问题
I've recently started attempting to solve problems on project Euler using python, and have hit this road bump when trying to calculate primes and append them to a list. I have written the following code, but I'm confused as to why it doesn't output anything when I run it.
import math
primes = []
def isPrime(i):
if number<=1:
return False
if number==2:
return True
if number%2==0:
return False
for i in range(3,int(sqrt(number))+1):
if number%i==0:
return False
return True
for i in range (1, 9999999):
if isPrime(i) == True:
primes.append(i)
else:
continue
print(primes)
回答1:
Try :
import math
primes = []
def isPrime(number):
if number<=1:
return False
if number==2:
return True
if number%2==0:
return False
for i in range(3,int(math.sqrt(number))+1):
if number%i==0:
return False
return True
for i in range (1, 9999999):
if isPrime(i) == True:
primes.append(i)
print(primes)
回答2:
Try this:
import math
primes = []
def isPrime(number):
if number<=1:
return False
if number==2:
return True
if number%2==0:
return False
for ind in range(3,int(math.sqrt(number))+1):
if number%ind==0:
return False
return True
for i in range (1, 100):
if isPrime(i) == True:
primes.append(i)
else:
continue
print(primes)
To show it working, I print the first 100:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97]
回答3:
If you're building up a list of primes, it can be more efficient to use that list as part of your solution.
For example, this loop:
for i in range(3, int(math.sqrt(number)) + 1):
For the prime 1009 will test ~30 numbers but there are only 10 primes less than the square root of 1009 that actually need to be tested. And this difference just keeps increasing.
Using our growing prime list as part of the solution:
primes = [2]
for number in range(3, 9999999 + 1, 2): # only test odd numbers
for prime in primes:
if prime * prime > number: # we're past sqrt, a prime!
primes.append(number)
break
if number % prime == 0: # a composite
break
print(*primes[:10], '...', *primes[-10:])
Nowhere as fast as @ClockSlave's sieve, but will likely finish before many of the other solutions.
回答4:
The easiest way to do this is to use something called as Sieve. Here's how to get all the primes upto one million.
def mark_sieve(sieve, x):
for i in range(x+x, len(sieve), x):
sieve[i] = False
sieve = [True]*(10**7+1)
for x in range(2, int(len(sieve)**0.5 + 1)):
if sieve[x]: mark_sieve(sieve, x)
The idea is that we initially create a list called sieve and assign all values to True which means that we are for now considering all numbers to 1 million(inclusive) as primes. We will iterate over every number to one million and mark every multiple of it as False in the sieve list. Additionally, to optimize, we iterate only to the square root of 1 million. Why so? Because a number can't have two factors both of which are greater than the square root of the number. So if we divide a number by integers till the ceiling of its square root and its still indivisible, that means its a prime.
So if you want to check if a number is prime, you can simple use sieve[some_number]. If it returns False its not a prime. To get a list of primes you can use [x for x in range(len(sieve)) if sieve[x]]
EDIT Speed comparisons -
import timeit
import math
def isPrime(number):
if number<=1:
return False
if number==2:
return True
if number%2==0:
return False
for ind in range(3,int(math.sqrt(number))+1):
if number%ind==0:
return False
return True
def mark_sieve(sieve, x):
for i in range(x+x, len(sieve), x):
sieve[i] = False
# Other approaches
time_0 = timeit.default_timer()
primes = []
for i in range (1, 10**6+1):
if isPrime(i) == True:
primes.append(i)
else:
continue
# Sieve Approach
time_1 = timeit.default_timer()
sieve = [True]*(10**6+1)
sieve[0] = False #because we wont consider zero and one as primes
sieve[1] = False
for x in range(2, int(len(sieve)**0.5 + 1)):
if sieve[x]: mark_sieve(sieve, x)
primes_2 = [x for x in range(len(sieve)) if sieve[x]]
time_2 = timeit.default_timer()
time_1-time_0 # 12.423080921173096 seconds
time_2-time_1 # 0.9901950359344482 seconds
For a 100 thousand numbers, using sieves is more than 12 times faster. For a million that ratio becomes 90. Also, when using that many number of numbers, I would advise against appending lists. Instead, initiate a list and then assign values.
回答5:
Using a conditional list comprehension:
primes = [
i for i in range(1, 9999999)
if i == 2
or i > 2 # Anything less than 2 is not prime.
and i % 2 # No evens (except for 2 above)
and all(i % n for n in range(3, int(i ** 0.5) + 1))]
>>> primes[:10]
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29]
>>> primes[-10:]
[9999889,
9999901,
9999907,
9999929,
9999931,
9999937,
9999943,
9999971,
9999973,
9999991]
回答6:
Your algorithm to get all prime numbers in [0,9999999] is not very efficient. It need spend a long time to do it so that you cannot see the output when you execute it. It is just because it has not been done. For a faster algorithm, you might check this out
回答7:
Now it works, I have shortned the numbers to 999
import math
primes = []
def isPrime(number):
if number <= 1:
return False
for i in range(2, int(math.sqrt(number)) + 1):
if number % i == 0:
return False
return True
for i in range(1, 999):
if isPrime(i):
primes.append(i)
print(primes)
[OUTPUT]:
[2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997]
来源：https://stackoverflow.com/questions/45444072/calculating-primes-and-appending-to-a-list