def upper_bound_generating_set(n):
"""
Estimate an upper bound for the integer k such that any group of order n has
a generating subset of size at most k.
Estimated by prime factorising n and adding together the exponents of each
prime.
The correctness of this bound may be reasoned as follows:
Let G be a group of order n. Let g_0 be the identity 1 in G, and
inductively, for each i, set H_i = <g_1, ... g_i>. Choose a g_{i + 1} not in
H_i. Then the order of H_{i + 1} is a proper multiple of the order of H_i,
by Lagrange's theorem. Say that r is the least integer such that H_r = G.
We can write
|G| = (|H_r| / |H_{r - 1}|) (|H_{r - 1}| / |H_{r - 2}|)
... (|H_2| / |H_1|) (|H_1| / |H_0|)
ie |G| is a product of r nonunit integers, by construction. By considering
prime factorisations it follows that r is at most the number of prime
factors that |G| has, counted with multiplicity.
For p-groups, groups of order 2p, and probably some other classes, this
bound can be shown to be tight. Sadly we know that the only group of order
15 is Z/15Z, which is generated by one element.
"""
return sum(1 for _ in prime_factors(n))
def d(n):
if n in cache:
return cache[n]
f = factors.factors(primes.prime_factors(n,prime_list))
f.remove(n)
cache[n] = sum(f)
return sum(f)
def answer():
target = 15499 / 94744
start = 2 * 3 * 5 * 7 * 11 * 13 * 17 * 19 * 23
for n in count(start, start):
factors = prime_factors(n).items()
r = totient(factors) / (n - 1)
if r < target:
return n
def main():
current_sieve_limit = 100000
sieve = sieve_of_eratosthenes(current_sieve_limit)
n = 2*3*4*5 - 1
while True:
n += 1
# Re-compute the sieve
if n > current_sieve_limit:
current_sieve_limit *= 2
sieve = sieve_of_eratosthenes(current_sieve_limit)
if (len(prime_factors(n, sieve)) == 4
and len(prime_factors(n+1, sieve)) == 4
and len(prime_factors(n+2, sieve)) == 4
and len(prime_factors(n+3, sieve)) == 4):
print(n)
break
def answer():
consecutive = []
number = 646
while len(consecutive) < 4:
number += 1
if len(primes.prime_factors(number)) == 4:
consecutive.append(number)
else:
consecutive = []
return consecutive[0]
def least_common_multiple(m = []):
nums = []
mpfs = []
for n in m:
nums.append(primes.prime_factors(n))
# Build the list
for g in nums:
for f in set(g):
while mpfs.count(f) < g.count(f):
mpfs.append(f)
return primes.list_product(mpfs)
def main():
consecutive_count = 0
for i in range(1000,1000000):
factors = prime_factors(i)
if len(set(factors)) == 4:
consecutive_count += 1
else:
consecutive_count = 0
if consecutive_count == 4:
for j in range(i-3, i+1):
print('%s: %s' % (j, factors))
return
def factors(number):
"""
Given a number, return a list of all factors as (prime, exponent) tuples
"""
factors = {}
try:
for factor in prime_factors(number):
try:
factors[factor] += 1
except KeyError:
factors[factor] = 1
except TypeError:
# prime_numbers() is None
pass # No factors but itself
if len(factors) == 0: return [(number, 1)]
return [(prime, exp) for (prime, exp) in factors.items()]
def p69(n):
start = timer()
et = {}
phis = {}
for i in primorial(n):
et[i] = list(set(prime_factors(i)))
for i in et:
phi = i
for x in et[i]:
phi *= (1 - 1 / float(x))
phis[i] = [phi, float(i) / phi]
v = [x[1] for x in phis.values()]
k = list(phis.keys())
kmax = k[v.index(max(v))]
print(kmax, phis[kmax])
print('Elapsed time: ', timer() - start, 's')
def p69(n):
start=timer()
et={}
phis={}
for i in primorial(n):
et[i]=list(set(prime_factors(i)))
for i in et:
phi=i
for x in et[i]:
phi*=(1-1/float(x))
phis[i]=[phi,float(i)/phi]
v=[x[1] for x in phis.values()]
k=list(phis.keys())
kmax= k[v.index(max(v))]
print (kmax,phis[kmax])
print ('Elapsed time: ',timer()-start,'s')
def main():
relevant_ints = range(1, 21)
prime_factors = [primes.prime_factors(n) for n in relevant_ints]
# num_factors: { prime factor : number of occurrences }, e.g. 9 would consist of {3: 2}
num_factors = {}
for pf in prime_factors:
pf_count = Counter(pf)
for key, value in pf_count.items():
# check if the num_factors already include this specific prime factorization
# e.g. the prime factorization of 20 (2,2,5) and 18 (2,3,3) the prime factorization of 12 (2,2,3).
if key in num_factors:
if num_factors[key] < value:
num_factors[key] = value
else:
num_factors[key] = value
product = 1
for key, value in num_factors.items():
product *= (key**value)
return product
assert (is_prime(13))
assert (is_prime(17))
assert (is_prime(23))
assert (is_prime(29))
assert (is_prime(31))
assert (is_prime(37))
assert (not is_prime(-1))
assert (not is_prime(0))
assert (not is_prime(4))
assert (not is_prime(6))
assert (not is_prime(9))
assert (not is_prime(15))
assert (not is_prime(25))
with timeit("prime_factors test suite"):
assert (list(prime_factors(2)) == [2])
assert (list(prime_factors(3)) == [3])
assert (list(prime_factors(5)) == [5])
assert (list(prime_factors(2 * 3 * 5 * 7 * 11 *
13)) == [2, 3, 5, 7, 11, 13])
assert (list(prime_factors(2 * 2 * 3 * 3 * 5 * 5 * 7 *
7)) == [2, 2, 3, 3, 5, 5, 7, 7])
with timeit("all_factors test suite"):
assert (all_factors(512) == {1, 2, 4, 8, 16, 32, 64, 128, 256, 512})
assert (all_factors(2 * 3 *
5) == {1, 2, 3, 5, 2 * 3, 2 * 5, 3 * 5, 2 * 3 * 5})
with timeit("is_prime perf test"):
assert (is_prime(2**61 - 1))
"""Solve problem 12 of Project Euler and store the answer in answer."""
from primes import prime_factors
target = 500
temp, divisors, i = 0, 0, 1
while divisors <= target:
temp += i
i += 1
divisors, factors = 0, prime_factors(temp)
for j in range(len(factors)):
if j == 0:
divisors += 1
increase = 1
elif factors[j] == factors[j - 1]:
divisors += increase
elif factors[j] != factors[j - 1]:
divisors, increase = 2*divisors + 1, divisors + 1
divisors += 1
answer = temp
# numbers from 1 to 20?
import primes
def is_subset(la, lb):
sa = set(la)
#sb = set(lb)
for i in sa:
if la.count(i) > lb.count(i):
return False
return True
def least_common_multiple(m = []):
nums = []
mpfs = []
for n in m:
nums.append(primes.prime_factors(n))
# Build the list
for g in nums:
for f in set(g):
while mpfs.count(f) < g.count(f):
mpfs.append(f)
return primes.list_product(mpfs)
p8 = primes.prime_factors(8)
p16 = primes.prime_factors(16)
print(least_common_multiple(list(range(1, 21))))
def divisors(n):
pf = prime_factors(n)
divisors = divisors_from_factorization(pf)
return list(divisors)
"""
Problem :
What is the largest prime factor of the number 600851475143 ?
Performance time: ~0.0014s
"""
from timer import timer
from primes import prime_factors
timer.start()
print(max(prime_factors(600851475143)))
timer.stop()
#!/usr/bin/env python3
#https://projecteuler.net/problem=5
import primes
prime_list = primes.primes(20)
factors = {}
for i in range(2, 21):
tmp = primes.prime_factors(i, prime_list)
tmp = { x: tmp.count(x) for x in set(tmp) }
for j in tmp.keys():
if tmp[j] > factors.get(j, 0):
factors[j] = tmp[j]
ans = 1
for i in factors:
ans *= (i ** factors[i])
print(ans)
def main():
print(prime_factors(num)[-1])
#Let us list the factors of the first seven triangle numbers:
#1: 1
#3: 1,3
#6: 1,2,3,6
#10: 1,2,5,10
#15: 1,3,5,15
#21: 1,3,7,21
#28: 1,2,4,7,14,28
#We can see that 28 is the first triangle number to have over five divisors.
#
#What is the value of the first triangle number to have over five hundred divisors?
import primes
import factors
def triangle_generator():
tmp = 1
while True:
yield tmp * (tmp+1) // 2
tmp += 1
prime_list = primes.primes(1000000)
for t in triangle_generator():
tmp = factors.factors(primes.prime_factors(t, prime_list))
if len(tmp) > 500:
print(t)
exit()
def count_factors(x):
pf = prime_factors(x)
n = 1
for prime, count in pf:
n *= (count+1)
return n
def test_prime_factors(self):
self.assertEqual(primes.prime_factors(15), [3, 5])
self.assertEqual(primes.prime_factors(460), [2, 2, 5, 23])
self.assertEqual(primes.prime_factors(13195), [5, 7, 13, 29])
def check_abundant(n):
pf = factors.proper_factors(primes.prime_factors(n, prime_list))
return sum(pf) > n
import primes print(primes.isprime(18)) print(primes.all_primes(18)) print(primes.prime_factors(18))
def test_prime_factors(self):
self.assertEqual(tuple(prime_factors(2)), (2, ))
self.assertEqual(tuple(prime_factors(6)), (2, 3))
self.assertEqual(tuple(prime_factors(2000)), (2, 2, 2, 2, 5, 5, 5))
self.assertEqual(tuple(prime_factors(2017)), (2017, ))
self.assertEqual(tuple(prime_factors(2021)), (43, 47))
def test_prime_factors():
print('test_prime_factors')
assert prime_factors(2) == [2]
assert prime_factors(12) == [2, 2, 3]
from time import process_time from primes import prime_factors print(process_time()) print(prime_factors(600851475143)[-1]) print(process_time())
"""
Problem :
How many fractions lie between 1/3 and 1/2 in the sorted set of reduced
proper fractions for d ≤ 12,000?
Performance time : ~3.8s
"""
from timer import timer
from primes import prime_factors
timer.start()
#mark all possible denominators with their prime factors
compositeness = {n: set(prime_factors(n)) for n in range(12001)}
answer = 0
for d in range(5, 12001):
d_factors = compositeness[d]
for n in range(d // 3, d // 2):
n_factors = compositeness[n]
if not n_factors & d_factors:
answer += 1
print(answer)
timer.stop()