def prime_equivalences(bound):
"""
Let x,y be positive integers, with prime factorizations
x = p_1^a_1 * ... * p_n ^ a_n
y = q_1^b_1 * ... * q_n ^ b_n
"""
primecap = int(log(bound) **2 / log(10) ) + 10
primes = primes_and_mask(primecap)[0]
temp = bound
max_primes = 0
for i, p in enumerate(primes):
temp /= p
if not temp:
max_primes = i
break
res = []
state = [0 for i in xrange(max_primes)]
def recur(pind, bound, exp_bound):
if bound == 0: return
if exp_bound == 0: return
if pind >= max_primes: return
for exp in xrange(1, exp_bound+1):
state[pind] = exp
new_bound = bound / primes[pind]**exp
if new_bound <= 0: break
res.append(copy(state))
recur(pind+1, new_bound, exp)
#cleanup
state[pind] = 0
recur(0, bound, 100)
return res, primes[:max_primes]
def solve_1(cap,mod=10**9+7):
primes = primes_and_mask(cap)[0]
R = [1 for i in xrange(cap+1)]
N = [i**4+4 for i in xrange(cap+1)]
sq = [0,0,0,0]
for p in primes:
if p == 2:
for i in xrange(1,cap+1):
mul = 1
while N[i]%2==0:
N[i]/=2
mul*=2
if mul>1:
R[i] = (R[i]*(mul+1))%mod
continue
sq[0] = mod_sqrt(p-4,p)
if not sq[0]:
continue
sq[2] = mod_sqrt(sq[0],p)
sq[3] = mod_sqrt(p-sq[0],p)
sq[0] = (p-sq[3])%p
sq[1] = (p-sq[2])%p
# print p,sq
for q in xrange(0,cap+1,p):
for r in sq:
if r and r <= cap-q:
mul=1
while N[q+r]%p==0:
mul *= p
N[q+r]/=p
if mul > 1:
R[q+r] = (R[q+r]*(mul+1))%mod
print "Completed seiving."
c=0
s = 0
for i,r in enumerate(R):
if i==0:
continue
n = N[i]
if n ==1:
s = (s + r -i**4-4)%mod
continue
factors = shanks_factorize(n)
for f in factors:
if f>1:
r = (r*(f**factors[f]+1))%mod
if len(factors)>=2:
c +=1
s = (s + r -i**4-4)%mod
print s
print c
return s
def solve(n):
primes = np.array(pe.primes_and_mask(n)[0], dtype = np.int64)
pi = len(primes)
#product of first k primes mod p, for each prime
mod_prods = 2*np.ones(pi, dtype=np.int64)
a_mods = np.ones(pi, dtype = np.int64)
#a_0 is 0, I guess
a = 1
mask = primes == -1
for i in xrange(1, pi):
#we want to find the smallest k with k*p_1*...*p_i + A_i = t*p_(i+1) + (i+1)
#solving this equation mod p_(i+1) for k yields the answer
p = primes[i]
k = ((i+1 - a_mods[i]) * pe.mod_inv(mod_prods[i], p)) % p
a_mods[i:] = (a_mods[i:] + k*mod_prods[i:]) % primes[i:]
mod_prods[i:] *= p
mod_prods[i:] %= primes[i:]
mask[a_mods==0] = True
print np.sum(primes[mask])
def solve(n, k, mod = 10**9+7):
primes = pe.primes_and_mask(n)[0]
pi = len(primes) + 1
nstates = 2 ** (len(bin(pi-1))-2)
nimber_counts = np.zeros(nstates, dtype = np.int64)
mask = np.zeros(n, dtype = np.int64)
# pcounts = np.cumsum(mask.astype(int))
for i, p in enumerate(primes, 1):
mask[p] = i
for j in xrange(p*p, n, p):
if mask[j]:
mask[j] = min(mask[j], i)
else:
mask[j] = i
print "Starting exponentiation."
bcounts = np.bincount(mask)
nimber_counts[2:len(bcounts)] = bcounts[2:]
nimber_counts[1] = 1
nimber_counts[0] = bcounts[1]
nimber_counts = nimber_exp(k, nimber_counts, mod)
print "Solution " + str(nimber_counts[0])
first_power_primes = np.load("e565primes.npy")
for p in first_power_primes:
if not p%2017 == 2016: print p
plist = [(p,p) for p in first_power_primes]
"""
for k in xrange(1, (n/d)/6):
p = d*k*6-1
if pe.MillerRabin(p): first_power_primes.append(p)
p += 2*d
if pe.MillerRabin(p): first_power_primes.append(p)
print len(first_power_primes)
np.save("e565primes",np.array(first_power_primes))
"""
print len(first_power_primes)
sq = pe.isqrt(n)
primes = pe.primes_and_mask(sq)[0]
#print plist[0]
max_power = int(np.log(n)/np.log(2))
powers = {}
for power in xrange(2, max_power):
powers[power] = []
for p in primes:
exp = p**power
if exp > n:
break
# if p%2017 == 2016 and po: continue
if ((exp*p-1)/(p-1)) % 2017 == 0:
plist.append((exp, p))
# print plist[-1]
plist = sorted(plist)
from proj_euler import primes_and_mask
from proj_euler import MillerRabin
import sys
primes, mask = primes_and_mask(10 ** 4)
def concat(a, b):
return int(str(a) + str(b))
def slower_is_prime(n):
global primes
for p in primes:
if p * p > n:
return 1
if not n % p:
return 0
return 1
prime_dicts = {p: set() for p in primes}
for p1 in primes:
for p2 in primes:
if MillerRabin(concat(p1, p2)) and MillerRabin(concat(p2, p1)):
prime_dicts[p1].add(p2)
best_sum = 10 ** 20
for p1 in primes:
for p2 in primes:
from proj_euler import primes_and_mask
cap = 10**9
t = 100
primes = primes_and_mask(t)[0]
print primes
def nHamming(minInd,maxNum):
global primes
if maxNum <= 1:
return maxNum
s = 1
i = minInd
for p in primes[minInd:]:
if p > maxNum:
return s
s += nHamming(i,maxNum/p)
i += 1
return s
print nHamming(0,cap)
from proj_euler import primes_and_mask from math import log import numpy as np n = 2*10**7 logs = np.zeros(n) mods = np.ones(n,dtype=int) mod = 10**9 primes = primes_and_mask(n)[0] def carmichael(p,k): if p == 2: if k <=2: return 2**(k-1) else: return 2**(k-2) else: return (p-1)*p**(k-1) for p in primes: lp = log(p) k = 1 while True: d = carmichael(p,k) if d > n: break lg = k*lp for i in xrange(d,n,d): logs[i] += lp mods[i] = (mods[i]*p) % mod k += 1
import proj_euler as pe
import numpy as np
cap = 2*10**9
primes,mask = pe.primes_and_mask(int(cap**.5))
def f_old(cap):
global primes, mask
sq = set()
sqList = []
for i in xrange(1,int(cap**.5)+1):
sq.add(i*i)
sqList.append(i*i)
s1 = set()
s2 = set()
s3 = set()
s7 = set()
for i in sqList:
for j in sqList:
t1 = i+j
if t1 > cap:
break
s1.add(t1)
t2 = i + 2*j
t3 = i + 3*j
t7 = i + 7*j
if t2 <= cap:
s2.add(t2)
if t3 <= cap:
s3.add(t3)
import proj_euler as pe
pcap = 2*10**5
d = 19
cap = 10**12
pow10 = [10**i for i in xrange(d+2)]
primes = pe.primes_and_mask(pcap)[0]
def prime_proof(t,st):
global pow10
digs = len(st) #number of digits
for i in xrange(digs):
for j in xrange(10):
temp = t + (j-int(st[-i-1]))*pow10[i]
if pe.MillerRabin(temp):
return False
return True
pp_squbes = []
for p in primes:
if p > cap/8:
break
for q in primes:
if q == p:
continue
t = p*p*q**3
if t > cap:
break
st = str(t)
if st.find("200") >= 0:
import proj_euler as pe
import numpy as np
# cap = 100000000
cap = 1000000
s = 0
op = 0
primes, mask = pe.primes_and_mask(cap)
factorDict = {}
for p in primes:
j = 2 * p
while j <= cap:
if mask[j - 1]:
t = j
e = 0
while t % p == 0:
t /= p
e += 1
if j not in factorDict:
factorDict[j] = [(p, 2 * e)]
else:
factorDict[j].append((p, 2 * e))
j += p
factorDict[3] = [(3, 2)]
nump = len(primes)
print "nump: " + str(nump)
from proj_euler import primes_and_mask
import numpy as np
k = 20000
mod=1004535809
primes = np.array(primes_and_mask(5*10**5)[0][:k+1])
A1 = np.zeros(k+1,dtype=np.int64)
A1[1:] = primes[1:]-primes[:-1]
A1[0] = 1
def seq_mult(a,b,mod):
'''
Given two polynomials of degree at most n with coefficients given by a and b, numpy arrays.
This function returns the first n coefficients of a*b, modulo mod.
'''
n=a.size
if not n == b.size:
raise ValueError("Input arrays must have the same shape in seq_mult.")
res = np.zeros(n,dtype=np.int64)
for i in xrange(n):
res[i:] = np.remainder(res[i:] + b[i]*a[:n-i],mod)
return res
t=k
accum = np.zeros(k+1,dtype=np.int64)
accum[0] = 1
temp = np.copy(A1)
#print temp, accum
while t:
if t&1:
accum = seq_mult(accum,temp,mod)
import proj_euler as pe
N = 50
primes = pe.primes_and_mask(N)[0]
def comb_squarefree(n,k):
global primes
for p in primes:
t = pe.prime_fact_ord(p,n)-pe.prime_fact_ord(p,n-k)-pe.prime_fact_ord(p,k)
if t > 1:
return 0
return 1
s = set()
prev = [0 for i in xrange(N+1)]
curr = []
for i in xrange(N+1):
curr = [1]
for j in xrange(N):
curr.append(prev[j]+prev[j+1])
if curr[-1] and comb_squarefree(i,j+1):
s.add(curr[-1])
prev = curr
print sum(s)
from proj_euler import primes_and_mask
from math import log
primes = primes_and_mask(10**4)[0]
def is_counterexample(p,q):
if p == q:
return False
mod = (p-1)*(pow(q,p)-1) / (q-1)
res = pow(p,q,mod)-1
if not res:
return True
else:
return False
exp_bound = 10000
for i,p in enumerate(primes):
if p < exp_bound:
if i%100==0:
print "On %d" % i
for q in primes:
if is_counterexample(p,q):
print "Counterexample found at %d,%d" % (p,q)
else:
break
from proj_euler import primes_and_mask,isqrt,MillerRabin
from bisect import bisect_left
import sys
sys.setrecursionlimit(10**6)
primes = primes_and_mask(10**6)[0]
f = [0,1]
for i in xrange(43):
f.append(f[-1]+f[-2])
def pi(n):
pos = bisect_left(primes,n)
return pos+1 if primes[pos] == n else pos
cache = {}
def phi(x, a):
if (x, a) in cache:
return cache[(x, a)]
if a == 1:
return (x + 1) // 2
t = (x+1) // 2
for i in xrange(1,a):
t -= phi(x // primes[i], i)
# t = phi(x, a-1) - phi(x // primes[a-1], a-1)
cache[(x, a)] = t
return t
def fpi(n):
if n <= len(primes):
import numpy as np
import proj_euler as pe
maxSum = 1548136
cap = 5000
mod = 10**16
primes,mask = pe.primes_and_mask(maxSum)
old = np.zeros(maxSum+1).astype(np.int64)
new = np.zeros(maxSum+1).astype(np.int64)
old[0] = 1
s = 0
for p in primes:
if p > cap:
break
s += p
for j in xrange(p):
new[j] = old[j]
for j in xrange(p,s+1):
new[j] = (old[j] + old[j-p]) % mod
temp = old
old = new
new = temp
res = 0
for p in primes:
res = (res + old[p]) % mod
print res
from CR_NEWTON_LAGRANGE import Lagrange_cr as chin
import proj_euler as pe
cap = 5000
lower = 1000
N = 10**18
K = 10**9
primes = pe.primes_and_mask(cap)[0][168:]
np=len(primes)
print np
s = 0
def binom(n,k,p):
if k > n:
return 0
if not k:
return 1
if k < p:
r = 1
for i in xrange(k):
r = ( r * pow(i+1,p-2,p) * (n-i))%p
return r
return (binom(n/p,k/p,p) * binom(n%p,k%p,p))%p
pc = {p:binom(N,K,p) for p in primes}
for i,p in enumerate(primes):
import proj_euler as pe
import timeit
import numpy as np
from bisect import bisect_left
import sys
cap = 10**int(sys.argv[1])
mP = min(5*10**7,int(cap**(2./3)))
primes,mask = pe.primes_and_mask(mP)
NP = len(primes)
print "NUMP " +str(NP)
print "PCAP " + str(mP)
f = lambda n : n*(n+1) / 2
s = f(cap)
def prods(prod,sign, startNum,primes,NP):
global s,cap
if prod > cap:
return
for i in xrange(startNum,NP):
if primes[i] > cap/prod:
break
s += sign * (prod*primes[i] * f(cap/(prod*primes[i])) - primes[i] * (cap/(prod*primes[i])))
prods(prod*primes[i], -1*sign,i+1,primes,NP)
return
import proj_euler as pe
import numpy as np
primes= pe.primes_and_mask(25)[0]
cap = 10**8
cands = sorted([el for el in pe.HarshadGen(primes, cap, exact=False)])
print len(cands)
seen = {}
for i, el in enumerate(cands):
for other in cands[i:]:
if other > 1.1*el:
break
else:
s = el*other
reps = 1
if s not in seen:
seen[s] = reps
else: seen[s] += reps
m_dict = {i:np.inf for i in xrange(2,101)}
for el in seen:
if seen[el] > 100 or seen[el]==1: continue
m_dict[seen[el]] = min(m_dict[seen[el]], el)
print m_dict
print sum([m_dict[i] for i in xrange(2,101)])
from proj_euler import primes_and_mask
from math import log,floor
cap = 10**6+4
p1_cap = 10**6
primes = primes_and_mask(cap)[0]
l = log(10)
pow10 = [10**i for i in xrange(7)]
s = 0
for i,p in enumerate(primes):
if p > p1_cap:
break
if p < 5:
continue
digits = int(floor(log(p)/l)) + 1
p10 = pow10[digits]
p2 = primes[i+1]
n = p + p10 * (((p2-p) * pow(p10,p2-2,p2)) % p2)
# print n,p,p2,p10
s += n
print s
def init(n): global pcap,primes,mask pcap = int(n**.5)+1 primes,mask = primes_and_mask(pcap)