B [l@sdZddlmZmZddlZddlmZddlmZddlm Z ddl m Z m Z d d Z d d Zd dZGdddZeZddZddZd'ddZddZddZddZd(ddZd)d!d"Zd#d$Zd%d&ZdS)*z" Generating and counting primes. )print_functiondivisionN)bisect)array)isprime)as_intrangecCstddg|S)Nlr)_array)nr 5lib/python3.7/site-packages/sympy/ntheory/generate.py_azerossrcGs td|S)Nr )r )vr r r_asetsrcCstdt||S)Nr )r r )abr r r_arangesrc@sjeZdZdZddZddZdddZd d Zd d Zd dZ ddZ ddZ ddZ ddZ ddZdS)SieveaAn infinite list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes. When a lookup is requested involving an odd number that has not been sieved, the sieve is automatically extended up to that number. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> 25 in sieve False >>> sieve._list array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23]) csld_tdddddd_tdd d ddd _tdd d d dd _tfd d jjjfDshtdS)N rrc3s|]}t|jkVqdS)N)len_n).0i)selfr r 4sz!Sieve.__init__..)r r_list_tlist_mlistallAssertionError)r#r )r#r__init__/s zSieve.__init__cCsddt|j|jd|jd|jd|jd|jddt|j|jd|jd|jd|jd|jdd t|j|jd|jd|jd|jd|jdfS) Nzs<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>primerrrrtotientmobius)rr%r&r')r#r r r__repr__6s   zSieve.__repr__NcCsjtdd|||fDr$d}}}|r:|jd|j|_|rP|jd|j|_|rf|jd|j|_dS)z]Reset all caches (default). To reset one or more set the desired keyword to True.css|]}|dkVqdS)Nr )r!r"r r rr$GszSieve._reset..TN)r(r%r r&r')r#r+r-r.r r r_resetDs z Sieve._resetcCst|}||jdkrdSt|dd}|||jdd}t||d}x@|d|D]0}| |}x t|t||D] }d||<q~Wq^W|jtddd |D7_dS) aGrow the sieve to cover all primes <= n (a real number). Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True rNg?rrrr cSsg|] }|r|qSr r )r!xr r r ssz Sieve.extend..)intr%extendr primeranger rr )r#r ZmaxbaseZbeginZnewsievep startindexr"r r rr4Ps   z Sieve.extendcCs8t|}x*t|j|kr2|t|jddq WdS)aExtend to include the ith prime number. i must be an integer. The list is extended by 50% if it is too short, so it is likely that it will be longer than requested. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend_to_no(9) >>> sieve._list array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23]) rg?N)rrr%r4r3)r#r"r r r extend_to_nouszSieve.extend_to_noccsddlm}tdt||}t||}||kr6dS||||d}t|jd}x6||kr|j|d}||kr|V|d7}q^dSq^WdS)zGenerate all prime numbers in the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.primerange(7, 18)]) [7, 11, 13, 17] r)ceilingrNr)#sympy.functions.elementary.integersr9maxrr4searchrr%)r#rrr9r"Zmaxir6r r rr5s     zSieve.primerangec csDddlm}tdt||}t||}t|j}||kr@dS||krjxt||D]}|j|VqTWn|jt||7_xftd|D]X}|j|}||d||}x&t|||D]}|j||8<qW||kr|VqWxXt||D]J}|j|}x,td|||D]}|j||8<qW||kr|VqWdS)zGenerate all totient numbers for the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.totientrange(7, 18)]) [6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16] r)r9rNr)r:r9r;rrr&r r) r#rrr9r r"Ztir7jr r r totientranges.      zSieve.totientrangec csFddlm}tdt||}t||}t|j}||kr@dS||krjxt||D]}|j|VqTWn|jt||7_xftd|D]X}|j|}||d||}x&t|||D]}|j||8<qW||kr|VqWxXt||D]J}|j|}x,td|||D]}|j||8<qW||kr|VqWdS)zGenerate all mobius numbers for the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.mobiusrange(7, 18)]) [-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1] r)r9rNr)r:r9r;rrr'r r) r#rrr9r r"Zmir7r=r r r mobiusranges.      zSieve.mobiusrangecCsddlm}t||}t|}|dkr4td|||jdkrL||t|j|}|j|d|krr||fS||dfSdS)a~Return the indices i, j of the primes that bound n. If n is prime then i == j. Although n can be an expression, if ceiling cannot convert it to an integer then an n error will be raised. Examples ======== >>> from sympy import sieve >>> sieve.search(25) (9, 10) >>> sieve.search(23) (9, 9) r)r9rzn should be >= 2 but got: %srrN)r:r9r ValueErrorr%r4r)r#r r9Ztestrr r rr<s     z Sieve.searchc Cs\yt|}|dkstWnttfk r0dSX|ddkrF|dkS||\}}||kS)NrFr)rr)r@r<)r#r rrr r r __contains__s zSieve.__contains__cCsXt|tr4||j|j|jd|jd|jSt|}|||j|dSdS)zReturn the nth prime numberrN) isinstanceslicer8stopr%startstepr)r#r r r r __getitem__!s    zSieve.__getitem__)NNN)__name__ __module__ __qualname____doc__r*r/r0r4r8r5r>r?r<rArGr r r rrs %''! rcCst|}|dkrtd|ttjkr.t|Sddlm}ddlm}d}t ||||||}x2||kr||d?}|||kr|}qh|d}qhWt |d}x$||krt |r|d7}|d7}qW|dS)a5 Return the nth prime, with the primes indexed as prime(1) = 2, prime(2) = 3, etc.... The nth prime is approximately n*log(n). Logarithmic integral of x is a pretty nice approximation for number of primes <= x, i.e. li(x) ~ pi(x) In fact, for the numbers we are concerned about( x<1e11 ), li(x) - pi(x) < 50000 Also, li(x) > pi(x) can be safely assumed for the numbers which can be evaluated by this function. Here, we find the least integer m such that li(m) > n using binary search. Now pi(m-1) < li(m-1) <= n, We find pi(m - 1) using primepi function. Starting from m, we have to find n - pi(m-1) more primes. For the inputs this implementation can handle, we will have to test primality for at max about 10**5 numbers, to get our answer. References ========== - https://en.wikipedia.org/wiki/Prime_number_theorem#Table_of_.CF.80.28x.29.2C_x_.2F_log_x.2C_and_li.28x.29 - https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number - https://en.wikipedia.org/wiki/Skewes%27_number Examples ======== >>> from sympy import prime >>> prime(10) 29 >>> prime(1) 2 >>> prime(100000) 1299709 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n rz-nth must be a positive integer; prime(1) == 2r)li)logr) rr@rsiever%'sympy.functions.special.error_functionsrL&sympy.functions.elementary.exponentialrMr3primepir)nthr rLrMrrmidZn_primesr r rr+/s(1         r+c Cst|}|dkrdS|tjdkr0t|dSt|d}|d8}t|d}x|||krf|d7}qPW|d8}dg|d}dg|d}x2td|dD] }|d||<||d||<qWxtd|dD]}||||dkrq||d}xntdt||||dD]N}||}||krD|||||8<n||||||8<qWt|||d}x2t||dD]"}||||||8<qWqW|dS)aC Return the value of the prime counting function pi(n) = the number of prime numbers less than or equal to n. Algorithm Description: In sieve method, we remove all multiples of prime p except p itself. Let phi(i,j) be the number of integers 2 <= k <= i which remain after sieving from primes less than or equal to j. Clearly, pi(n) = phi(n, sqrt(n)) If j is not a prime, phi(i,j) = phi(i, j - 1) if j is a prime, We remove all numbers(except j) whose smallest prime factor is j. Let x= j*a be such a number, where 2 <= a<= i / j Now, after sieving from primes <= j - 1, a must remain (because x, and hence a has no prime factor <= j - 1) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes <= j - 1 Now, if a is a prime less than equal to j - 1, x= j*a has smallest prime factor = a, and has already been removed(by sieving from a). So, we don't need to remove it again. (Note: there will be pi(j - 1) such x) Thus, number of x, that will be removed are: phi(i / j, j - 1) - phi(j - 1, j - 1) (Note that pi(j - 1) = phi(j - 1, j - 1)) => phi(i,j) = phi(i, j - 1) - phi(i / j, j - 1) + phi(j - 1, j - 1) So,following recursion is used and implemented as dp: phi(a, b) = phi(a, b - 1), if b is not a prime phi(a, b) = phi(a, b-1)-phi(a / b, b-1) + phi(b-1, b-1), if b is prime Clearly a is always of the form floor(n / k), which can take at most 2*sqrt(n) values. Two arrays arr1,arr2 are maintained arr1[i] = phi(i, j), arr2[i] = phi(n // i, j) Finally the answer is arr2[1] Examples ======== >>> from sympy import primepi >>> primepi(25) 9 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime rrrg?r)r3rNr%r<r;r min) r ZlimZarr1Zarr2r"r6r=stZlim2r r rrQzs:C     " "&rQcCsBt|}t|}|dkrD|}d}xt|}|d7}||kr"Pq"W|S|dkrPdS|dkrldddddd|S|tjdkrt|\}}||krt|dSt|Sd|d}||kr|d7}t|r|S|d 7}n4||dkr|d7}t|r|S|d 7}n|d}x2t|r|S|d7}t|r0|S|d 7}q Wd S) aB Return the ith prime greater than n. i must be an integer. Notes ===== Potential primes are located at 6*j +/- 1. This property is used during searching. >>> from sympy import nextprime >>> [(i, nextprime(i)) for i in range(10, 15)] [(10, 11), (11, 13), (12, 13), (13, 17), (14, 17)] >>> nextprime(2, ith=2) # the 2nd prime after 2 5 See Also ======== prevprime : Return the largest prime smaller than n primerange : Generate all primes in a given range rrrrr)rrrrrr,rrN)r3r nextprimerNr%r<r)r Zithr"Zprr=r unnr r rrVsN      rVcCsddlm}t||}|dkr(td|dkrDdddddd|S|tjd kr|t|\}}||krtt|d St|Sd |d }||d kr|d }t|r|S|d 8}n|d }x,t|r|S|d8}t|r|S|d 8}qWd S)a Return the largest prime smaller than n. Notes ===== Potential primes are located at 6*j +/- 1. This property is used during searching. >>> from sympy import prevprime >>> [(i, prevprime(i)) for i in range(10, 15)] [(10, 7), (11, 7), (12, 11), (13, 11), (14, 13)] See Also ======== nextprime : Return the ith prime greater than n primerange : Generates all primes in a given range r)r9rzno preceding primesrr)rrrrrrrrrN)r:r9rr@rNr%r<r)r r9r rWrXr r r prevprime#s2      rZccsddlm}||krdS|tjdkrFxt||D] }|Vq4WdSt||d}t||}x t|}||kr||VqddSqdWdS)a Generate a list of all prime numbers in the range [a, b). If the range exists in the default sieve, the values will be returned from there; otherwise values will be returned but will not modify the sieve. Notes ===== Some famous conjectures about the occurrence of primes in a given range are [1]: - Twin primes: though often not, the following will give 2 primes an infinite number of times: primerange(6*n - 1, 6*n + 2) - Legendre's: the following always yields at least one prime primerange(n**2, (n+1)**2+1) - Bertrand's (proven): there is always a prime in the range primerange(n, 2*n) - Brocard's: there are at least four primes in the range primerange(prime(n)**2, prime(n+1)**2) The average gap between primes is log(n) [2]; the gap between primes can be arbitrarily large since sequences of composite numbers are arbitrarily large, e.g. the numbers in the sequence n! + 2, n! + 3 ... n! + n are all composite. References ========== 1. http://en.wikipedia.org/wiki/Prime_number 2. http://primes.utm.edu/notes/gaps.html Examples ======== >>> from sympy import primerange, sieve >>> print([i for i in primerange(1, 30)]) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] The Sieve method, primerange, is generally faster but it will occupy more memory as the sieve stores values. The default instance of Sieve, named sieve, can be used: >>> list(sieve.primerange(1, 30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] See Also ======== nextprime : Return the ith prime greater than n prevprime : Return the largest prime smaller than n randprime : Returns a random prime in a given range primorial : Returns the product of primes based on condition Sieve.primerange : return range from already computed primes or extend the sieve to contain the requested range. r)r9Nrr)r:r9rNr%r5rrV)rrr9r"r r rr5Vs;   r5cCsZ||kr dStt||f\}}t|d|}t|}||krFt|}||krVtd|S)a Return a random prime number in the range [a, b). Bertrand's postulate assures that randprime(a, 2*a) will always succeed for a > 1. References ========== - http://en.wikipedia.org/wiki/Bertrand's_postulate Examples ======== >>> from sympy import randprime, isprime >>> randprime(1, 30) #doctest: +SKIP 13 >>> isprime(randprime(1, 30)) True See Also ======== primerange : Generate all primes in a given range Nrz&no primes exist in the specified range)mapr3randomZrandintrVrZr@)rrr r6r r r randprimesr]TcCsx|rt|}nt|}|dkr&tdd}|rTxDtd|dD]}|t|9}q>Wn xtd|dD] }||9}qdW|S)a* Returns the product of the first n primes (default) or the primes less than or equal to n (when ``nth=False``). >>> from sympy.ntheory.generate import primorial, randprime, primerange >>> from sympy import factorint, Mul, primefactors, sqrt >>> primorial(4) # the first 4 primes are 2, 3, 5, 7 210 >>> primorial(4, nth=False) # primes <= 4 are 2 and 3 6 >>> primorial(1) 2 >>> primorial(1, nth=False) 1 >>> primorial(sqrt(101), nth=False) 210 One can argue that the primes are infinite since if you take a set of primes and multiply them together (e.g. the primorial) and then add or subtract 1, the result cannot be divided by any of the original factors, hence either 1 or more new primes must divide this product of primes. In this case, the number itself is a new prime: >>> factorint(primorial(4) + 1) {211: 1} In this case two new primes are the factors: >>> factorint(primorial(4) - 1) {11: 1, 19: 1} Here, some primes smaller and larger than the primes multiplied together are obtained: >>> p = list(primerange(10, 20)) >>> sorted(set(primefactors(Mul(*p) + 1)).difference(set(p))) [2, 5, 31, 149] See Also ======== primerange : Generate all primes in a given range rzprimorial argument must be >= 1r)rr3r@r r+r5)r rRr6r"r r r primorials/  r^Fc cst|pd}d}}|||}}d}xR||krx|r<||krx|d7}||kr\|}|d9}d}|rf|V||}|d7}q(W|r||kr|rdS|dfVdS|sd} |}}xt|D] }||}qWx$||kr||}||}| d7} qW| r| d8} || fVdS)aFor a given iterated sequence, return a generator that gives the length of the iterated cycle (lambda) and the length of terms before the cycle begins (mu); if ``values`` is True then the terms of the sequence will be returned instead. The sequence is started with value ``x0``. Note: more than the first lambda + mu terms may be returned and this is the cost of cycle detection with Brent's method; there are, however, generally less terms calculated than would have been calculated if the proper ending point were determined, e.g. by using Floyd's method. >>> from sympy.ntheory.generate import cycle_length This will yield successive values of i <-- func(i): >>> def iter(func, i): ... while 1: ... ii = func(i) ... yield ii ... i = ii ... A function is defined: >>> func = lambda i: (i**2 + 1) % 51 and given a seed of 4 and the mu and lambda terms calculated: >>> next(cycle_length(func, 4)) (6, 2) We can see what is meant by looking at the output: >>> n = cycle_length(func, 4, values=True) >>> list(ni for ni in n) [17, 35, 2, 5, 26, 14, 44, 50, 2, 5, 26, 14] There are 6 repeating values after the first 2. If a sequence is suspected of being longer than you might wish, ``nmax`` can be used to exit early (and mu will be returned as None): >>> next(cycle_length(func, 4, nmax = 4)) (4, None) >>> [ni for ni in cycle_length(func, 4, nmax = 4, values=True)] [17, 35, 2, 5] Code modified from: http://en.wikipedia.org/wiki/Cycle_detection. rrrN)r3r ) fZx0ZnmaxvaluesZpowerZlamZtortoiseZharer"Zmur r r cycle_length s>4       rac Cst|}|dkrtdddddddd d d d g }|dkrD||dSdtjd }}||t|dkrx:||dkr||d?}|t|d|kr|}qj|}qjWt|r|d8}|Sddlm}ddlm }d}t ||||||}x>||kr.||d?}|||d|kr$|}q|d}qW|t|d}x*||krjt|s^|d8}|d8}qBWt|r~|d8}|S)a Return the nth composite number, with the composite numbers indexed as composite(1) = 4, composite(2) = 6, etc.... Examples ======== >>> from sympy import composite >>> composite(36) 52 >>> composite(1) 4 >>> composite(17737) 20000 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range primepi : Return the number of primes less than or equal to n prime : Return the nth prime compositepi : Return the number of positive composite numbers less than or equal to n rz1nth must be a positive integer; composite(1) == 4rrrY rr)rL)rM) rr@rNr%rQrrOrLrPrMr3) rRr Z composite_arrrrrSrLrMZ n_compositesr r r compositefsB          ricCs$t|}|dkrdS|t|dS)ak Return the number of positive composite numbers less than or equal to n. The first positive composite is 4, i.e. compositepi(4) = 1. Examples ======== >>> from sympy import compositepi >>> compositepi(25) 15 >>> compositepi(1000) 831 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime primepi : Return the number of primes less than or equal to n composite : Return the nth composite number rrr)r3rQ)r r r r compositepisrj)r)T)NF)rKZ __future__rrr\rrr Z primetestrZsympy.core.compatibilityrr rrrrrNr+rQrVrZr5r]r^rarirjr r r rs.   Ke D3R& ? YB