def test_goldbach(n):
if n % 2 == 0 or eulerlib.is_prime(n):
return True
for i in itertools.count(1):
k = n - 2 * i * i
if k <= 0:
return False
elif eulerlib.is_prime(k):
return True
def test_goldbach(n):
if n % 2 == 0 or eulerlib.is_prime(n):
return True
for i in itertools.count(1):
k = n - 2 * i * i
if k <= 0:
return False
elif eulerlib.is_prime(k):
return True
def answer():
ans = 1487
while (True):
ans += 2
b = ans + 3330
c = ans + 6660
if (eulerlib.is_prime(ans) and eulerlib.is_prime(b)
and eulerlib.is_prime(c) and eulerlib.is_perm(ans, b)
and eulerlib.is_perm(b, c)):
return str(ans) + str(b) + str(c)
def compute():
lim = 10000
for base in range(1489, lim, 2):
if eulerlib.is_prime(base):
a = base + 3330
b = a + 3330
if b < lim and eulerlib.is_prime(a) and is_perm(a, base) \
and eulerlib.is_prime(b) and is_perm(b, base):
return str(base) + str(a) + str(b)
raise RuntimeError("Not found")
def is_truncatable_prime(n):
i = 10
while i <= n:
if not eulerlib.is_prime(n % i):
return False
i *= 10
while n > 0:
if not eulerlib.is_prime(n):
return False
n //= 10
return True
def find_harshad_primes(n, digitsum, isstrong):
# Shift left by 1 digit, and try all 10 possibilities for the rightmost digit
m = n * 10
s = digitsum
for i in range(10):
if m >= LIMIT:
break
if isstrong and eulerlib.is_prime(m):
ans[0] += m
if m % s == 0:
find_harshad_primes(m, s, eulerlib.is_prime(m // s))
m += 1
s += 1
def find_harshad_primes(n, digitsum, isstrong):
# Shift left by 1 digit, and try all 10 possibilities for the rightmost digit
m = n * 10
s = digitsum
for i in range(10):
if m >= LIMIT:
break
if isstrong and eulerlib.is_prime(m):
ans[0] += m
if m % s == 0:
find_harshad_primes(m, s, eulerlib.is_prime(m // s))
m += 1
s += 1
def prime_factors(n):
a = []
t = n
if eulerlib.is_prime(t):
return [t]
for p in eulerlib.list_primes(t):
while t % p == 0:
t /= p
a.append(p)
if eulerlib.is_prime(t):
a.append(t)
break
return a
def is_truncatable_prime(n): # Test if left-truncatable i = 10 while i <= n: if not eulerlib.is_prime(n % i): return False i *= 10 # Test if right-truncatable while n > 0: if not eulerlib.is_prime(n): return False n //= 10 return True
def is_truncatable_prime(n):
# Test if left-truncatable
i = 10
while i <= n:
if not eulerlib.is_prime(n % i):
return False
i *= 10
# Test if right-truncatable
while n > 0:
if not eulerlib.is_prime(n):
return False
n //= 10
return True
def find(n: int):
primes = [p for p in range(1237, 9874) if lib.is_prime(p)]
primes = [((p, set(lib.digits(p))), (q, set(lib.digits(q)))) for p in primes for q in primes if q > p]
arithm = {}
answer = set()
for P, Q in primes:
p, dp = P
q, dq = Q
if dp != dq:
continue
else:
key = (tuple(sorted(dp)), q - p)
if key in arithm:
arithm[key].update({p, q})
else:
arithm[key] = {p, q}
if len(arithm[key]) == 3:
s = int("".join(map(str, sorted(arithm[key]))))
answer.add(s)
else:
return list(answer - {148748178147})[0]
def ysec(n):
# можно ли убрать слева
i = 10
while i <= n:
if not eulerlib.is_prime(n % i): #is_prime-проверка первичности
return False
i *= 10
# можно ли убрать справа
while n > 0:
if not eulerlib.is_prime(n):
return False
n //= 10
return True
print(calc())
def compute(): TARGET = fractions.Fraction(15499, 94744) totient = 1 denominator = 1 p = 2 while True: totient *= p - 1 denominator *= p # Note: At this point in the code, denominator is the product of one copy of each # prime number up to and including p, totient is equal to totient(denominator), # and totient/denominator = R'(2 * 3 * ... * p) (the pseudo-resilience). # Advance to the next prime while True: p += 1 if eulerlib.is_prime(p): break # If the lower bound is below the target, there might be a suitable solution d such that # d's factorization only contains prime factors strictly below the current (advanced) value of p if fractions.Fraction(totient, denominator) < TARGET: # Try to find the lowest factor i such that R(i*d) < TARGET, if any. # Note that over this range of i, we have R'(d) = R'(i*d) < R(i*d). for i in range(1, p): numer = i * totient denom = i * denominator if fractions.Fraction(numer, denom - 1) < TARGET: return str(denom)
def is_prime(n): if n < 0: return False elif n < len(isprimecache): return isprimecache[n] else: return eulerlib.is_prime(n)
def compute():
TARGET = fractions.Fraction(15499, 94744)
totient = 1
denominator = 1
p = 2
while True:
totient *= p - 1
denominator *= p
# Note: At this point in the code, denominator is the product of one copy of each
# prime number up to and including p, totient is equal to totient(denominator),
# and totient/denominator = R'(2 * 3 * ... * p) (the pseudo-resilience).
# Advance to the next prime
while True:
p += 1
if eulerlib.is_prime(p):
break
# If the lower bound is below the target, there might be a suitable solution d such that
# d's factorization only contains prime factors strictly below the current (advanced) value of p
if fractions.Fraction(totient, denominator) < TARGET:
# Try to find the lowest factor i such that R(i*d) < TARGET, if any.
# Note that over this range of i, we have R'(d) = R'(i*d) < R(i*d).
for i in range(1, p):
numer = i * totient
denom = i * denominator
if fractions.Fraction(numer, denom - 1) < TARGET:
return str(denom)
def is_prime(n):
if n < 0:
return False
elif n < len(isprimecache):
return isprimecache[n]
else:
return eulerlib.is_prime(n)
def answer():
ans = 7654321 # biggest pandigital 7-digit number
while not (eulerlib.is_prime(ans) and isPandigital(ans,len(str(ans)))):
ans -= 2
if(ans < 1000000):
ans = 4321
return str(ans)
def compute():
numprimes = 0
for i in itertools.count(1, 2):
for j in range(4):
if eulerlib.is_prime(i * i - j * (i - 1)):
numprimes += 1
if i > 1 and numprimes * 10 < i * 2 - 1:
return str(i)
def test_is_prime():
is_prime_cases = [
(-1, False), (0, False), (1, False), (2, True), (3, True), (4, False),
(5, True), (6, False), (7, True), (8, False), (9, False), (10, False),
(11, True), (12, False)
]
for inp, output in is_prime_cases:
assert is_prime(inp) == output
def prime_length(a: int, b: int, sieve: lib.Sieve = None):
n = 0
while True:
k = f(n, a, b)
if not lib.is_prime(k, sieve):
return n
else:
n += 1
def compute():
numprimes = 0
for i in itertools.count(1, 2):
for j in range(4):
if eulerlib.is_prime(i * i - j * (i - 1)):
numprimes += 1
if i > 1 and numprimes * 10 < i * 2 - 1:
return str(i)
def count_consecutive_primes(ab):
a, b = ab
for i in itertools.count():
n = i * i + i * a + b
if n < 0:
return i
elif not eulerlib.is_prime(n):
return i
def compute(): n = 1 count = 0 while count < 10001: n += 1 if eulerlib.is_prime(n): count += 1 return str(n)
def compute():
n = 1
count = 0
while count < 10001:
n += 1
if eulerlib.is_prime(n):
count += 1
return str(n)
def find(n):
primes = [p for p in range(n) if lib.is_prime(p)]
m = 0
r = 0
for i in range(len(primes)):
k = 0
s = 0
for j in range(i, len(primes)):
s += primes[j]
if s > n:
break
if lib.is_prime(s):
if k > m:
m = k
r = s
k += 1
return r
def fun(n):
if (is_prime(n) or math.gcd(n, 10) != 1):
return 0
repunit = 1
len_rep = 1
while (repunit % n != 0):
repunit = (repunit * 10 + 1) % n
len_rep += 1
return len_rep
def pe0041():
print 'Project Euler Problem 41 https://projecteuler.net/problem=41'
for i in range(7, 0, -1):
pan_perms = tuple(permutations(range(1, i + 1), i))[::-1]
for pan in pan_perms:
p = int(''.join(str(c) for c in pan))
if eulerlib.is_prime(p):
print 'Answer is %d' % p
return
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())
def compute(): TARGET = fractions.Fraction(1, 10) numprimes = 0 for n in itertools.count(1, 2): for i in range(4): if eulerlib.is_prime(n * n - i * (n - 1)): numprimes += 1 if n > 1 and fractions.Fraction(numprimes, n * 2 - 1) < TARGET: return str(n)
def compute():
TARGET = fractions.Fraction(1, 10)
numprimes = 0
for i in itertools.count(1, 2):
for j in range(4):
if eulerlib.is_prime(i * i - j * (i - 1)):
numprimes += 1
if i > 1 and fractions.Fraction(numprimes, i * 2 - 1) < TARGET:
return str(i)
def compute(): ans = 0 count = 0 for i in itertools.count(2): if eulerlib.is_prime(i) and repunit_mod(10**9, i) == 0: ans += i count += 1 if count == 40: return str(ans)
def calc():
a = list(reversed(range(1, n + 1)))
while True:
if a[-1] not in last:
n = int("".join(str(x) for x in a))
if eulerlib.is_prime(n):
return str(n)
if not per(a):
break
def compute():
TARGET = fractions.Fraction(1, 10)
numprimes = 0
for n in itertools.count(1, 2):
for i in range(4):
if eulerlib.is_prime(n * n - i * (n - 1)):
numprimes += 1
if n > 1 and fractions.Fraction(numprimes, n * 2 - 1) < TARGET:
return str(n)
def compute():
ans = 0
found = 0
for i in itertools.count(7, 2):
if i % 5 != 0 and not eulerlib.is_prime(i) and (i - 1) % find_least_divisible_repunit(i) == 0:
ans += i
found += 1
if found == 25:
return str(ans)
def compute():
ans = 0
found = 0
for i in itertools.count(7, 2):
if i % 5 != 0 and not eulerlib.is_prime(i) and (
i - 1) % find_least_divisible_repunit(i) == 0:
ans += i
found += 1
if found == 25:
return str(ans)
def compute():
for i in range(9, 0, -1):
arr = list(range(i, 0, -1))
while True:
if i == 1 or arr[-1] not in NONPRIME_LAST_DIGITS:
n = int("".join([str(x) for x in arr]))
if eulerlib.is_prime(n):
return str(n)
if not prev_permutation(arr):
break
def num_prime_sum_ways(n):
for i in range(primes[-1] + 1, n + 1):
if eulerlib.is_prime(i):
primes.append(i)
ways = [1] + [0] * n
for p in primes:
for i in range(n + 1 - p):
ways[i + p] += ways[i]
return ways[n]
def compute(): sum = 0 count = 0 i = 2 while count < 40: if eulerlib.is_prime(i) and repunit_mod(10**9, i) == 0: sum += i count += 1 i += 1 return str(sum)
def compute():
for i in range(9, 0, -1):
arr = list(range(i, 0, -1))
while True:
if i == 1 or arr[-1] not in NONPRIME_LAST_DIGITS:
n = int("".join(str(x) for x in arr))
if eulerlib.is_prime(n):
return str(n)
if not prev_permutation(arr):
break
def num_prime_sum_ways(n): for i in range(primes[-1] + 1, n + 1): if eulerlib.is_prime(i): primes.append(i) ways = [1] + [0] * n for p in primes: for i in range(n + 1 - p): ways[i + p] += ways[i] return ways[n]
def p007():
i = 2
primes = []
while len(primes) < 10001:
if eulerlib.is_prime(i):
primes.append(i)
i = i + 1
return primes
def get_family_count(mk, n, d):
count = 1
num = list(str(n))
for i in range(d + 1, 10):
for j in range(0, len(mk)):
if not mk[j]:
num[j] = str(i)
if eulerlib.is_prime(int(''.join(x for x in num))):
count += 1
return count
def divisor(n):
cond=False
for i in range(1,int(math.sqrt(n))+1):
if(n%i==0):
if(l.is_prime(i+n//i)):
cond=True
else:
print(n)
cond=False
break
return cond
def compute(): TARGET = 2000 # Must be at least 3 count = 2 # Because n = 1 and 2 satisfy PD(n) = 3 for ring in itertools.count(2): if eulerlib.is_prime(ring * 6 - 1) and eulerlib.is_prime(ring * 6 + 1) and eulerlib.is_prime(ring * 12 + 5): count += 1 if count == TARGET: return str(ring * (ring - 1) * 3 + 2) if eulerlib.is_prime(ring * 6 - 1) and eulerlib.is_prime(ring * 6 + 5) and eulerlib.is_prime(ring * 12 - 7): count += 1 if count == TARGET: return str(ring * (ring + 1) * 3 + 1)
def is_prime(n): if n < len(isprime): return isprime[n] else: return eulerlib.is_prime(n)
def compute():
# Among the integers starting from 2, take the sum of
# the first 40 integers satisfying the filter condition
cond = lambda i: eulerlib.is_prime(i) and repunit_mod(10 ** 9, i) == 0
ans = sum(itertools.islice(filter(cond, itertools.count(2)), 40))
return str(ans)
def compute(): cond = lambda i: (i % 5 != 0) and (not eulerlib.is_prime(i)) \ and ((i - 1) % find_least_divisible_repunit(i) == 0) ans = sum(itertools.islice(filter(cond, itertools.count(7, 2)), 25)) return str(ans)
