def compute():
isprime = eulerlib.list_primality(10000) # Cache for small numbers
digits = list(range(1, 10))
def count_prime_sets(startindex, prevnum):
if startindex == len(digits):
return 1
else:
result = 0
for split in range(startindex + 1, len(digits) + 1):
num = int("".join(map(str, digits[startindex : split])))
if num > prevnum and is_prime(num):
result += count_prime_sets(split, num)
return result
def is_prime(n):
if n < len(isprime):
return isprime[n]
else:
return eulerlib.is_prime(n)
ans = 0
while True:
ans += count_prime_sets(0, 0)
if not next_permutation(digits):
break
return str(ans)
def compute():
LIMIT = 10**8
ans = 0
isprime = eulerlib.list_primality(LIMIT - 1)
# Search all possible x's. Note that a + 1 = x * y * y >= x. Furthermore, a < b, so a + 1 <= b.
# Thus if x >= LIMIT, then LIMIT <= a + 1 <= b. With b >= LIMIT, no candidates are possible.
for x in range(1, LIMIT):
# Search all possible y's. Notice that when y increases, 'a' strictly increases.
# So when some y generates an 'a' such that a >= LIMIT, no candidates are possible with higher values of y.
for y in itertools.count(1):
a = x * y * y - 1
if a >= LIMIT:
break
if not isprime[a]:
continue
# Search all valid z's. We require z > y and gcd(y, z) = 1. Notice that when z increases, c strictly increases.
# So when some z generates a c such that c >= LIMIT, no candidates are possible with higher values of z.
for z in itertools.count(y + 1):
if fractions.gcd(y, z) != 1:
continue
c = x * z * z - 1
if c >= LIMIT:
break
# Check whether (a, b, c) is a solution
b = x * y * z - 1
if isprime[b] and isprime[c]:
ans += a + b + c
return str(ans)
def compute():
LIMIT = 10**8
ans = 0
isprime = eulerlib.list_primality(LIMIT - 1)
# Search all possible x's. We know that c = x * z * z - 1. With the requirement c < LIMIT, we have x * z * z <= LIMIT.
# Because z > y > 0, we know z >= 2. So at the very least we require x * 4 <= LIMIT. This implies x <= floor(LIMIT/4).
for x in range(1, LIMIT // 4 + 1):
# Search all possible y's. Notice that when y increases, 'a' strictly increases.
# So when some y generates an 'a' such that a >= LIMIT, no candidates are possible with higher values of y.
for y in itertools.count(1):
a = x * y * y - 1
if a >= LIMIT:
break
if not isprime[a]:
continue
# Search all valid z's. We require z > y and gcd(y, z) = 1. Notice that when z increases, c strictly increases.
# So when some z generates a c such that c >= LIMIT, no candidates are possible with higher values of z.
for z in itertools.count(y + 1):
if fractions.gcd(y, z) != 1:
continue
c = x * z * z - 1
if c >= LIMIT:
break
# Check whether (a, b, c) is a solution
if isprime[c]:
b = x * y * z - 1
if isprime[b]:
ans += a + b + c
return str(ans)
def compute():
isprime = eulerlib.list_primality(10000) # Cache for small numbers
digits = list(range(1, 10))
def count_prime_sets(startindex, prevnum):
if startindex == len(digits):
return 1
else:
result = 0
for split in range(startindex + 1, len(digits) + 1):
num = int("".join(map(str, digits[startindex:split])))
if num > prevnum and is_prime(num):
result += count_prime_sets(split, num)
return result
def is_prime(n):
if n < len(isprime):
return isprime[n]
else:
return eulerlib.is_prime(n)
ans = 0
while True:
ans += count_prime_sets(0, 0)
if not eulerlib.next_permutation(digits):
break
return str(ans)
def compute(): LIMIT = 10**8 ans = 0 isprime = eulerlib.list_primality(LIMIT - 1) # Search all possible x's. We know that c = x * z * z - 1. With the requirement c < LIMIT, we have x * z * z <= LIMIT. # Because z > y > 0, we know z >= 2. So at the very least we require x * 4 <= LIMIT. This implies x <= floor(LIMIT/4). for x in range(1, LIMIT // 4 + 1): # Search all possible y's. Notice that when y increases, 'a' strictly increases. # So when some y generates an 'a' such that a >= LIMIT, no candidates are possible with higher values of y. for y in itertools.count(1): a = x * y * y - 1 if a >= LIMIT: break if not isprime[a]: continue # Search all valid z's. We require z > y and gcd(y, z) = 1. Notice that when z increases, c strictly increases. # So when some z generates a c such that c >= LIMIT, no candidates are possible with higher values of z. for z in itertools.count(y + 1): if fractions.gcd(y, z) != 1: continue c = x * z * z - 1 if c >= LIMIT: break # Check whether (a, b, c) is a solution if isprime[c]: b = x * y * z - 1 if isprime[b]: ans += a + b + c return str(ans)
def compute(): START_NUM = 1 END_NUM = 500 CROAK_SEQ = "PPPPNNPPPNPPNPN" NUM_JUMPS = len(CROAK_SEQ) - 1 NUM_TRIALS = 2**NUM_JUMPS globalnumerator = 0 isprime = eulerlib.list_primality(END_NUM) for i in range(START_NUM, END_NUM + 1): for j in range(NUM_TRIALS): pos = i trialnumerator = 1 if isprime[pos] == (CROAK_SEQ[0] == 'P'): trialnumerator *= 2 for k in range(NUM_JUMPS): if pos <= START_NUM: pos += 1 elif pos >= END_NUM: pos -= 1 elif (j >> k) & 1 == 0: pos += 1 else: pos -= 1 if isprime[pos] == (CROAK_SEQ[k + 1] == 'P'): trialnumerator *= 2 globalnumerator += trialnumerator globaldenominator = (END_NUM + 1 - START_NUM) * 2**NUM_JUMPS * 3**len(CROAK_SEQ) ans = fractions.Fraction(globalnumerator, globaldenominator) return str(ans)
def compute():
primes = eulerlib.list_primality(999999)
def is_circular_prime(n):
s = str(n)
return all(primes[int(s[i:] + s[:i])] for i in range(len(s)))
return sum(1 for i in range(1000000) if is_circular_prime(i))
def compute():
primes = eulerlib.list_primality(999999)
def is_truncatable(num):
return all(primes[int(str(num)[:i])]
for i in range(1, len(str(num)))) and all(
primes[int(str(num)[i:])] for i in range(len(str(num))))
return sum(i for i in range(10, 999999) if is_truncatable(i))
def compute():
isprime = eulerlib.list_primality(999999)
def is_circular_prime(n):
s = str(n)
return all(isprime[int(s[i:] + s[:i])] for i in range(len(s)))
ans = sum(1 for i in range(len(isprime)) if is_circular_prime(i))
return str(ans)
def compute(): isprime = eulerlib.list_primality(999999) def is_circular_prime(n): s = str(n) return all(isprime[int(s[i : ] + s[ : i])] for i in range(len(s))) ans = sum(1 for i in range(len(isprime)) if is_circular_prime(i)) return str(ans)
def calc():
isprime = eulerlib.list_primality(999999)
def s(n):
k = str(n)
return all(isprime[int(k[i:] + k[:i])] for i in range(len(k)))
a = sum(1 for i in range(len(isprime)) if s(i))
return str(a)
print(calc())
def compute(): LIMIT = 10**8 isprime = eulerlib.list_primality(LIMIT + 1) def is_prime_generating(n): return all( n % d != 0 or isprime[d + n // d] for d in range(2, eulerlib.sqrt(n) + 1)) ans = sum(n for n in range(LIMIT + 1) if isprime[n + 1] and is_prime_generating(n)) return str(ans)
def compute():
primes = eulerlib.list_primality(1000000)
for i in range(33, 1000000, 2):
if primes[i]:
continue
isOk = True
for j in range(1, int(pow(i / 2, 1 / 2)) + 1):
if primes[i - j * j * 2]:
isOk = False
break
if isOk:
print(i)
break
def compute():
LIMIT = 10000
isprime = eulerlib.list_primality(LIMIT - 1)
for base in range(1000, LIMIT):
if isprime[base]:
for step in range(1, LIMIT):
a = base + step
b = a + step
if a < LIMIT and isprime[a] and has_same_digits(a, base) \
and b < LIMIT and isprime[b] and has_same_digits(b, base) \
and (base != 1487 or a != 4817):
return str(base) + str(a) + str(b)
raise RuntimeError("Not found")
def compute():
isprime = eulerlib.list_primality(
999999) # List of True, False,.... for whether each number is prime.
def is_circular_prime(n):
s = str(n)
# Returns true if all elements pass a test:
return all(isprime[int(s[i:] + s[:i])]
for i in range(len(s))) # Cycling
ans = sum(
1 # Ahhh instead of i for i in .... we're just adding 1 each time
for i in range(len(isprime)) if is_circular_prime(i))
return str(ans)
def compute():
primes = eulerlib.list_primality(10000)
for i in range(1000, 10000):
if not primes[i]:
continue
changeList = createList(str(i))
for j in changeList:
if j <= i:
continue
if list.__contains__(
changeList, i +
(j - i) * 2) and primes[j] and primes[i + (j - i) * 2]:
print(str(i) + str(j) + str(i + (j - i) * 2))
break
def compute(): LIMIT = 5000 MODULUS = 10**16 # Use dynamic programming. count[i] is the number of subsets of primes with the sum of i, modulo MODULUS. count = [0] * (LIMIT**2 // 2) count[0] = 1 s = 0 # Sum of all primes seen so far, and thus the highest index among nonzero entries in 'count' for p in eulerlib.list_primes(LIMIT): for i in range(s, -1, -1): count[i + p] = (count[i + p] + count[i]) % MODULUS s += p isprime = eulerlib.list_primality(len(count)) ans = sum(count[i] for i in range(len(count)) if isprime[i]) % MODULUS return str(ans)
def compute():
LIMIT = 5000
MODULUS = 10**16
# Use dynamic programming. count[i] is the number of subsets of primes with the sum of i, modulo MODULUS.
count = [0] * (LIMIT**2 // 2)
count[0] = 1
s = 0 # Sum of all primes seen so far, and thus the highest index among nonzero entries in 'count'
for p in eulerlib.list_primes(LIMIT):
for i in range(s, -1, -1):
count[i + p] = (count[i + p] + count[i]) % MODULUS
s += p
isprime = eulerlib.list_primality(len(count))
ans = sum(count[i] for i in range(len(count)) if isprime[i]) % MODULUS
return str(ans)
def compute():
ans = 0
isprime = eulerlib.list_primality(999999)
primes = eulerlib.list_primes(999999)
consecutive = 0
for i in range(len(primes)):
sum = primes[i]
consec = 1
for j in range(i + 1, len(primes)):
sum += primes[j]
consec += 1
if sum >= len(isprime):
break
if isprime[sum] and consec > consecutive:
ans = sum
consecutive = consec
return str(ans)
def compute(): ans = 0 isprime = eulerlib.list_primality(999999) primes = eulerlib.list_primes(999999) consecutive = 0 for i in range(len(primes)): sum = primes[i] consec = 1 for j in range(i + 1, len(primes)): sum += primes[j] consec += 1 if sum >= len(isprime): break if isprime[sum] and consec > consecutive: ans = sum consecutive = consec return str(ans)
def compute():
START_NUM = 1
END_NUM = 500
CROAK_SEQ = "PPPPNNPPPNPPNPN"
assert 0 <= START_NUM < END_NUM
assert 1 <= len(CROAK_SEQ)
NUM_JUMPS = len(CROAK_SEQ) - 1
NUM_TRIALS = 2**NUM_JUMPS
globalnumerator = 0
isprime = eulerlib.list_primality(END_NUM)
# For each starting square
for i in range(START_NUM, END_NUM + 1):
# For each sequence of jumps
for j in range(NUM_TRIALS):
# Set initial position and croak
pos = i
trialnumerator = 1
if isprime[pos] == (CROAK_SEQ[0] == 'P'):
trialnumerator *= 2
# Simulate each jump and croak
for k in range(NUM_JUMPS):
if pos <= START_NUM:
pos += 1 # Forced move
elif pos >= END_NUM:
pos -= 1 # Forced move
elif (j >> k) & 1 == 0:
pos += 1 # Chosen move
else:
pos -= 1 # Chosen move
# Multiply the running probability by 2/3 if primeness of current position
# matches croak sequence at current index, otherwise multiply by 1/3
if isprime[pos] == (CROAK_SEQ[k + 1] == 'P'):
trialnumerator *= 2
globalnumerator += trialnumerator
# Calculate final probability fraction
globaldenominator = (END_NUM + 1 -
START_NUM) * 2**NUM_JUMPS * 3**len(CROAK_SEQ)
ans = fractions.Fraction(globalnumerator, globaldenominator)
return str(ans)
def compute():
primes = eulerlib.list_primes(1000000)
isPrimes = eulerlib.list_primality(1000000)
max = 0
maxConsecutive = 0
for i in range(len(primes)):
sum = 0
length = 0
for j in range(i, len(primes)):
sum += primes[j]
length += 1
if sum > 1000000:
break
if isPrimes[sum]:
if length > maxConsecutive:
max = sum
maxConsecutive = length
print(max)
def compute():
primes = eulerlib.list_primality(7071)
helo = [False] * (5 * 10**7)
for i in range(7071):
if primes[i]:
for j in range(368):
if i**2 + j**3 > len(helo) - 1:
break
if primes[j]:
for k in range(84):
if i**2 + j**3 + k**4 > len(helo) - 1:
break
if primes[k]:
helo[i**2 + j**3 + k**4] = True
ans = 0
for i in range(len(helo)):
if helo[i]:
ans += 1
return ans
def compute():
isprime = eulerlib.list_primality(1000000)
for i in range(len(isprime)):
if not isprime[i]:
continue
n = [int(c) for c in str(i)]
for mask in range(1 << len(n)):
digits = do_mask(n, mask)
count = 0
for j in range(10):
if digits[0] != 0 and isprime[to_number(digits)]:
count += 1
digits = add_mask(digits, mask)
if count == 8:
digits = do_mask(n, mask)
for j in range(10):
if digits[0] != 0 and isprime[to_number(digits)]:
return str(to_number(digits))
digits = add_mask(digits, mask)
raise AssertionError("Not found")
def compute():
LIMIT = 10**7
isprime = eulerlib.list_primality(LIMIT)
# pathmax[i] = None if i is not prime or i is not connected to 2.
# Otherwise, considering all connection paths from 2 to i and for each path computing
# the maximum number, pathmax[i] is the minimum number among all these maxima.
pathmax = [None] * len(isprime)
# Process paths in increasing order of maximum number
queue = [(2, 2)]
while len(queue) > 0:
pmax, n = heapq.heappop(queue)
if pathmax[n] is not None and pmax >= pathmax[n]:
# This happens if at the time this update was queued, a better
# or equally good update was queued ahead but not processed yet
continue
# Update the target node and explore neighbors
pathmax[n] = pmax
# Try all replacements of a single digit, including the leading zero.
# This generates exactly all (no more, no less) the ways that a number m is connected to n.
digits = to_digits(n)
tempdigits = list(digits)
for i in range(len(tempdigits)): # For each digit position
for j in range(10): # For each digit value
tempdigits[i] = j
m = to_number(tempdigits)
nextpmax = max(m, pmax)
if m < len(isprime) and isprime[m] and (pathmax[m] is None or
nextpmax < pathmax[m]):
heapq.heappush(queue, (nextpmax, m))
tempdigits[i] = digits[i] # Restore the digit
ans = sum(i for i in range(len(isprime))
if (isprime[i] and (pathmax[i] is None or pathmax[i] > i)))
return str(ans)
def compute(): LIMIT = 10**7 isprime = eulerlib.list_primality(LIMIT) # pathmax[i] = None if i is not prime or i is not connected to 2. # Otherwise, considering all connection paths from 2 to i and for each path computing # the maximum number, pathmax[i] is the minimum number among all these maxima. pathmax = [None] * len(isprime) # Process paths in increasing order of maximum number queue = [(2, 2)] while len(queue) > 0: pmax, n = heapq.heappop(queue) if pathmax[n] is not None and pmax >= pathmax[n]: # This happens if at the time this update was queued, a better # or equally good update was queued ahead but not processed yet continue # Update the target node and explore neighbors pathmax[n] = pmax # Try all replacements of a single digit, including the leading zero. # This generates exactly all (no more, no less) the ways that a number m is connected to n. digits = to_digits(n) tempdigits = list(digits) for i in range(len(tempdigits)): # For each digit position for j in range(10): # For each digit value tempdigits[i] = j m = to_number(tempdigits) nextpmax = max(m, pmax) if m < len(isprime) and isprime[m] and (pathmax[m] is None or nextpmax < pathmax[m]): heapq.heappush(queue, (nextpmax, m)) tempdigits[i] = digits[i] # Restore the digit ans = sum(i for i in range(len(isprime)) if isprime[i] and (pathmax[i] is None or pathmax[i] > i)) return str(ans)
def compute(): TARGET = 500500 MODULUS = 500500507 isprime = eulerlib.list_primality(7376507) # 500500th (1-based) prime number queue = [] nextprime = 2 heapq.heappush(queue, nextprime) ans = 1 for _ in range(TARGET): item = heapq.heappop(queue) ans *= item ans %= MODULUS heapq.heappush(queue, item**2) if item == nextprime: nextprime += 1 while not isprime[nextprime]: nextprime += 1 heapq.heappush(queue, nextprime) return str(ans)
import eulerlib, itertools def compute(): ans = max(((a, b) for a in range(-999, 1000) for b in range(2, 1000)), key=count_consecutive_primes) return str(ans[0] * ans[1]) def count_consecutive_primes(ab): a, b = ab for i in itertools.count(): n = i * i + i * a + b if not is_prime(n): return i isprimecache = eulerlib.list_primality(1000) def is_prime(n): if n < 0: return False elif n < len(isprimecache): return isprimecache[n] else: return eulerlib.is_prime(n) if __name__ == "__main__": print(compute())
import eulerlib, itertools
def calc():
ans = max(((a, b) for a in range(-999, 1000) for b in range(2, 1000)),
key=ppc)
return str(ans[0] * ans[1])
def ppc(ab): #счет простых чисел
a, b = ab
for i in itertools.count():
n = i * i + i * a + b
if not is_prime(n):
return i
main_c = eulerlib.list_primality(1000) #основной кеш
def is_prime(n):
if n < 0:
return False
elif n < len(main_c):
return main_c[n]
else:
return eulerlib.is_prime(n)
print(calc())
import eulerlib, sys
if sys.version_info.major == 2:
range = xrange
isprime = eulerlib.list_primality(999999)
def compute():
ans = sum(1 for i in range(len(isprime)) if is_circular_prime(i))
return str(ans)
def is_circular_prime(n):
s = str(n)
return all(isprime[int(s[i:] + s[:i])] for i in range(len(s)))
if __name__ == "__main__":
print(compute())
# # Solution to Project Euler problem 35 # by Project Nayuki # # http://www.nayuki.io/page/project-euler-solutions # https://github.com/nayuki/Project-Euler-solutions # import eulerlib, sys if sys.version_info.major == 2: range = xrange isprime = eulerlib.list_primality(999999) def compute(): ans = 0 for i in range(len(isprime)): if is_circular_prime(i): ans += 1 return str(ans) def is_circular_prime(n): s = str(n) for i in range(len(s)): if not isprime[int(s[i : ] + s[ : i])]: return False return True
def compute():
isprime = eulerlib.list_primality(20000000)
ans = sum(sam_transitions_minus_max_transitions(i) for (i, p) in enumerate(isprime) if i >= 10000000 and p)
return str(ans)
def compute():
isprime = eulerlib.list_primality(20000000)
ans = sum(
sam_transitions_minus_max_transitions(i)
for (i, p) in enumerate(isprime) if i >= 10000000 and p)
return str(ans)
def compute():
ans = max(((a, b) for a in range(-999, 1000) for b in range(2, 1000)),
key=count_consecutive_primes)
return str(ans[0] * ans[1])
def count_consecutive_primes(ab):
a, b = ab
for i in itertools.count():
n = i * i + i * a + b
if not is_prime(n):
return i
isprimecache = eulerlib.list_primality(1000)
def is_prime(n):
if n < 0:
return False
elif n < len(isprimecache):
return isprimecache[n]
else:
return eulerlib.is_prime(n)
if __name__ == "__main__":
print(compute())