def search(s):
for n in nt.primeList(1000000):
n = str(n)
for i in n:
if rep(n.replace(str(i), "*"), int(i == n[0])) == s:
return n.replace(str(i), "*")
return 0
def randprime(bits, safeness = 0):
rng = random.SystemRandom()
i = 0
while len('{0:b}'.format(i)) != bits:
i = rng.getrandbits(bits)
i += 1 - (i % 2)
pList = list(primeList(1000))
while True:
if all(i % p not in ([0] + [(p-(2**n-1))//2**n for n in range(safeness + 1)]) for p in pList) and all(isPrime(2**n*i+2**n-1) for n in range(safeness + 1)):
return i
i += 2
def squarefree(n):
d = 1
for p in numbertheory.primeList(int(math.sqrt(n))):
d *= p ** 2
n *= p ** 2 - 1
return n/d
import numbertheory
def n(p1, p2):
size = len(str(p1))
k = ((p2-p1)*pow(10**size, p2 - 2, p2))%p2
return k*10**size+p1
primes = numbertheory.primeList(10**9)
next(primes)#2
next(primes)#3
p1 = next(primes)#5
total = 0
for p2 in primes:
total += n(p1, p2)
p1 = p2
print(total)
import numbertheory
LIMIT = 100000
l = [0]*LIMIT
l[0] = 1
plist = numbertheory.primeList(LIMIT*2+1)
next(plist)
for x in plist:
for y in range(int((LIMIT - x//2)**.5)):
l[x//2+y**2] = 1
for x in range(LIMIT):
if l[x] == 0:
print(2*x+1)
break
import numbertheory as asdf
import math
p = list(asdf.primeList(10**8))
s = 0
for prime in p:
if prime * 2 < 10 ** 8:
foo = list(asdf.primeList(10**8//prime))
for n in foo:
if n >= prime:
s += 1
print(s)
##684288 (809, 31, 2) (827, 7, 2)
##105418 (191, 41, 2) (317, 17, 2)
##148902 (3, 53, 2) (383, 13, 2)
##149014 (11, 53, 2) (353, 29, 2)
##159094 (101, 53, 2) (367, 29, 2)
##171094 (149, 53, 2) (383, 29, 2)
##211894 (251, 53, 2) (433, 29, 2)
##334654 (431, 53, 2) (557, 29, 2)
##1142902 (997, 53, 2) (1069, 5, 2)
##1520134 (1171, 53, 2) (1223, 29, 2)
##2770054 (1619, 53, 2) (1657, 29, 2)
##12044494 (3449, 53, 2) (3467, 29, 2)
##334276 (359, 59, 2) (577, 11, 2)
##1068436 (929, 59, 2) (1033, 11, 2)
##2112556 (1381, 59, 2) (1453, 11, 2)
##12685048 (3511, 71, 2) (3547, 47, 2)
t, r = 0, 0
trim = {}
for q in nt.primeList(84):
for c in nt.primeList(int(pow(5*10**7 - q**4, 1/3))):
for s in nt.primeList(int(pow(5*10**7 - q**4 - c**3, 1/2))):
x = s**2+c**3+q**4
if x in trim:
#print(x, (s, c, q), *trim[x])
r += 1
else:
t += 1
trim[x] = {(s, c, q)}
print(t)
from numbertheory import primeList
import numbertheory
def residues(n, m):
for i in range(m):
yield pow(i, n, m)
n = 13
for m in primeList(1000):
t = numbertheory.totient(m)
r = list(residues(n, m))
r = list(set(r))
r.sort()
if m % n == 1 and numbertheory.isPrime(m):
print(m , len(r), r)
import math
import numbertheory
#accounts for 2 and 5
count = 2
primes = list(numbertheory.primeList(1000000))
for prime in primes:
cyclic = True
prime = str(prime)
if '0' not in prime and '2' not in prime and '4' not in prime and '5' not in prime and '6' not in prime and '8' not in prime:
for x in range(1,len(prime)):
cyclic = cyclic and numbertheory.isPrime(int(prime[x:]+prime[:x]))
if cyclic:
count += 1
print(prime)
else:
pass
print(count)
import numbertheory
import itertools
plist = [x for x in numbertheory.primeList(10000)]
for x in plist:
i = list(map(int, [''.join(a) for a in itertools.permutations(str(x))]))
for y in plist:
if x < y and y != x and y in i and y in plist and (2 * y - x) in i and (2 * y - x) in plist:
print(x, y, (2 * y - x))
import numbertheory
LIMIT = 1000
c = [0]*(LIMIT + 1)
p = numbertheory.primeList(LIMIT)
for x in p:
for k in range(LIMIT//x + 1):
c[x*k] += x
b = [0]*(LIMIT + 1)
for n in range(1, LIMIT + 1):
b[n] = (c[n] + sum(c[k] * b[n - k] for k in range(1, n)))//n
for x in range(1, LIMIT):
if b[x] > 5000:
print(x, b[x])
break
