一些基本且低效的素数生成算法
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在任何典型的编程语言课程中，学生都会得到一个项目，要求他们编写一个生成素数的程序。这被认为是一项相对简单的任务，会在课程的前几周内分配。我相信您也知道，一些简单而有效的算法可以在短短几分钟内完成作业。在以下示例中，我将使用 Python2.5 编程语言来演示此类算法并比较它们的效率。
我们将考虑的第一个算法将从整数 2 开始，并继续选择每个连续的整数作为潜在的素数 (pp)，通过测试它是否可以被任何先前识别的素数分解来检查素性，然后将每个新验证的素数存储在一个素数集 (ps) 数组中。
pp = 2
ps = [pp]
lim = raw_input("\nGenerate prime numbers up to what number? : ")
while pp < int(lim):
pp += 1
for a in ps:
if pp % a == 0:
break
else:
if pp not in ps:
ps.append(pp)
print ps
注意：上面给出的代码不构成完整的程序。请参阅附录 A，了解包含用户界面的完整程序。
这样一种简单的算法采用严格的暴力方法有效地实现了识别每个素数并将其存储在数组中的目标。我相信您会同意，这也是生成素数最不高效的方法。正如我们将看到的，使用筛选过程的元素将提高我们程序的效率，同时避免真正埃拉托斯特尼筛法的耗时属性，埃拉托斯特尼筛法在识别为可分解并将其从素数中排除之前，会选择每个连续的整数作为潜在的素数，就像 PGsimple1 所做的那样。
运行时数据
primes pgsimple1
up to takes
100 .00100 sec
1000 .02800 sec
10000 1.6980 sec
100000 130.44 sec
1000000 10732 sec
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这些是在每个限制下 5 次测试运行中获得的最佳时间结果。
此表记录了 pgsimple1 在每个指示的限制下的运行时间。请接受，这些运行时间以及本文档中给出的所有运行时间可能与您可能在具有不同硬件或软件的计算机上运行相同程序时获得的运行时间有所不同，我的计算机配备 AMD Turion 64 1.9 GHz，2GB RAM，160GB HDD 和 Windows Vista。
改进此算法的第一步可能是有效地选择潜在的素数。在算法中，最常见的是从整数 3 开始，通过仅选择连续的奇数作为潜在的素数来进行。这将必须测试的潜在素数总数减少了一半。
pp = 2
ps = [pp]
pp += 1
ps.append(pp)
lim = raw_input("\nGenerate prime numbers up to what number? : ")
while pp < int(lim):
pp += 2
test = True
for a in ps:
if pp % a == 0:
test = False
break
if test:
ps.append(pp)
return ps
现在，暴力方法已通过一些简单的逻辑进行增强，从而显着提高了效率，将潜在素数的数量减少了一半。
运行时数据
primes pgsimple2 times faster compared
up to takes to pgsimple1
100 0.0 ~
1000 .01400 2.00
10000 .85700 1.98
100000 65.240 2.00
1000000 5392.9 1.99
10000000 458123 ~
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这些是在每个限制下 5 次测试运行中获得的最佳时间结果。
此表记录了 pgsimple2 的运行时间，以及与 pgsimple1 相比，它在每个限制下完成运行的速度快了多少倍。请注意，效率在任何限制下都保持接近 pgsimple1 的两倍。即使以这种速度，生成 8 位或更多位的素数仍然很不切实际。但是，我这样做只是为了看看需要多长时间。
下一个最明显的改进可能是将测试过程限制为仅检查潜在的素数是否可以被小于或等于潜在素数平方根的素数分解，因为大于潜在素数平方根的素数将是至少一个小于潜在素数平方根的素数的互补因子。
from math import sqrt
pp = 2
ps = [pp]
pp += 1
ps.append(pp)
lim = raw_input("\nGenerate prime numbers up to what number? : ")
while pp < int(lim):
pp += 2
test = True
sqrtpp = sqrt(pp)
for a in ps:
if a > sqrtpp:
break
if pp % a == 0:
test = False
break
if test:
ps.append(pp)
return ps
此算法在效率方面取得了真正显著的进步，并且，在这一点上，大多数程序员已经用尽了继续提高素数生成器效率的能力或愿望，但我们将继续前进。
运行时数据
primes pgsimple3 times faster compared times faster compared
up to takes to pgsimple1 to pgsimple2
100 0.0 ~ ~
1000 .00300 9.33 4.67
10000 .06200 27.4 13.8
100000 1.1220 116 58.1
1000000 26.979 398 200
10000000 705.37 ~ 649
100000000 18780 ~ ~
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这些是在每个限制下 5 次测试运行中获得的最佳时间结果。
此表记录了 pgsimple3 的运行时间，以及与 pgsimple1 和 pgsimple2 相比，它在每个限制下完成运行的速度快了多少倍。请注意，程序运行的时间越长，效率就越显着。
认识到通过使用跳数 2 来仅选择奇数潜在素数，不再需要针对小于素数平方根的所有素数测试潜在的素数，因为它们都不能被 2 分解。因此，我们可以从我们测试潜在素数的素数集中移除第一个素数。这需要将素数集 (ps) 数组划分为例外素数 (ep) 和测试素数 (tp) 数组，然后在最后将它们重新组合以将完整的集合发送回函数调用。
from math import sqrt
pp = 2
ep = [pp]
pp += 1
tp = [pp]
ss = [2]
lim = raw_input("\nGenerate prime numbers up to what number? : ")
while pp < int(lim):
pp += ss[0]
test = True
sqrtpp = sqrt(pp)
for a in tp:
if a > sqrtpp:
break
if pp % a == 0:
test=False
break
if test:
tp.append(pp)
ep.reverse()
[tp.insert(0,a) for a in ep]
return tp
在下一个版本中，将展示为什么我们将跳数 (2) 放入跳集 (ss) 数组中。
运行时数据
primes pgsimple4 times faster compared
up to takes to pgsimple3
100 0.0 ~
1000 .00300 1.00
10000 .05200 1.19
100000 1.1140 1.01
1000000 26.734 1.01
10000000 702.54 1.00
100000000 18766 1.00
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这些是在每个限制下 5 次测试运行中获得的最佳时间结果。
此表记录了 pgsimple4 的运行时间，以及与 pgsimple3 相比，它在每个限制下完成运行的速度快了多少倍。
效率有什么改进？请注意，与 pgsimple3 相比，效率只有微弱的提高。不用担心，随着我在接下来向您展示的程序更高级版本中从测试过程中排除更多素数，效率的提高将成倍增加。
此算法通过从考虑因素中排除先前识别的素数的倍数来有效地选择潜在的素数，并将必须执行以验证每个潜在素数的素性的测试次数降至最低。虽然选择潜在素数的效率允许程序在程序运行的时间越长，每秒筛选的数字范围越大，但必须对每个潜在素数执行的测试次数确实会继续上升（但与其他算法相比，上升速度较慢）。总而言之，这些过程提高了生成素数的效率，使即使在个人计算机上也可以在合理的时间内生成 10 位经过验证的素数。
可以开发更多跳集，以排除可以被每个已识别的素数分解的潜在素数的选择。尽管此过程更复杂，但它可以被概括并变得优雅。同时，我们可以继续从测试素数集中删除跳集排除倍数的每个素数，从而最大限度地减少必须对每个潜在素数执行的测试次数。此示例已逐行完全注释，并提供了一些解释，以帮助读者充分理解算法的工作原理。附录 B 中提供了包含用户界面但没有注释的完整程序。
请忽略用户界面中出现的语法错误，例如“第 1 个素数”，而不是“第 1 个”，以及在完成的数组中包含最后一个生成的素数，即使它可能大于用户定义的限制。这些错误可以方便地由学生程序员轻松修正，但对于说明算法的性能来说没有必要。对于由此给读者带来的任何混乱或不便，我深感抱歉。
from math import sqrt
lim = raw_input("\nGenerate prime numbers up to what number? : ")
""" Get an upper limit from the user to determine the generator's termination point. """
sqrtlim = sqrt(float(lim))
""" Get the square root of the upper limit. This will be the upper limit of the test prime array
for primes used to verify the primacy of any potential primes up to (lim). Primes greater than
(sqrtlim) will be placed in an array for extended primes, (xp), not needed for the verification
test. The use of an extended primes array is technically unnecessary, but helps to clarify that we
have minimized the size of the test prime array. """
pp = 2
""" Initialize the variable for the potential prime, setting it to begin with the first prime
number, (2). """
ss = [pp]
""" Initialize the array for the skip set, setting it at a single member, being (pp=2). Although
the value of the quantity of members in the skip set is never needed in the program, it may be
useful to understand that future skip sets will contain more than one member, the quantity of which
can be calculated, and is the quantity of members of the previous skip set multiplied by one less
than the value of the prime which the new skip set will exclude multiples of. Example - the skip
set which eliminates multiples of primes up through 3 will have (3-1)*1=2 members, since the
previous skip set had 1 member. The skip set which eliminates multiples of primes up through 5 will
have (5-1)*2=8 members, since the previous skip set had 2 members, etc. """
ep = [pp]
""" Initialize the array for primes which the skip set eliminate multiples of, setting the first
member as (pp=2) since the first skip set will eliminate multiples of 2 as potential primes. """
pp += 1
""" Advance to the first potential prime, which is 3. """
rss = [ss[0]]
""" Initialize an array for the ranges of each skip set, setting the first member to be the range
of the first skip set, which is (ss[0]=2). Future skip sets will have ranges which can be
calculated, and are the sum of the members of the skip set. Another method of calculating the range
will also be shown below. """
tp = [pp]
""" Initialize an array for primes which are needed to verify potential primes against, setting the
first member as (pp=3), since we do not yet have a skip set that excludes multiples of 3. Also note
that 3 is a verified prime, without testing, since there are no primes less than the square root of
3. """
i = 0
""" Initialize a variable for keeping track of which skip set range is current. """
rss.append(rss[i] * tp[0])
""" Add a member to the array of skip set ranges, the value being the value of the previous skip
set range, (rss[0]=2), multiplied by the value of the largest prime which the new skip set will
exclude multiples of, (tp[0]=3), so 2*3=6. This value is needed to define when to begin
constructing the next skip set. """
xp = []
""" Initialize an array for extended primes which are larger than the square root of the user
defined limit (lim) and not needed to verify potential primes against, leaving it empty for now.
Again, the use of an extended primes array is technically unnecessary, but helps to clarify that we
have minimized the size of the test prime array. """
pp += ss[0]
""" Advance to the next potential prime, which is the previous potential prime, (pp=3), plus the
value of the next member of the skip set, which has only one member at this time and whose value is
(ss[0]=2), so 3+2=5. """
npp = pp
""" Initialize a variable for the next potential prime, setting its value as (pp=5). """
tp.append(npp)
""" Add a member to the array of test primes, the member being the most recently identified prime,
(npp=5). Note that 5 is a verified prime without testing, since there are no TEST primes less than
the square root of 5. """
while npp < int(lim):
""" Loop until the user defined upper limit is reached. """
i += 1
""" Increment the skip set range identifier. """
while npp < rss[i] + 1:
""" Loop until the next skip set range is surpassed, since data through that range is
needed before constructing the next skip set. """
for n in ss:
""" Loop through the current skip set array, assigning the variable (n) the value
of the next member of the skip set. """
npp = pp + n
""" Assign the next potential prime the value of the potential prime plus
the value of the current member of the skip set. """
if npp > int(lim):
break
""" If the next potential prime is greater than the user defined limit,
then end the 'for n' loop. """
if npp <= rss[i] + 1:
pp = npp
""" If the next potential prime is still within the range of the next skip
set, then assign the potential prime variable the value of the next
potential prime. Otherwise, the potential prime variable is not changed
and the current value remains the starting point for constructing the next
skip set. """
sqrtnpp = sqrt(npp)
""" Get the square root of the next potential prime, which will be the
limit for the verification process. """
test = True
""" Set the verification flag to True. """
for q in tp:
""" Loop through the array of the primes necessary for verification of the
next potential prime. """
if sqrtnpp < q:
break
""" If the test prime is greater than the square root of the next
potential prime, then end testing through the 'for q' loop. """
elif npp % q == 0:
""" If the test prime IS a factor of the next potential prime. """
test = False
""" Then set the verification flag to False since the next
potential prime is not a prime number. """
break
""" And end testing through the 'for q' loop. """
""" Otherwise, continue testing through the 'for q' loop. """
if test:
""" If the next potential prime has been verified as a prime number. """
if npp <= sqrtlim:
tp.append(npp)
""" And if the next potential prime is less than or equal to the
square root of the user defined limit, then add it to the array of
primes which potential primes must be tested against. """
else:
xp.append(npp)
""" Otherwise, add it to the array of primes not needed to verify
potential primes against. """
""" Then continue through the 'for n' loop. """
if npp > int(lim):
break
""" If the next potential prime is greater than the user defined limit, then end
the 'while npp<rss[i]+1' loop. """
""" Otherwise, continue the 'while npp<rss[i]+1' loop. """
if npp > int(lim):
break
""" If the next potential prime is greater than the user defined limit, then end the 'while
npp<int(lim)' loop. """
""" At this point, the range of the next skip set has been reached, so we may begin
constructing a new skip set which will exclude multiples of primes up through the value of
the first member of the test prime set, (tp[0]), from being selected as potential
primes. """
lrpp = pp
""" Initialize a variable for the last relevant potential prime and set its value to the
value of the potential prime. """
nss = []
""" Initialize an array for the next skip set, leaving it empty for now. """
while pp < (rss[i] + 1) * 2-1:
""" Loop until the construction of the new skip set has gone through the range of the new
skip set. """
for n in ss:
""" Loop through the current skip set array. """
npp = pp + n
""" Assign the next potential prime the value of the potential prime plus
the value of the current member of the skip set. """
if npp > int(lim):
break
""" If the next potential prime is greater than the user defined limit,
then end the 'for n' loop. """
sqrtnpp = sqrt(npp)
""" Get the square root of the next potential prime, which will be the
limit for the verification process. """
test = True
""" Set the verification flag to True. """
for q in tp:
""" Loop through the array of the primes necessary for verification of the
next potential prime. """
if sqrtnpp < q:
break
""" If the test prime is greater than the square root of the next
potential prime, then end testing through the 'for q' loop. """
elif npp % q == 0:
""" If the test prime IS a factor of the next potential prime. """
test = False
""" Then set the verification flag to False since the next
potential prime is not a prime number. """
break
""" And end testing through the 'for q' loop. """
""" Otherwise, continue testing through the 'for q' loop. """
if test:
""" If the next potential prime has been verified as a prime number. """
if npp <= sqrtlim:
tp.append(npp)
""" And if the next potential prime is less than or equal to the
square root of the user defined limit, then add it to the array of
primes which potential primes must be tested against. """
else:
xp.append(npp)
""" Otherwise, add it to the array of primes not needed to verify
potential primes against. """
if npp % tp[0] != 0:
""" If the next potential prime was NOT factorable by the first member of
the test array, then it is relevant to the construction of the new skip set
and a member must be included in the new skip set for a potential prime to
be selected. Note that this is the case regardless of whether the next
potential prime was verified as a prime, or not. """
nss.append(npp-lrpp)
""" Add a member to the next skip set, the value of which is the
difference between the last relevant potential prime and the next
potential prime. """
lrpp = npp
""" Assign the variable for the last relevant potential prime the
value of the next potential prime. """
pp = npp
""" Assign the variable for the potential prime the value of the next
potential prime. """
""" Then continue through the 'for n' loop. """
if npp > int(lim):
break
""" If the next potential prime is greater than the user defined limit, then end
the 'while npp<(rss[i]+1)*2-1' loop. """
""" Otherwise, continue the 'while npp<(rss[i]+1)*2-1' loop. """
if npp > int(lim):
break
""" If the next potential prime is greater than the user defined limit, then end the 'while
npp<int(lim)' loop. """
ss=nss
""" Assign the skip set array the value of the new skip set array. """
ep.append(tp[0])
""" Add a new member to the excluded primes array, since the newly constructed skip set
will exclude all multiples of primes through the first member of the test prime array. """
del tp[0]
""" Delete the first member from the test prime array since future potential primes will
not have to be tested against this prime. """
rss.append(rss[i] * tp[0])
""" Add a member to the skip set range array with the value of the range of the next skip
set. """
npp = lrpp
""" Assign the next potential prime the value of the last relevant potential prime. """
""" Then continue through the 'while npp<int(lim)' loop. """
""" At this point the user defined upper limit has been reached and the generator has completed
finding all of the prime numbers up to the user defined limit. """
ep.reverse()
""" Flip the array of excluded primes. """
[tp.insert(0, a) for a in ep]
""" Add each member of the flipped array into the beginning of the test primes array. """
tp.reverse()
""" Flip the array of test primes. """
[xp.insert(0, a) for a in tp]
""" Add each member of the flipped array into the beginning of the extended primes array. """
return xp
""" Send the completed array of all primes up to the user defined limit back to the function call. """
运行时数据
primes pg7.8 times faster compared times faster compared
up to takes to pgsimple1 to pgsimple4
100 0.0 ~ ~
1000 .00100 28.0 3.00
10000 .01800 94.3 2.89
100000 .28400 459 3.92
1000000 5.6220 1909 4.76
10000000 120.53 ~ 5.83
100000000 2752.1 ~ 6.82
1000000000 65786 ~ ~
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这些是在每个限制下 5 次测试运行中获得的最佳时间结果。
此表记录了 pg7.8 的运行时间，以及与 pgsimple1 和 pgsimple4 相比，它在每个限制下完成运行的速度快了多少倍。请注意，程序运行的时间越长，效率就越显着。
感谢您抽出时间研究该算法，我希望它能给您带来启发。如果您选择将此算法翻译成其他编程语言，请将您的作品副本发送电子邮件至 [email protected]。
嗯，所有步骤都很好，但你可以在一开始就停止；只要你能够说明所有 N>1 的 2N 都不是素数，你也可以说明除了 3 之外的所有素数都不在 3N 的形式中，等等。
它在第一步中导致了已知的 6N±1 素数形式（这很优雅但错误，真正的形式是 6N+1 或 6N+5），但你可以做得更好，只要你使用 30N+1、30N+7、30N+11 ..... 30N+29 或者甚至更好，使用 210N+1、210N+11、210N+13 ...... 210N+209。
简而言之，使用算法以智能方式缩减“搜索范围”，使用 1*1, 1*2, 1*2*3, 1*2*3*5, 1*2*3*5*7 .... 作为“基础”以及一个“位移”列表来添加。
但请注意，速度仍然受到多个不断增长的“sqrt”和“div”（或更好的“mod”）运算的影响。
“sqrt”运算可以“反转”，只需执行一个 N^2 即可限制测试的有效性：时间效益非常大！
同样可以避免“模数”运算，但技巧有点长，以后再说。
因此，只要你能猜到素数，并能区分“真实”素数和“伪”素数，就没有必要计算素数。
如果你已经研究过这个算法，你会发现它通过生成跳跃集来智能地缩减“搜索范围”，这些跳跃集以已知素数为基础，无限地过滤 6N、30N、210N 等。就像你建议的那样，即使如此，只有大于跳跃集基础素数的因子才能被过滤，需要测试大于过滤素数但小于被测试数字的平方根的素数。因此，需要至少生成所有小于最大测试数字的平方根的素数。也就是说，区分“真实”素数和“伪”素数的唯一方法。
#! /usr/bin/env python
from math import sqrt
from time import time
def pg():
# pgsimple1, the least efficient algorithm
lim = raw_input("\nGenerate prime numbers up to what number? : ")
pp = 2
ps = [pp]
bt = time()
while pp<int(lim):
pp += 1
test = True
for a in ps:
if pp % a == 0:
test = False
if test:
ps.append(pp)
et = time()
tt = et - bt
a = test = et = bt = pp = 0
print "\nIt took",tt,"seconds to generate the prime set up to: ", lim, "\nwith", len(ps), "members."
tt = lim = 0
return ps
def ui(a):
m = "\nDo you wish to review the numbers? Enter y for yes or q to quit. "
n = "From: "
o = "To: "
p = "or"
q = "is out of the range of the set generated."
r = "and"
s = "There are none between"
t = "\nPlease pick between 0"
u = "\nThey are the"
v = "th thru"
w = "th members."
x = "\nPlease choose a number that is less than"
y = "a prime number."
z = "\nThere are"
A = "members of the prime set"
C = "Make a selection or enter q to quit. "
f = raw_input(m)
while f != 'q':
if f == 'y':
print "\nChoose a category:"
print "a) Pick a range of indexes of members of the prime number set."
print "b) Pick a range of numbers to view prime numbers in that range."
print "d) Input a number to check its membership in the prime number set."
print "e) Get the number of members in the prime set up to a particular number."
print "f) Get the number of members in the prime set between a range of numbers."
print "v) View 100 primes at a time."
f = raw_input(C)
if f == 'a':
print t, r, len(a)
f = raw_input(n)
g = raw_input(o)
if int(g) < int(f):
h = f
f = g
g = h
if int(f) < 0 or int(g) > len(a):
print f, p, g, q
elif f == g:
print s, f, r, g
else:
print [a[h] for h in range(int(f), int(g))], "\n", u, str(int(f) + 1) + v, str(g) + w
if f == 'b':
print t, r, a[len(a) - 1] + 1
f = raw_input(n)
g = raw_input(o)
if int(g) < int(f):
h = f
f = g
g = h
if int(f) < 0 or int(g) > a[len(a) - 1] + 1:
print f, p, g, q
elif f == g:
print s, f, r, g
else:
i = 0
while a[i] < int(f):
i += 1
j = i
while i < len(a) and a[i] <= int(g):
print a[i],
i += 1
print u, str(int(j) + 1) +v,str(i) + w
if f == 'd':
print x, a[len(a) - 1] + 1
f = raw_input("What number do you want to check? ")
for g in a:
if int(g) == int(f):
print f, "is", y
if int(g) > int(f):
print f, "is not", y
if int(f) < 0 or int(g) >= int(f):
break
if int(f) > g + 1 or int(f) < 0:
print f, q
if f == 'e':
print x, a[len(a) - 1] + 2
f = raw_input(o)
if -1 < int(f) <a[len(a) - 1] + 2:
g = 0
while a[g] <= int(f):
g += 1
if g == len(a):
break
print z, g, A, "up to", f
else:
print f, q
if f == 'f':
print t, r, a[len(a) - 1] + 1
f = raw_input(n)
g = raw_input(o)
if int(g) < int(f):
h = f
f = g
g = h
i = 0
if int(f) < 0 or int(g) > a[len(a) - 1] + 1:
print f, p, g, q
elif f == g:
print s, f, r, g
else:
for j in a:
if int(f) <= int(j) <= int(g):
i += 1
elif int(j) > int(g):
break
print z, i, A, "from", f, "thru", g
if f == 'v':
g = 0
h = 1
while f != 'q' and g < len(a):
i = h * 100
for g in range(100*(h - 1),i):
if g == len(a):
i = len(a)
break
print a[g],
print u, str(100 * (h - 1) + 1) + v, str(i) + w
h += 1
if g == len(a):
break
f = raw_input("\nView the next 100 members or enter q to quit. ")
f = raw_input(m)
def run(a = 'r'):
while a is 'r':
a = raw_input("\nEnter r to run prime generator. ")
if a != 'r':
b = pg()
ui(b)
if __name__ == "__main__":
run()
print "\n" * 5, "Don't go away mad...Just go away.", "\n" * 5
#! /usr/bin/env python
from math import sqrt
from time import time
def pg():
lim = raw_input("\nGenerate prime numbers up to what number? : ")
sqrtlim = sqrt(float(lim))
pp = 2
ep = [pp]
ss = [pp]
pp += 1
i = 0
rss = [ss[0]]
tp = [pp]
xp = []
pp += ss[0]
npp = pp
tp.append(npp)
rss.append(rss[i] * tp[0])
bt = time()
while npp < int(lim):
i += 1
while npp < rss[i] + 1:
for n in ss:
npp = pp + n
if npp > int(lim):
break
if npp <= rss[i] + 1:
pp = npp
sqrtnpp = sqrt(npp)
test = True
for q in tp:
if sqrtnpp < q:
break
elif npp % q == 0:
test = False
break
if test:
if npp <= sqrtlim:
tp.append(npp)
else:
xp.append(npp)
if npp > int(lim):
break
if npp > int(lim):
break
lrpp = pp
nss = []
while pp < (rss[i] + 1) * 2 - 1:
for n in ss:
npp = pp + n
if npp > int(lim):
break
sqrtnpp = sqrt(npp)
test = True
for q in tp:
if sqrtnpp < q:
break
elif npp % q == 0:
test = False
break
if test:
if npp <= sqrtlim:
tp.append(npp)
else:
xp.append(npp)
if npp % tp[0] != 0:
nss.append(npp - lrpp)
lrpp = npp
pp = npp
if npp > int(lim):
break
if npp > int(lim):
break
ss = nss
ep.append(tp[0])
del tp[0]
rss.append(rss[i] * tp[0])
npp = lrpp
et = time()
i = nss = npp = n = sqrtnpp = test = q = r = lrpp = rss = ss = pp = sqrtlim = 0
tt = et - bt
ep.reverse()
[tp.insert(0, a) for a in ep]
tp.reverse()
[xp.insert(0, a) for a in tp]
print "\nIt took", tt, "seconds to generate the prime set up to: ", lim, "\nwith", len(xp), "members."
et = bt = ep = tp = a = tt = lim = 0
return xp
def ui(a):
m = "\nDo you wish to review the numbers? Enter y for yes or q to quit. "
n = "From: "
o = "To: "
p = "or"
q = "is out of the range of the set generated."
r = "and"
s = "There are none between"
t = "\nPlease pick between 0"
u = "\nThey are the"
v = "th thru"
w = "th members."
x = "\nPlease choose a number that is less than"
y = "a prime number."
z = "\nThere are"
A = "members of the prime set"
C = "Make a selection or enter q to quit. "
f = raw_input(m)
while f != 'q':
if f == 'y':
print "\nChoose a category:"
print "a) Pick a range of indexes of members of the prime number set."
print "b) Pick a range of numbers to view prime numbers in that range."
print "d) Input a number to check its membership in the prime number set."
print "e) Get the number of members in the prime set up to a particular number."
print "f) Get the number of members in the prime set between a range of numbers."
print "v) View 100 primes at a time."
f = raw_input(C)
if f== 'a':
print t, r, len(a)
f = raw_input(n)
g = raw_input(o)
if int(g) < int(f):
h = f
f = g
g = h
if int(f) < 0 or int(g) > len(a):
print f, p, g, q
elif f==g: print s,f,r,g
else:
print [a[h] for h in range(int(f), int(g))], "\n" , u, str(int(f) + 1) + v, str(g) + w
if f == 'b':
print t, r, a[len(a) - 1] + 1
f = raw_input(n)
g = raw_input(o)
if int(g) < int(f):
h = f
f = g
g = h
if int(f) < 0 or int(g) > a[len(a) - 1] + 1:
print f, p, g, q
elif f == g:
print s, f, r, g
else:
i = 0
while a[i] < int(f):
i += 1
j = i
while i < len(a) and a[i] <= int(g):
print a[i],
i += 1
print u, str(int(j) + 1) + v, str(i) + w
if f == 'd':
print x, a[len(a) - 1] + 1
f = raw_input("What number do you want to check? ")
for g in a:
if int(g) == int(f):
print f, "is", y
if int(g) > int(f):
print f, "is not", y
if int(f) < 0 or int(g) >= int(f):
break
if int(f) > g + 1 or int(f) < 0:
print f, q
if f == 'e':
print x, a[len(a) - 1] + 2
f = raw_input(o)
if -1 < int(f) < a[len(a) - 1] + 2:
g = 0
while a[g] <= int(f):
g += 1
if g == len(a):
break
print z, g, A, "up to", f
else:
print f, q
if f == 'f':
print t, r, a[len(a) - 1] + 1
f = raw_input(n)
g = raw_input(o)
if int(g) < int(f):
h = f
f = g
g = h
i = 0
if int(f) < 0 or int(g) > a[len(a) - 1] + 1:
print f, p, g, q
elif f == g:
print s, f, r, g
else:
for j in a:
if int(f) <= int(j) <= int(g):
i += 1
elif int(j) > int(g):
break
print z, i, A, "from", f, "thru", g
if f == 'v':
g = 0
h = 1
while f != 'q' and g < len(a):
i = h * 100
for g in range(100 * (h - 1), i):
if g == len(a):
i = len(a)
break
print a[g],
print u, str(100 * (h - 1) + 1) + v, str(i) + w
h += 1
if g == len(a):
break
f = raw_input("\nView the next 100 members or enter q to quit. ")
f = raw_input(m)
def run(a = 'r'):
while a is 'r':
a = raw_input("\nEnter r to run prime generator. ")
if a != 'r':
b = pg()
ui(b)
if __name__ == "__main__":
run()
print "\n" * 5, "Don't go away mad...Just go away.", "\n" * 5