes += 1
if (primes + 0.0) / total < 0.1:
print n
break
#26241
from euler import isPrime
from math import sqrt
number = 600851475143
i = int(sqrt(number))
while i >= 2:
if number % i == 0:
if isPrime(i) == True:
print i
break
else:
i = i-1
else:
i = i -1
def isP(x):
return isPrime(x)
from euler import isPrime
# Upper bound is hackery, probably can be made formal
N = 2000000000
cubes = [x**3 for x in range(1, 2 + int(N**(1/3)))]
# Crucial observation: For n^3 + n^2 p to be a perfect cube,
# n^2 (thus n) and n + p must be perfect cubes
ps = set()
for i,x in enumerate(cubes):
for j in range(i+1, len(cubes)):
if isPrime(cubes[j] - x):
if cubes[j] - x < 1e6:
ps.add(cubes[j] - x)
print(ps)
print(len(ps))
from euler import isPrime
from itertools import permutations
nums = []
def getPermutations(perm):
for i in permutations(perm):
yield ''.join(i)
for i in range(1, 10):
nums.append(str(i))
for perm in getPermutations(nums):
if isPrime(int(perm)):
max = perm
print max
#7652413
def testFailure(self):
self.assertFalse(euler.isPrime(0))
self.assertFalse(euler.isPrime(1))
self.assertFalse(euler.isPrime(4))
self.assertFalse(euler.isPrime(9))
self.assertFalse(euler.isPrime(15))
import math
from euler import isPrime
# What is the 10,001st prime number?
primesFound = 1
x = 3
while True:
if isPrime(x):
primesFound += 1
if primesFound == 10001:
print(x)
break
x += 2
import euler
import sys
primeset = set()
for i in range(1001, 10000, 1):
if euler.isPrime(i) is True: primeset = primeset | {str(i)}
diffdict = {}
primedict= {}
# Get list of permuted numbers that are primes
for prime in primeset:
tperms=[]
xperms=euler.getPerms(prime)
for perm in xperms:
if perm in primeset: tperms.append(perm)
tperms.sort()
tperms = tperms[tperms.index(prime):]
primedict[prime]=tperms
for i in range(len(tperms)-1, 0, -1):
sub = int(tperms[i]) - int(tperms[0])
if sub in diffdict: diffdict[sub].append({tperms[0], tperms[i]})
else: diffdict[sub] = [{tperms[0], tperms[i]}]
def numPrimes(lista):
return len([x for x in lista if isPrime(x)])
from euler import isPrime
def highestExp(x,n):
result = x
count = 0
while result <= n:
count += 1
result *= x
return count
output = 1
for i in range(2,20):
if isPrime(i):
exponent= highestExp(i,20)
output *= i**exponent
print output
count = 1 validprimeset = set([2]) invalidset = set() for i in range(3, 10000, 2): if i not in validprimeset and i not in invalidset: notprime = False perms = euler.getPerms(str(i)) perms = set(map(int, perms)) tmp = [] for perm in perms: for j in range(2, int((math.sqrt(perm)))): tmp.append(j*perm) if not euler.isPrime(perm): invalidset = invalidset | set([perm]) notprime = True break # invalidset = invalidset | set(tmp) if notprime is False: count += len(perms) validprimeset = validprimeset | perms print count, len(validprimeset), len(invalidset)
# 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?
import itertools
from euler import isPrime
largest_prime = 0
for size in range(1,10):
# generate all the pandigital numbers of 'size'
for i in itertools.permutations([str(x) for x in range(1,size+1)]):
k = int("".join(i)) # convert tuple to int
if isPrime(k) and k > largest_prime:
largest_prime = k
print(k)
print(largest_prime)
from euler import primes,isPrime
primes=primes(10**6)
realmax=0
for c in range(2,1000):
max=0
for i in range(0,len(primes)):
tot=sum(primes[i:i+c])
if tot>10**6:
break
elif isPrime(tot):
if max<tot:
max=tot
if max>0:
realmax=max
print realmax
#997651
increment = 2
while True:
yield num, increment
cycle += 1
num += increment
if cycle == 4:
cycle = 0
increment += 2
primeCount = 0
sideLength = 0
for i, values in enumerate(incrementSpiralDiagonals()):
num = values[0]
inc = values[1]
if euler.isPrime(num):
primeCount += 1
if primeCount > 0 and primeCount / (i + 1) < TARGET_PERCENTAGE:
sideLength = inc - 1
break
print(
"The side length of a square spiral with a prime ratio on the diagonals where first less than {0}% is {1}."
.format(TARGET_PERCENTAGE * 100, sideLength))
print("Program execution took {0} seconds.".format(time.clock() - startTime))
#!/usr/bin/python
from euler import isPrime, appendToPrimes
PRIMES = []
UPPER = 10**5
appendToPrimes(UPPER, PRIMES)
twoSquares = [2 * n * n for n in range(0, UPPER)]
n = 3
while True:
while isPrime(n, PRIMES):
n += 2
bad = True
for j in twoSquares:
if n <= j:
break
if isPrime(n - j, PRIMES):
bad = False
n += 2
break
if bad:
break
print(n)
def generateNextA(previousVal, position):
return previousVal + 2 * position
def generateNextB(previousVal, position):
return previousVal + 4 * ((position + 1) // 2)
# represents the values with grid-size 1.
currA = 1
currB = 1
gridSize = 1
numPrimes = 0
while True:
for i in range(2): #grow both dimensions
nextA, nextB = generateNextA(currA,
gridSize), generateNextB(currB, gridSize)
if isPrime(nextA):
numPrimes += 1
if isPrime(nextB):
numPrimes += 1
gridSize += 1
currA, currB = nextA, nextB
percentage = numPrimes * 100 / (gridSize // 2 * 4 + 1)
if percentage < 10.0:
print("{} grid size = {} primes. {}%".format(gridSize, numPrimes,
percentage))
break
from math import log
from math import sqrt
from euler import appendToPrimes
from euler import isPrime
from euler import appendDigitsUnsorted
N = 1000000 # N = 100 #
BASE = 10
primes = []
appendToPrimes(int(sqrt(N)), primes)
# print(primes) # [2, 3, 5, 7]
a = int(sqrt(N))
while a < N:
if isPrime(a, primes):
primes.append(a)
a += 1
# primes contains all primes less than N.
# print(primes) # [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
possibles = [
] # potentially right truncatable: no internal {0,2,4,5,6,8} and no left {0,4,6,8}
for p in primes:
digits = appendDigitsUnsorted(p, [])
if 0 in digits or 4 in digits or 6 in digits or 8 in digits:
continue
digits.pop(0)
if 2 in digits or 5 in digits:
continue
#
# How many circular primes are there below one million?
from euler import isPrime
# if there are even digits, skip it! one of the rotations will be even, therefore non prime.
def contains_even_digits(number):
even_digits = set('24680')
return any((d in even_digits) for d in str(number))
def find_rotations(number):
as_str = str(number)
result = [number]
for i in range(0, len(as_str) - 1):
rotation = as_str[1:] + as_str[:1]
result.append(int(rotation))
as_str = rotation
return result
if __name__ == '__main__':
UPPER_LIMIT = 1000000
# skip 2, the only EVEN prime...
number_found = 1 # conting 2
for i in range(3, UPPER_LIMIT):
if contains_even_digits(i):
continue
if all(isPrime(x) for x in find_rotations(i)):
number_found += 1
print(number_found)
from euler import isPrime, range_infinite, primes
primes = primes(10**5)
for i in range_infinite(35, 2):
if not isPrime(i):
flag = True
for p in primes:
if p < i:
found = False
for j in range_infinite(1, 1):
if p + 2 * j * j > i:
break
elif p + 2 * j * j == i:
found = True
break
if found:
break
else:
flag = False
break
if not flag:
print i
break
#5777
from euler import isPrime
curr = 1
diff = 2
primecount = 0
side = 1
while (primecount / (side * 2.0 - 1)) > 0.1 or primecount == 0:
for k in range(4):
curr = curr + diff
if isPrime(curr):
primecount += 1
diff += 2
side += 2
print side