def run():
limit = 10_000 # We're interested in 4-digit primes, i.e. primes < 10000.
sieve = prime_sieve(limit)
# Loop through all 4-digit primes:
for primeCandidate in range(1000, limit - 1):
if sieve[primeCandidate]:
continue
# We're looking for three evenly spaced prime numbers below 10000.
# So we can at most add ((limit - 1) - primeCandidate) / 2, twice.
upperLimit = int(((limit - 1) - primeCandidate) / 2)
for j in range(1, upperLimit):
# Calculate the two candidates, then check whether they're prime:
primeCandidate2 = primeCandidate + j
primeCandidate3 = primeCandidate + (2 * j)
if sieve[primeCandidate2] or sieve[primeCandidate3]:
continue
# Now we know all three numbers are prime, we still have to test
# whether they're also permutations of one another:
else:
sortedPrime = sorted(str(primeCandidate))
if sortedPrime == sorted(
str(primeCandidate2)) and sortedPrime == sorted(
str(primeCandidate3)):
# Now we concatenate the three primes into the answer string,
# and make sure that it's not the answer already provided in the problem description.
answer_string = str(primeCandidate) + str(
primeCandidate2) + str(primeCandidate3)
if answer_string != "148748178147":
return answer_string
def run():
# Generate the prime sieve, and create a generator for all primes below the limit.
limit = 1_000_000 # arbitrary, but it yields the correct result
sieve = prime_sieve(limit)
primes = (i for i in range(1, limit) if not sieve[i])
for prime in primes:
prime_length = len(str(prime))
# Loop through all possibilities of how many digits to replace - at most all but one.
# Then use itertools.combinations to yield all combinations of which *digit positions* to replace.
# The construction below yields all the combinations of the integers 0 ... len(prime)
# of length amount_to_replace, in sorted order, i.e. there could be a "0129",
# but then there'd be no "9012". Which is exactly what I want.
for amount_to_replace in range(1, prime_length):
for digitsToReplace in combinations(range(prime_length),
amount_to_replace):
primeFamily = []
newNumberList = [digit for digit in str(prime)]
for newDigit in range(
10
): # Replace the chosen digits with the numbers 0...9.
for index in digitsToReplace:
newNumberList[index] = str(newDigit)
newNumber = int("".join(newNumberList))
# If newNumber begins with a zero, disregard it:
if len(str(newNumber)) < prime_length:
continue
# If newNumber is prime, add it to the primeFamily.
if not sieve[newNumber]:
primeFamily.append(newNumber)
# If we've found a primeFamily of length 8, we're done:
if len(primeFamily) == 8:
return min(primeFamily)
def run():
limit = 800_000 # arbitrary, but it yields the correct result
sieve = prime_sieve(limit)
count_trunc = 0
truncSum = 0
for primeCandidate in range(
11, limit -
1): # The single-digit primes aren't considered truncatable.
if not sieve[primeCandidate]: # sieve[4] is True if 4 is *not* prime.
truncLeft = str(primeCandidate)
truncRight = str(primeCandidate)
while True:
# Truncate the primes. Then if the truncated numbers
# aren't prime anymore, exit the loop.
truncLeft = truncLeft[1:]
truncRight = truncRight[:-1]
if sieve[int(truncLeft)] or sieve[int(truncRight)]:
break
# Otherwise, check whether the numbers have been truncated to 1 digit,
# in which case we've found a new truncatable prime.
if len(truncLeft) == 1:
count_trunc += 1
truncSum += primeCandidate
break
# According to the description, there are only 11 truncatable primes.
# So we're done once we've found them all.
if count_trunc >= 11:
return truncSum
def run():
limit = 150_000 # According to https://primes.utm.edu/howmany.html, there are ~10000 primes below ~100000.
sieve = prime_sieve(limit)
counter = 0
for n in range(limit):
if not sieve[n]: # False = prime
counter += 1
if counter == 10001:
return n
def run():
# Generate the prime sieve, and create a list of all primes below the limit.
limit = 1_000_000 # Primes below one million.
sieve = prime_sieve(limit)
primeList = [i for i in range(1, limit) if not sieve[i]]
# Initialize some variables:
lowerLimit = 0
max_summands = 0
prime_with_most_summands = 0
# Loop through all primes in descending order.
primeIndex = len(primeList) - 1 # Begin with the largest prime in the list. If a list has 5 elements, it has length 5, but its largest index is 4.
prime = primeList[primeIndex]
while prime > lowerLimit:
prime_sum = 0
summandList = deque()
# Loop through all primes <= prime, then sum up consecutive primes
# and check whether the sum equals prime.
# Because we begin the sum with the smallest prime, once prime_sum == prime,
# we've automatically found the sum with the most summands for this prime.
for summandIndex in range(primeIndex):
if prime_sum < prime:
summand = primeList[summandIndex]
prime_sum += summand
summandList.append(summand)
elif prime_sum == prime:
# We've constructed the given prime as a sum of consecutive primes.
# Now we count the number of summands sumCounter and test if it's a new maximum.
# If it is, we increase the lower limit accordingly, so that it always corresponds to
# the sum of the sumCounter lowest primes.
sumCounter = len(summandList)
if sumCounter > max_summands:
# Increase the lower limit by however many summands have been added due to the increase in max:
for i in range(max_summands, sumCounter):
lowerLimit += primeList[i]
max_summands = sumCounter
prime_with_most_summands = prime
# Otherwise, prime_sum > prime, and we have to subtract summands from the left
# until prime_sum <= prime again.
else:
while prime_sum > prime:
prime_sum -= summandList.popleft()
# Finally, we go to the next-largest prime.
primeIndex -= 1
prime = primeList[primeIndex]
return prime_with_most_summands
def run():
limit = 1_000_000 # primes below 1000000
sieve = prime_sieve(limit)
# Loop through all primes. Rotate the primes,
# then check if their rotations are also prime.
count_circular = 0
for primeCandidate in range(1, limit - 1):
if not sieve[primeCandidate]: # sieve[4] is True if 4 is *not* prime.
candidateList = rotateNumber(primeCandidate)
# If all the rotated candidates are prime, we've found a circular prime.
for rotated_number in candidateList:
if sieve[rotated_number]:
break
else: # Triggers if the for-loop wasn't stopped early.
count_circular += 1
return count_circular
def run():
limit = 1_000_000 # Check primes below 1000000
sieve = prime_sieve(limit)
# Loop through the composite numbers:
for compositeCandidate in range(4, limit - 1):
# Check whether the current number and the next three consecutive numbers in the sieve
# are all composite. If they aren't, skip to the next compositeCandidate.
for i in range(3 + 1):
if not sieve[compositeCandidate + i]:
break
# Now that we've found four consecutive composite numbers,
# we count their distinct prime factors.
else:
for j in range(3 + 1):
if count_unique_prime_factors(compositeCandidate + j,
sieve) < 4:
break
# Once we've found four consecutive composites with
# at least 4 distinct prime factors each, we're done.
else:
return compositeCandidate
def run():
limit = 1_000_000 # arbitrary, but it yields the correct result
sieve = prime_sieve(limit)
# Loop through all odd composites, so begin with 9 (all smaller odd numbers are prime),
# and use a step size of 2.
for compositeCandidate in range(9, limit - 1, 2):
if sieve[
compositeCandidate]: # sieve[4] is True if 4 is composite, i.e. not prime.
# Once we've found an odd composite, we try to test the conjecure,
# i.e. we try to write the composite as the sum of a prime and twice a square.
# First, we loop through all valid squares for the "twice a square" part of the conjecture.
# The upper limit comes from setting the prime to 0, so composite = 0 + 2*a**2
# -> a = sqrt(composite/2) is the upper limit for a.
for i in range(1, int(sqrt(compositeCandidate / 2)) + 1):
doubleSquare = 2 * (i**2)
# Finally, we test if the composite minus the doubled square is still prime:
# composite = potentialPrime + doubleSquare -> potentialPrime = composite - doubleSquare.
if not sieve[compositeCandidate - doubleSquare]:
break
# We're done once we've found an odd composite which doesn't fulfill the condition:
else:
return compositeCandidate
def run():
limit = 2_000_000 # Numbers *below* 2,000,000.
sieve = prime_sieve(limit)
prime_sum = sum(n for n in range(limit) if not sieve[n])
return prime_sum
