def Euler_41():
high = [0]
for i in range(1,8):
# primes = [str(x) for x in maths.primes(10**(i)//4,10**(i-1)//4)]
alist =["".join(x) for x in itertools.permutations("".join([str(x) for x in range(1,i+1)]))]
high=[x for x in alist if maths.isprime2(int(x))]
return high[-1]
def e146(top=10 ** 6):
ret = 0
x1 = lambda n: n ** 2 + 1
x2 = lambda n: n ** 2 + 3
x3 = lambda n: n ** 2 + 7
x4 = lambda n: n ** 2 + 9
x5 = lambda n: n ** 2 + 13
x6 = lambda n: n ** 2 + 27
test = [x1, x2, x3, x4, x5, x6]
mini = 1000
for i in range(1, top + 1):
if i > mini:
print(i)
mini += 1000
z = 0
yesno = i
for j in test:
y = j(i)
if z:
if y != z:
yesno = 0
break
if not isprime2(y):
continue
z = nextprime(y)
ret += yesno
return ret
def e146(top=10**6):
ret = 0
x1 = lambda n: n**2 + 1
x2 = lambda n: n**2 + 3
x3 = lambda n: n**2 + 7
x4 = lambda n: n**2 + 9
x5 = lambda n: n**2 + 13
x6 = lambda n: n**2 + 27
test = [x1, x2, x3, x4, x5, x6]
mini = 1000
for i in range(1, top + 1):
if i > mini:
print(i)
mini += 1000
z = 0
yesno = i
for j in test:
y = j(i)
if z:
if y != z:
yesno = 0
break
if not isprime2(y): continue
z = nextprime(y)
ret += yesno
return ret
def Euler_5(top=20):
val = 1
for i in range(1,top+1):
if maths.isprime2(i):
num = i
while num<=top:
num = num*i
val *=(num//i)
return val
def e131(top=10**6):
i = 1
j = 2
count = 0
while True:
p = j**3 - i**3
if p >= top: break
if isprime2(p): count += 1
i, j = j, j + 1
return count
def e131(top = 10**6):
i = 1
j = 2
count = 0
while True:
p = j**3-i**3
if p>=top:break
if isprime2(p):count+=1
i,j = j,j+1
return count
def factors(x,i=0):
if isprime2(x):return [x]
p = sieve(int(x**.5))
for j,prime in enumerate(p[i:]):
tmp = x%prime
if tmp==0:
tp = x//prime
print(tmp,prime)
return [prime]+factors(tp,j)
return [x]
def Euler_3(num=600851475143):
primes = []
i = 2
while i <= num:
if num % i == 0 and isprime2(i):
num = num // i
primes = primes + [i]
i = 2
i += 1
return max(primes)
def Euler_41():
high = [0]
for i in range(1, 8):
# primes = [str(x) for x in maths.primes(10**(i)//4,10**(i-1)//4)]
alist = [
"".join(x) for x in itertools.permutations("".join(
[str(x) for x in range(1, i + 1)]))
]
high = [x for x in alist if maths.isprime2(int(x))]
return high[-1]
def Euler_3(num=600851475143):
primes = []
i=2
while i <= num:
if num%i ==0 and isprime2(i):
num = num//i
primes = primes +[i]
i = 2
i+=1
return max(primes)
def factors(x, i=0):
if isprime2(x): return [x]
p = sieve(int(x**.5))
for j, prime in enumerate(p[i:]):
tmp = x % prime
if tmp == 0:
tp = x // prime
print(tmp, prime)
return [prime] + factors(tp, j)
return [x]
def factors(n):
global primes
p = primes
if n ==1:return [1]
if n in p:return [n]
else:
for i in range(len(p)):
if n%p[i]==0:
result =[p[i]]+factors(n//p[i])#,primes)
return result
if isprime2(n):return [n]
else:
primes+=[nextprime(primes[-1])]
return factors(n)#,primes)
def factors(n):
global primes
p = primes
if n == 1:
return [1]
if n in p:
return [n]
else:
for i in range(len(p)):
if n % p[i] == 0:
result = [p[i]] + factors(n // p[i]) # ,primes)
return result
if isprime2(n):
return [n]
else:
primes += [nextprime(primes[-1])]
return factors(n) # ,primes)
def nextprime(p):
if p%2==0:p+=1
while not isprime2(p):
p+=2
if p>=top:return []
return [p]+nextprime(p+1)
def nextprime(p):
if p % 2 == 0: p += 1
while not isprime2(p):
p += 2
if p >= top: return []
return [p] + nextprime(p + 1)