def main(max_n):
divisors = dict()
prime_object = ProjectEulerPrime()
for n in xrange(max_n, 1, -1):
# from:
# http://stackoverflow.com/questions/110344/algorithm-to-
# calculate-the-number-of-divisors-of-a-given-number
# we know that:
# divisors(n) = product(each factor's multiplicity + 1)
factorization = prime_object.factorize(n)
multiplicities = Counter(factorization)
divisors[n] = 1
for factor in multiplicities:
divisors[n] *= multiplicities[factor] + 1
count = 0
for n in divisors:
if n + 1 in divisors and divisors[n] == divisors[n + 1]:
count += 1
print "Adjacent pairs with the same number of divisors: %d" % count
def main():
start = time()
primeObject = ProjectEulerPrime()
solutionSet = set([])
squareList = [number ** 2 for number in xrange(2, int(LIMIT ** 0.5) + 1) if primeObject.isPrime(number)]
cubeList = [number ** 3 for number in xrange(2, int(LIMIT ** (1./3)) + 1) if primeObject.isPrime(number)]
fourthList = [number ** 4 for number in xrange(2, int(LIMIT ** (1./4)) + 1) if primeObject.isPrime(number)]
for square in squareList:
for cube in cubeList:
if square + cube >= LIMIT:
break
for fourth in fourthList:
resultingSum = square + cube + fourth
if resultingSum >= LIMIT:
break
else:
solutionSet.add(resultingSum)
print "Number of sums formed from squared, cubed, and fourthed primes that are < ",LIMIT, " : ", len(solutionSet)
end = time()
print "Runtime: ", end - start, " seconds."
def main():
start = time()
primeObject = ProjectEulerPrime()
# so, my first attempt at this used Euler's product formula and was really slow.
# Thanks to the people in the forum of P69 of Project Euler for help with this.
# we want to maximize n / phi(n). phi(n) = n * product((prime - 1)/prime) for each prime
# therefore, we want to maximize 1 / product((prime - 1)/prime) for each prime
# let's think about the most naive strategy to get these primes: get the first x primes whose product <= LIMIT
# if we were to swap even one of the small primes in this list with a slightly larger prime, then
# product((prime - 1)/prime) for each prime would actually get bigger.
# thus making 1 / product((prime - 1)/prime) for each prime smaller.
# therefore, we just get the first couple primes whose product is <= LIMIT.
product = 2
for i in xrange(3, LIMIT + 1, 2):
if product * i > LIMIT:
break
if primeObject.isPrime(i):
product *= i
end = time()
print "Value for which ratio is maximized: ", product
print "Runtime: ", end - start, " seconds."
def main():
start = time()
primeObject = ProjectEulerPrime()
# the question is actually asking for the output of the totient function
# from 2 to the problem's LIMIT.
phiList = [1 for number in xrange(LIMIT + 1)]
# defaults.
phiList[0] = 0
phiList[1] = 0
for i in xrange(2, LIMIT + 1):
if primeObject.isPrime(i):
phiList[i] = i - 1
for j in xrange(i * 2, len(phiList), i):
phiList[j] *= float(i - 1) / i
else:
phiList[i] *= i
print "Number of reduced, proper fractions where the denominator <= ", LIMIT, " : ", sum([int(round(number, 0)) for number in phiList])
end = time()
print "Runtime: ", end - start, " seconds."
def main():
start = time()
primeObject = ProjectEulerPrime()
# the question is actually asking for the output of the totient function
# from 2 to the problem's LIMIT.
phiList = [1 for number in xrange(LIMIT + 1)]
# defaults.
phiList[0] = 0
phiList[1] = 0
for i in xrange(2, LIMIT + 1):
if primeObject.isPrime(i):
phiList[i] = i - 1
for j in xrange(i * 2, len(phiList), i):
phiList[j] *= float(i - 1) / i
else:
phiList[i] *= i
print "Number of reduced, proper fractions where the denominator <= ", LIMIT, " : ", sum(
[int(round(number, 0)) for number in phiList])
end = time()
print "Runtime: ", end - start, " seconds."
def main():
start = time()
primeObject = ProjectEulerPrime()
primeList = [2] # 2 is the only even prime.
for i in xrange(3, LIMIT / 183, 2): # interesting pattern:
# all primes will be in the range of numbers
if primeObject.isPrime(
i
): # LIMIT / (number of prime terms in the series for LIMIT / 10 )
primeList.append(
i
) # so 10 = 10 / 1, 100 = 100 / 2, 1000 = 1000 / 6, 10000 = 10000 / 21,
# 100000 = 100000 / 65, 1000000 = 1000000 / 183 and so on for at least 100M.
maxPrime = 0
maxNumberOfPrimes = 0
for i in xrange(len(primeList)):
currentPrimeSequence = [primeList[i]]
currentSum = primeList[i]
for j in xrange(i + 1, len(primeList)):
currentSum += primeList[j]
currentPrimeSequence.append(primeList[j])
if currentSum < LIMIT and primeObject.isPrime(currentSum):
if len(currentPrimeSequence) > maxNumberOfPrimes:
maxNumberOfPrimes = len(currentPrimeSequence)
maxPrime = sum(currentPrimeSequence)
elif currentSum >= LIMIT:
break
print "Longest number of sequential primes that sum to a prime: ", maxNumberOfPrimes
print "Sum: ", maxPrime
end = time()
print "Runtime: ", end - start, " seconds."
def main():
start = time()
primeObject = ProjectEulerPrime()
primeList = [2] # 2 is the only even prime.
for i in xrange(3, LIMIT / 183, 2): # interesting pattern:
# all primes will be in the range of numbers
if primeObject.isPrime(i): # LIMIT / (number of prime terms in the series for LIMIT / 10 )
primeList.append(i) # so 10 = 10 / 1, 100 = 100 / 2, 1000 = 1000 / 6, 10000 = 10000 / 21,
# 100000 = 100000 / 65, 1000000 = 1000000 / 183 and so on for at least 100M.
maxPrime = 0
maxNumberOfPrimes = 0
for i in xrange(len(primeList)):
currentPrimeSequence = [primeList[i]]
currentSum = primeList[i]
for j in xrange(i + 1, len(primeList)):
currentSum += primeList[j]
currentPrimeSequence.append(primeList[j])
if currentSum < LIMIT and primeObject.isPrime(currentSum):
if len(currentPrimeSequence) > maxNumberOfPrimes:
maxNumberOfPrimes = len(currentPrimeSequence)
maxPrime = sum(currentPrimeSequence)
elif currentSum >= LIMIT:
break
print "Longest number of sequential primes that sum to a prime: ", maxNumberOfPrimes
print "Sum: ", maxPrime
end = time()
print "Runtime: ", end - start, " seconds."
def main():
'''
If you do the algebra, you find that:
R(k) % n = 0
=> R(k) = a * n, where a is a natural number.
=> (10 ** k - 1) / 9 = a * n
=> 10 ** k = 9 * a * n + 1
And so 10 ** k mod 9 * n == 1
You also see:
(n - 1) = b * k, where b is a natural number.
=> (n - 1) % k == 0
'''
current = 91 # start at first composite example.
p = ProjectEulerPrime()
results = set()
while len(results) < 25:
# obviously, 10 ** i > (9 * current).
# note that since current is odd, current - 1 is even.
for i in xrange(int(log10(9 * current) + 1), int((current - 1) / 2.) + 1):
if (current - 1) % i == 0 and 10 ** i % (9 * current) == 1:
results.add(current)
break
while True:
current += 2
if current % 5 == 0: continue # check for multiples of 5
if p.isPrime(current): continue # test for primality.
break
print "Sum: {}".format(sum(results))
def main():
start = time()
primeObject = ProjectEulerPrime()
# so, my first attempt at this used Euler's product formula and was really slow.
# Thanks to the people in the forum of P69 of Project Euler for help with this.
# we want to maximize n / phi(n). phi(n) = n * product((prime - 1)/prime) for each prime
# therefore, we want to maximize 1 / product((prime - 1)/prime) for each prime
# let's think about the most naive strategy to get these primes: get the first x primes whose product <= LIMIT
# if we were to swap even one of the small primes in this list with a slightly larger prime, then
# product((prime - 1)/prime) for each prime would actually get bigger.
# thus making 1 / product((prime - 1)/prime) for each prime smaller.
# therefore, we just get the first couple primes whose product is <= LIMIT.
product = 2
for i in xrange(3, LIMIT + 1, 2):
if product * i > LIMIT:
break
if primeObject.isPrime(i):
product *= i
end = time()
print "Value for which ratio is maximized: ", product
print "Runtime: ", end - start, " seconds."
def main():
start = time()
primeObject = ProjectEulerPrime()
minRatio = LIMIT
minValue = LIMIT
sqrtOfLimit = int(LIMIT ** 0.5)
# since phi(n) = n - 1 when n == prime, phi(n) can't ever be a permutation of n.
# the next best thing is a pair of primes.
# since n / phi(n) means being as close to 1 as possible, look for numbers around sqrt(LIMIT)
rangeToConsider = RANGE_INCREMENT
while rangeToConsider < RANGE_LIMIT:
for i in xrange(sqrtOfLimit - rangeToConsider, sqrtOfLimit, 1):
if primeObject.isPrime(i):
for j in xrange(sqrtOfLimit + rangeToConsider, sqrtOfLimit, -1):
candidateProduct = i * j
if candidateProduct > LIMIT:
continue
if primeObject.isPrime(j):
phi = round(i * j * float(i - 1) / i * float(j - 1) / j, 0)
if i * j / phi < minRatio:
candidateCharList = [char for char in str(i * j)]
phiCharList = [char for char in str(int(phi))]
candidateCharList.sort()
phiCharList.sort()
if candidateCharList == phiCharList:
minRatio = i * j / phi
minValue = i * j
rangeToConsider += RANGE_INCREMENT
end = time()
print "Value for which ratio is minimized: ", minValue
print "Runtime: ", end - start, " seconds."
def main():
start = time()
primeObject = ProjectEulerPrime()
minRatio = LIMIT
minValue = LIMIT
sqrtOfLimit = int(LIMIT**0.5)
# since phi(n) = n - 1 when n == prime, phi(n) can't ever be a permutation of n.
# the next best thing is a pair of primes.
# since n / phi(n) means being as close to 1 as possible, look for numbers around sqrt(LIMIT)
rangeToConsider = RANGE_INCREMENT
while rangeToConsider < RANGE_LIMIT:
for i in xrange(sqrtOfLimit - rangeToConsider, sqrtOfLimit, 1):
if primeObject.isPrime(i):
for j in xrange(sqrtOfLimit + rangeToConsider, sqrtOfLimit,
-1):
candidateProduct = i * j
if candidateProduct > LIMIT:
continue
if primeObject.isPrime(j):
phi = round(
i * j * float(i - 1) / i * float(j - 1) / j, 0)
if i * j / phi < minRatio:
candidateCharList = [char for char in str(i * j)]
phiCharList = [char for char in str(int(phi))]
candidateCharList.sort()
phiCharList.sort()
if candidateCharList == phiCharList:
minRatio = i * j / phi
minValue = i * j
rangeToConsider += RANGE_INCREMENT
end = time()
print "Value for which ratio is minimized: ", minValue
print "Runtime: ", end - start, " seconds."
def generate_next_prime(start=2):
''' generator that spits out primes '''
# since 2 is the only even number,
# immediately yield it and start
# the below loop at the first odd prime.
if start == 2:
yield 2
start = 3
prime_object = ProjectEulerPrime()
while True:
if prime_object.isPrime(start):
yield start
start += 2
def main():
LIMIT = 10 ** 5
ITERATION = 10 ** 4
p = ProjectEulerPrime()
radDict = {} # product(rad(n)) = list of all n with that prime factorization
for i in xrange(1, LIMIT + 1):
product = reduce(mul, set(p.factorize(i)))
if product not in radDict:
radDict[product] = [i] # since we iterate from 1..LIMIT, use a list to store n in order.
else:
radDict[product].append(i)
print "Solution: ", getSortedValue(ITERATION, radDict)
def generate_next_prime(start=2):
''' generator that spits out primes '''
# since 2 is the only even number,
# immediately yield it and start
# the below loop at the first odd prime.
if start == 2:
yield 2
start = 3
prime_object = ProjectEulerPrime()
while True:
if prime_object.isPrime(start):
yield start
start += 2
def main():
start = time()
primeObject = ProjectEulerPrime()
LIMIT = 10000 # setting a limit too high enables finding 5-way pairs that have huge last numbers (eg, 20000)
# 10,000 found through trial and error to be sufficient.
primeList = [x for x in xrange(LIMIT) if primeObject.isPrime(x)]
solutionList = find5WayPrimes(primeList, primeObject)
print "Solutions: ", solutionList
print "Sum: ", sum(solutionList)
end = time()
print "Runtime: ", end - start, " seconds."
def get_primes_by_changing_two_digits(repeated_digit,
digits,
primes=ProjectEulerPrime()):
''' similar to the algo above, but alter two digits instead of 1 '''
source = list(str(repeated_digit) * digits)
for first_sub_place in xrange(len(source)):
for second_sub_place in (value for value in xrange(len(source)) \
if value != first_sub_place):
candidate = list(source)
for first_repl_number in \
(value for value in xrange(0, 9 + 1) if value != repeated_digit):
# first number can't be 0 as that would shrink the number of
# digits.
if first_repl_number == 0 and first_sub_place == 0:
continue
candidate[first_sub_place] = str(first_repl_number)
for second_repl_number in (value for value in \
xrange(0, 9 + 1) if value != repeated_digit):
if second_repl_number == 0 and second_sub_place == 0:
continue
candidate[second_sub_place] = str(second_repl_number)
candidate_number = int(''.join(candidate))
if primes.isPrime(candidate_number):
yield candidate_number
def getNumberOfDivisors(n, p=ProjectEulerPrime()):
'''
the formula to get the number of divisors is (n ** 2 + 1) / 2, as given
by Euler in http://projecteuler.net/thread=108
we first create a dict listing the frequency of a prime in the
factorization collection. the formula to get the number of divisors is:
(p_1 + 1) * (p_2 + 1) * ... * (p_n + 1)
where p_i is the frequency of a prime number in a number's factorization
'''
n = n**2
factors = p.factorize(n)
def getFrequency(factors):
frequencyDict = {}
for factor in factors:
if factor in frequencyDict:
frequencyDict[factor] += 1
else:
frequencyDict[factor] = 1
return frequencyDict
frequencyDict = getFrequency(factors)
divisorNumber = 1
for prime in frequencyDict:
divisorNumber *= (frequencyDict[prime] + 1)
return (divisorNumber + 1) / 2
def get_primes_by_changing_one_digit(repeated_digit,
digits,
primes=ProjectEulerPrime()):
'''
using a given digit to repeat and the length of the numbers we're
interested in, alter one place at a time with a digit and see
whether that results in a prime number. if so, yield it.
'''
source = list(str(repeated_digit) * digits)
for substitution_place in xrange(len(source)):
candidate = list(source)
for replacement_number in \
(value for value in xrange(0, 9 + 1) if value != repeated_digit):
# first number can't be 0 as that would shrink the number of
# digits.
if replacement_number == 0 and replacement_number == 0:
continue
candidate[substitution_place] = str(replacement_number)
candidate_number = int(''.join(candidate))
if primes.isPrime(candidate_number):
yield candidate_number
def main():
start = time()
primeObject = ProjectEulerPrime()
LIMIT = 10000 # setting a limit too high enables finding 5-way pairs that have huge last numbers (eg, 20000)
# 10,000 found through trial and error to be sufficient.
primeList = [x for x in xrange(LIMIT) if primeObject.isPrime(x)]
solutionList = find5WayPrimes(primeList, primeObject)
print "Solutions: ", solutionList
print "Sum: ", sum(solutionList)
end = time()
print "Runtime: ", end - start, " seconds."
def main():
LIMIT = 4 * 10 ** 6
p = ProjectEulerPrime()
# using an algo inspired by hk:
# http://projecteuler.net/thread=108
# first step: get a product of unique primes
# such that the number of divisors is >= LIMIT.
factors = getUniqueFactorization(LIMIT, p)
# second step: start minimizing the divisor number with a floor of LIMIT
# do this by removing the largest prime and replacing it with
# a group of primes s.t. their product is < the largest prime.
bestDivisorNumber = getNumberOfDivisors(factors)
bestFactorization = factors
for largestPrime in reversed(factors): # we need to try to replace each original prime once.
iterationBestDivisorNumber = (
bestDivisorNumber
) # these are used so as to not update the absolute minimized factorization
iterationBestFactorization = bestFactorization # until we finish a loop.
for i in xrange(4, largestPrime): # optimization: skip 2 and 3 as we can't replace primes with another prime.
if p.isPrime(i):
continue # skip primes.
newFactorization = p.factorize(i)
newFactorization.extend(bestFactorization)
newFactorization.remove(largestPrime)
newDivisorNumber = getNumberOfDivisors(newFactorization)
if newDivisorNumber >= LIMIT and newDivisorNumber < iterationBestDivisorNumber:
iterationBestFactorization = newFactorization
iterationBestDivisorNumber = newDivisorNumber
bestFactorization = iterationBestFactorization
bestDivisorNumber = iterationBestDivisorNumber
print "Lowest number with >=", LIMIT, "possible 2 unit fraction additions: ", reduce(mul, bestFactorization)
print "Number of additions: ", bestDivisorNumber
def main():
start = time()
primeObject = ProjectEulerPrime()
solutionSet = set([])
squareList = [
number**2 for number in xrange(2,
int(LIMIT**0.5) + 1)
if primeObject.isPrime(number)
]
cubeList = [
number**3 for number in xrange(2,
int(LIMIT**(1. / 3)) + 1)
if primeObject.isPrime(number)
]
fourthList = [
number**4 for number in xrange(2,
int(LIMIT**(1. / 4)) + 1)
if primeObject.isPrime(number)
]
for square in squareList:
for cube in cubeList:
if square + cube >= LIMIT:
break
for fourth in fourthList:
resultingSum = square + cube + fourth
if resultingSum >= LIMIT:
break
else:
solutionSet.add(resultingSum)
print "Number of sums formed from squared, cubed, and fourthed primes that are < ", LIMIT, " : ", len(
solutionSet)
end = time()
print "Runtime: ", end - start, " seconds."
def phi(number, primeObject=ProjectEulerPrime()):
''' return totient(n). this is a naive implementation.
look at problem 72 for an efficient way to do this
for multiple numbers you want phi() for.
'''
result = number
for prime in frozenset(primeObject.factorize(number)):
result *= 1 - float(1) / prime
return int(result)
def main():
start = time()
primeObject = ProjectEulerPrime()
print "First level with appropriate prime ratio (expressed as a side length): ",
print determineSpiralLevelWithCorrectRatio(primeObject)
end = time()
print "Time: ", end - start, " seconds."
def getNextPrime(beginning, p=ProjectEulerPrime()):
''' get the next prime after an input: beginning '''
i = beginning
nextPrime = 0
while nextPrime == 0:
i += 1
if p.isPrime(i):
nextPrime = i
return nextPrime
def main():
primeObject = ProjectEulerPrime()
longestChain = []
for i in xrange(2, LIMIT):
chainLinks = createAmicableChain(i, primeObject)
if chainLinks is not None and len(chainLinks) > len(longestChain):
longestChain = chainLinks
print "Lowest value:", min(longestChain)
def main():
primeObject = ProjectEulerPrime()
solution = 0
numberOfWays = [0 for i in xrange(LIMIT + 1)]
numberOfWays[0] = 1 # start it off
for i in xrange(2, LIMIT + 1):
if primeObject.isPrime(i):
for j in xrange(i, LIMIT + 1): numberOfWays[j] += numberOfWays[j - i]
if numberOfWays[i] >= SUM_LIMIT:
solution = i
break
print "First number to be produced", SUM_LIMIT, "different ways using sums of primes:", solution
def slowAlgorithm():
start = time()
primeObject = ProjectEulerPrime()
maxValue = LIMIT + 1
maxRatio = 0
for i in xrange(2, LIMIT + 1):
# Euler's product formula:
# phi(n) = n * product((1 - (1 / unique prime divisors))
phi = i
for prime in set(primeObject.factorize(i)):
phi *= 1 - float(1) / prime
if i / phi > maxRatio:
maxRatio = i / phi
maxValue = i
end = time()
print "Value for which ratio is maximized: ", maxValue
print "Runtime: ", end - start, " seconds."
def main():
primeObject = ProjectEulerPrime()
solution = 0
numberOfWays = [0 for i in xrange(LIMIT + 1)]
numberOfWays[0] = 1 # start it off
for i in xrange(2, LIMIT + 1):
if primeObject.isPrime(i):
for j in xrange(i, LIMIT + 1):
numberOfWays[j] += numberOfWays[j - i]
if numberOfWays[i] >= SUM_LIMIT:
solution = i
break
print "First number to be produced", SUM_LIMIT, "different ways using sums of primes:", solution
def main():
# generate primes.
p = ProjectEulerPrime()
primes = {2}
for i in xrange(3, 100000, 2):
if p.isPrime(i):
primes.add(i)
'''
We use the following facts:
R(n) mod(n)
=> ((10 ** n - 1) / 9) mod n
=> (10 ** n) mod (9 * n) = 1
x ** y (mod n) == x ** (y mod phi(n)) (mod n)
We stop once we detect a repeated residual
(which means that we've entered a period).
By sheer luck, I used a similar technique in problem 282
to collapse much larger power towers.
'''
results = set()
for prime in primes:
phi_mod = phi(9 * prime)
mod = 9 * prime
seen_residuals = set()
j = 1
while True:
residual = pow(10, pow(10, j, phi_mod), mod)
if residual in seen_residuals:
break
if residual == 1:
results.add(prime)
break
seen_residuals.add(residual)
j += 1
print "Sum: {}".format(sum(primes - results))
def main(ceiling):
primes = ProjectEulerPrime()
special_primes = 0
# n ** 3 + p * n ** 2 = x ** 3 can be re-written as:
# n ** 2 * (n + p) = x ** 3
# since x, p, and n are positive integers, we know that
# n ** 2 and (n + p) must be themselves perfect cubes.
#
# note that the expression can also be re-written as:
# n * (n ** 2 + n * p) = x ** 3, meaning that n must be a perfect cube.
#
# furthermore, if you inspect the first few primes that
# have the special property of the problem, you see this pattern:
# p + n = perfect cube (root)
# 7 + 1 = 8 (2)
# 19 + 8 = 27 (3)
# 37 + 27 = 64 (4)
# 61 + 64 = 125 (5)
# 127 + 216 = 343 (7)
# you see that the prime # is that which pushes x ** 3 -> (x + 1) ** 3
# it's also not 100% in effect (n=125, perfect cube = 216)
root = 1
while True:
difference = (root + 1) ** 3 - root ** 3
# you can see that the gulf between two adjacent perfect cubes
# is going to grow larger and larger. eventually, the prime falls out
# of our scope.
if difference >= ceiling:
break
elif primes.isPrime(difference):
special_primes += 1
root += 1
print "Number of special primes below %d: %d" % (ceiling, special_primes)
def slowAlgorithm():
start = time()
primeObject = ProjectEulerPrime()
maxValue = LIMIT + 1
maxRatio = 0
for i in xrange(2, LIMIT + 1):
# Euler's product formula:
# phi(n) = n * product((1 - (1 / unique prime divisors))
phi = i
for prime in set(primeObject.factorize(i)):
phi *= 1 - float(1) / prime
if i / phi > maxRatio:
maxRatio = i / phi
maxValue = i
end = time()
print "Value for which ratio is maximized: ", maxValue
print "Runtime: ", end - start, " seconds."
def main():
# generate primes.
p = ProjectEulerPrime()
primes = {2}
for i in xrange(3, 100000, 2):
if p.isPrime(i):
primes.add(i)
'''
We use the following facts:
R(n) mod(n)
=> ((10 ** n - 1) / 9) mod n
=> (10 ** n) mod (9 * n) = 1
x ** y (mod n) == x ** (y mod phi(n)) (mod n)
We stop once we detect a repeated residual
(which means that we've entered a period).
By sheer luck, I used a similar technique in problem 282
to collapse much larger power towers.
'''
results = set()
for prime in primes:
phi_mod = phi(9 * prime)
mod = 9 * prime
seen_residuals = set()
j = 1
while True:
residual = pow(10, pow(10, j, phi_mod), mod)
if residual in seen_residuals:
break
if residual == 1:
results.add(prime)
break
seen_residuals.add(residual)
j += 1
print "Sum: {}".format(sum(primes - results))
def main():
start = time()
primeObject = ProjectEulerPrime()
solutionSet = set([])
primeDict = {}
constructPrimeMap(primeObject, primeDict)
discoverArithmeticSequences(primeDict, solutionSet)
for solution in solutionSet:
print "Concatenated tuple: ", ''.join(
[str(number) for number in solution])
end = time()
print "Runtime: ", end - start, " seconds."
def main():
p = ProjectEulerPrime()
solutions = set()
LIMIT = 12000
for k in xrange(2, LIMIT + 1):
for i in xrange(
k, 2 * k + 1
): # found via trial and error to the the range in which this number exists for a given k.
factorization_generator = FactorizationGenerator(
k=k, original_number=i, factorization=p.factorize(i))
if factorization_generator.can_we_select_correct_factorization_using_k_numbers(
):
solutions.add(i)
break
print "Solutions:", sum(solutions)
def get_primes_from_repeated_zeroes(digits, primes=ProjectEulerPrime()):
'''
if zero, then we know (through empirical analysis) that we have
to change the first and last digits.
'''
source = list('0' * digits)
for first_replacement_number in xrange(1, 9 + 1):
source[0] = str(first_replacement_number)
for second_replacement_number in xrange(1, 9 + 1):
source[len(source) - 1] = str(second_replacement_number)
candidate_number = int(''.join(source))
if primes.isPrime(candidate_number):
yield candidate_number
def main():
p = ProjectEulerPrime()
solutions = set()
LIMIT = 12000
for k in xrange(2, LIMIT + 1):
for i in xrange(k, 2 * k + 1): # found via trial and error to the the range in which this number exists for a given k.
factorization_generator = FactorizationGenerator(k = k, original_number = i, factorization = p.factorize(i))
if factorization_generator.can_we_select_correct_factorization_using_k_numbers():
solutions.add(i)
break
print "Solutions:", sum(solutions)
def main(digits):
'''
through trial and error, I've found that most primes can be created
by just altering one number (at least for digits=2..13).
the rest can be created by altering two places.
'''
primes = ProjectEulerPrime()
special_primes = dict()
for repeated_digit in xrange(0, 9 + 1):
special_primes[repeated_digit] = set()
# zero is a special case as we just need to alter the first and
# last numbers.
if repeated_digit == 0:
for prime in get_primes_from_repeated_zeroes(digits, primes):
special_primes[repeated_digit].add(prime)
else:
# swap one number.
for prime in get_primes_by_changing_one_digit(
repeated_digit, digits, primes):
special_primes[repeated_digit].add(prime)
if len(special_primes[repeated_digit]) == 0:
# swap two numbers.
for prime in get_primes_by_changing_two_digits(
repeated_digit, digits, primes):
special_primes[repeated_digit].add(prime)
if len(special_primes[repeated_digit]) == 0:
# trouble.
raise Exception("%d requires > 2 swaps, but it's not supported" % \
repeated_digit)
total = 0
for digit in sorted(special_primes):
summation = sum(special_primes[digit])
print "S(%d, %d): %d" % (digits, digit, summation)
total += summation
print "Total: %d" % total
def main():
'''
To solve this problem, I use base-11 numbers instead of base-10; X is my 11th base.
I swap X characters to create prime numbers in base 10.
'''
start = time()
primeObject = ProjectEulerPrime()
solution = 0
candidateList = [
char for char in '56XX5'
] # has to be at least greater than the template for the smallest 7-member prime family
while solution == 0: # and odd, obviously.
if substitueAndCheckPrimality(primeObject, "".join(candidateList)):
solution = "".join(candidateList)
base11Increment(candidateList)
base11Increment(candidateList) # increment by 2 to keep number odd.
print "8-digit prime family template: ", solution
end = time()
print "Runtime: ", end - start, " seconds."
def main():
LIMIT = 10**10
p = ProjectEulerPrime()
# the algo is : remainder = p(n) * n * 2
# where p(n) is the nth prime
n = 3
prime = 5
solution = 0
while solution == 0:
prime = getNextPrime(prime, p)
n += 1
if n & 1 != 0: # the remainder for all even n from this formula is 2
if prime * 2 * n >= LIMIT:
solution = n
print "Solution: ", solution
def main():
# the strategy here is to first find
# all the primes composed of digits 1-9
# (with at most one of each digit) and to sort
# them by size.
# We then try all the various ways
# of adding up to 9 by adding variously-sized groups
# (ie, groups of 1 prime, 2 primes, .... 6 primes)
# There can only be at most 6 as primes must end with
# an odd number (thus, 5 numbers), and then there's 2.
prime_object = ProjectEulerPrime()
primes_sorted_by_size = dict()
digits = tuple(str(i) for i in xrange(1, 10))
for number_length in xrange(1, len(digits) + 1):
for permutation in permutations(digits, number_length):
number = int(''.join(permutation))
if prime_object.isPrime(number):
if len(permutation) not in primes_sorted_by_size:
primes_sorted_by_size[len(permutation)] = set()
primes_sorted_by_size[len(permutation)].add(tuple(permutation))
exclusively_prime_groups = 0
# 1-member groups
if 9 in primes_sorted_by_size:
exclusively_prime_groups += len(primes_sorted_by_size[9])
# 2-member:
def two_member_good_group_finder(x, y):
if x + y != 9:
raise Exception("%d + %d != 9" % (x, y))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
if len(uniques_with_second) == 9:
good_groups.add(frozenset([first, second]))
return len(good_groups)
exclusively_prime_groups += two_member_good_group_finder(8, 1)
exclusively_prime_groups += two_member_good_group_finder(7, 2)
exclusively_prime_groups += two_member_good_group_finder(6, 3)
exclusively_prime_groups += two_member_good_group_finder(5, 4)
# 3-member:
def three_member_good_group_finder(x, y, z):
if x + y + z != 9:
raise Exception("%d + %d + %d != 9" % (x, y, z))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
all_uniques = set(uniques_with_second)
for digit in third:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(frozenset([first, second, third]))
return len(good_groups)
exclusively_prime_groups += three_member_good_group_finder(7, 1, 1)
exclusively_prime_groups += three_member_good_group_finder(6, 2, 1)
exclusively_prime_groups += three_member_good_group_finder(5, 3, 1)
exclusively_prime_groups += three_member_good_group_finder(5, 2, 2)
exclusively_prime_groups += three_member_good_group_finder(4, 3, 2)
exclusively_prime_groups += three_member_good_group_finder(4, 4, 1)
exclusively_prime_groups += three_member_good_group_finder(3, 3, 3)
# 4-member:
def four_member_good_group_finder(x, y, z, xx):
if x + y + z + xx != 9:
raise Exception("%d + %d + %d + %d != 9" % (x, y, z, xx))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
uniques_with_third = set(uniques_with_second)
for digit in third:
uniques_with_third.add(digit)
for fourth in primes_sorted_by_size[xx]:
if fourth == third or fourth == second or \
fourth == first:
continue
all_uniques = set(uniques_with_third)
for digit in fourth:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(frozenset([first, second, third,
fourth]))
return len(good_groups)
exclusively_prime_groups += four_member_good_group_finder(6, 1, 1, 1)
exclusively_prime_groups += four_member_good_group_finder(5, 2, 1, 1)
exclusively_prime_groups += four_member_good_group_finder(4, 2, 2, 1)
exclusively_prime_groups += four_member_good_group_finder(4, 3, 1, 1)
exclusively_prime_groups += four_member_good_group_finder(3, 3, 2, 1)
exclusively_prime_groups += four_member_good_group_finder(3, 2, 2, 2)
# 5-member:
def five_member_good_group_finder(x, y, z, xx, yy):
if x + y + z + xx + yy != 9:
raise Exception("%d + %d + %d + %d + %d != 9" % (x, y, z, xx, yy))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
uniques_with_third = set(uniques_with_second)
for digit in third:
uniques_with_third.add(digit)
for fourth in primes_sorted_by_size[xx]:
if fourth == third or fourth == second or \
fourth == first:
continue
uniques_with_fourth = set(uniques_with_third)
for digit in fourth:
uniques_with_fourth.add(digit)
for fifth in primes_sorted_by_size[yy]:
if fifth == first or fifth == second or \
fifth == third or fifth == fourth:
continue
all_uniques = set(uniques_with_fourth)
for digit in fifth:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(frozenset([first, second,
third, fourth, fifth]))
return len(good_groups)
exclusively_prime_groups += five_member_good_group_finder(5, 1, 1, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(4, 2, 1, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(3, 3, 1, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(3, 2, 2, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(2, 2, 2, 2, 1)
# 6-member
def six_member_good_group_finder(x, y, z, xx, yy, zz):
if x + y + z + xx + yy + zz != 9:
raise Exception("%d + %d + %d + %d + %d != 9" %
(x, y, z, xx, yy, zz))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
uniques_with_third = set(uniques_with_second)
for digit in third:
uniques_with_third.add(digit)
for fourth in primes_sorted_by_size[xx]:
if fourth == third or fourth == second or \
fourth == first:
continue
uniques_with_fourth = set(uniques_with_third)
for digit in fourth:
uniques_with_fourth.add(digit)
for fifth in primes_sorted_by_size[yy]:
if fifth == first or fifth == second or \
fifth == third or fifth == fourth:
continue
uniques_with_fifth = set(uniques_with_fourth)
for digit in fifth:
uniques_with_fifth.add(digit)
for sixth in primes_sorted_by_size[zz]:
if sixth == first or sixth == second or \
sixth == third or sixth == fourth or \
sixth == fifth:
continue
all_uniques = set(uniques_with_fifth)
for digit in sixth:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(frozenset([first, second,
third, fourth, fifth, sixth]))
return len(good_groups)
exclusively_prime_groups += six_member_good_group_finder(2, 2, 2, 1, 1, 1)
exclusively_prime_groups += six_member_good_group_finder(2, 3, 1, 1, 1, 1)
print "Number of groups: %d" % exclusively_prime_groups
def main():
# the strategy here is to first find
# all the primes composed of digits 1-9
# (with at most one of each digit) and to sort
# them by size.
# We then try all the various ways
# of adding up to 9 by adding variously-sized groups
# (ie, groups of 1 prime, 2 primes, .... 6 primes)
# There can only be at most 6 as primes must end with
# an odd number (thus, 5 numbers), and then there's 2.
prime_object = ProjectEulerPrime()
primes_sorted_by_size = dict()
digits = tuple(str(i) for i in xrange(1, 10))
for number_length in xrange(1, len(digits) + 1):
for permutation in permutations(digits, number_length):
number = int(''.join(permutation))
if prime_object.isPrime(number):
if len(permutation) not in primes_sorted_by_size:
primes_sorted_by_size[len(permutation)] = set()
primes_sorted_by_size[len(permutation)].add(tuple(permutation))
exclusively_prime_groups = 0
# 1-member groups
if 9 in primes_sorted_by_size:
exclusively_prime_groups += len(primes_sorted_by_size[9])
# 2-member:
def two_member_good_group_finder(x, y):
if x + y != 9:
raise Exception("%d + %d != 9" % (x, y))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
if len(uniques_with_second) == 9:
good_groups.add(frozenset([first, second]))
return len(good_groups)
exclusively_prime_groups += two_member_good_group_finder(8, 1)
exclusively_prime_groups += two_member_good_group_finder(7, 2)
exclusively_prime_groups += two_member_good_group_finder(6, 3)
exclusively_prime_groups += two_member_good_group_finder(5, 4)
# 3-member:
def three_member_good_group_finder(x, y, z):
if x + y + z != 9:
raise Exception("%d + %d + %d != 9" % (x, y, z))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
all_uniques = set(uniques_with_second)
for digit in third:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(frozenset([first, second, third]))
return len(good_groups)
exclusively_prime_groups += three_member_good_group_finder(7, 1, 1)
exclusively_prime_groups += three_member_good_group_finder(6, 2, 1)
exclusively_prime_groups += three_member_good_group_finder(5, 3, 1)
exclusively_prime_groups += three_member_good_group_finder(5, 2, 2)
exclusively_prime_groups += three_member_good_group_finder(4, 3, 2)
exclusively_prime_groups += three_member_good_group_finder(4, 4, 1)
exclusively_prime_groups += three_member_good_group_finder(3, 3, 3)
# 4-member:
def four_member_good_group_finder(x, y, z, xx):
if x + y + z + xx != 9:
raise Exception("%d + %d + %d + %d != 9" % (x, y, z, xx))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
uniques_with_third = set(uniques_with_second)
for digit in third:
uniques_with_third.add(digit)
for fourth in primes_sorted_by_size[xx]:
if fourth == third or fourth == second or \
fourth == first:
continue
all_uniques = set(uniques_with_third)
for digit in fourth:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(
frozenset([first, second, third, fourth]))
return len(good_groups)
exclusively_prime_groups += four_member_good_group_finder(6, 1, 1, 1)
exclusively_prime_groups += four_member_good_group_finder(5, 2, 1, 1)
exclusively_prime_groups += four_member_good_group_finder(4, 2, 2, 1)
exclusively_prime_groups += four_member_good_group_finder(4, 3, 1, 1)
exclusively_prime_groups += four_member_good_group_finder(3, 3, 2, 1)
exclusively_prime_groups += four_member_good_group_finder(3, 2, 2, 2)
# 5-member:
def five_member_good_group_finder(x, y, z, xx, yy):
if x + y + z + xx + yy != 9:
raise Exception("%d + %d + %d + %d + %d != 9" % (x, y, z, xx, yy))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
uniques_with_third = set(uniques_with_second)
for digit in third:
uniques_with_third.add(digit)
for fourth in primes_sorted_by_size[xx]:
if fourth == third or fourth == second or \
fourth == first:
continue
uniques_with_fourth = set(uniques_with_third)
for digit in fourth:
uniques_with_fourth.add(digit)
for fifth in primes_sorted_by_size[yy]:
if fifth == first or fifth == second or \
fifth == third or fifth == fourth:
continue
all_uniques = set(uniques_with_fourth)
for digit in fifth:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(
frozenset(
[first, second, third, fourth, fifth]))
return len(good_groups)
exclusively_prime_groups += five_member_good_group_finder(5, 1, 1, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(4, 2, 1, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(3, 3, 1, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(3, 2, 2, 1, 1)
exclusively_prime_groups += five_member_good_group_finder(2, 2, 2, 2, 1)
# 6-member
def six_member_good_group_finder(x, y, z, xx, yy, zz):
if x + y + z + xx + yy + zz != 9:
raise Exception("%d + %d + %d + %d + %d != 9" %
(x, y, z, xx, yy, zz))
good_groups = set()
for first in primes_sorted_by_size[x]:
uniques = set(digit for digit in first)
for second in primes_sorted_by_size[y]:
if first == second:
continue
uniques_with_second = set(uniques)
for digit in second:
uniques_with_second.add(digit)
for third in primes_sorted_by_size[z]:
if third == second or third == first:
continue
uniques_with_third = set(uniques_with_second)
for digit in third:
uniques_with_third.add(digit)
for fourth in primes_sorted_by_size[xx]:
if fourth == third or fourth == second or \
fourth == first:
continue
uniques_with_fourth = set(uniques_with_third)
for digit in fourth:
uniques_with_fourth.add(digit)
for fifth in primes_sorted_by_size[yy]:
if fifth == first or fifth == second or \
fifth == third or fifth == fourth:
continue
uniques_with_fifth = set(uniques_with_fourth)
for digit in fifth:
uniques_with_fifth.add(digit)
for sixth in primes_sorted_by_size[zz]:
if sixth == first or sixth == second or \
sixth == third or sixth == fourth or \
sixth == fifth:
continue
all_uniques = set(uniques_with_fifth)
for digit in sixth:
all_uniques.add(digit)
if len(all_uniques) == 9:
good_groups.add(
frozenset([
first, second, third, fourth,
fifth, sixth
]))
return len(good_groups)
exclusively_prime_groups += six_member_good_group_finder(2, 2, 2, 1, 1, 1)
exclusively_prime_groups += six_member_good_group_finder(2, 3, 1, 1, 1, 1)
print "Number of groups: %d" % exclusively_prime_groups