MZdrdZddlZddlmZddlmZddlmZddlmZddl m Z dd l m Z dd l mZd Zd Zd ZdZGddZeZdZGddeZddZdZddZdZddZddZdZdZy) z" Generating and counting primes. N)bisectcount)array)Function)S)isprime)as_intc"tddg|zS)Nlr_arrayns 8/usr/lib/python3/dist-packages/sympy/ntheory/generate.py_azerosrs #s1u ctd|SNr r)vs r_asetrs #q>rc.tdt||Sr)rrangeabs r_arangers #uQ{ ##rc0ddlm}t||S)z Wrapping ceiling in as_int will raise an error if there was a problem determining whether the expression was exactly an integer or not.r)ceiling)#sympy.functions.elementary.integersr r )rr s r_as_int_ceilingr"s< '!* rc\eZdZdZdZdZddZdZdZddZ d Z d Z d Z d Z d ZdZy)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]) cd_tdddddd_tdd d ddd _tdd d d dd _t fd jjjfDsJy) N rr c3NK|]}t|jk(ywN)len_n).0iselfs r z!Sieve.__init__..=sU3q6TWW$Us"%)r2r_list_tlist_mlistallr5s`r__init__zSieve.__init__8ss1aAr2. Aq!Q1- Aq"b!R0 Utzz4;; .TUUUUrc.ddt|j|jd|jd|jd|jd|jddt|j|jd|jd|jd|jd|jdd t|j|jd|jd|jd|jd|jdfzS) Nzs<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>primerr r'r.totientmobius)r1r7r8r9r;s r__repr__zSieve.__repr__?s6c$**oA 1 tzz!}BB DKK(QQQR$++b/ s4;;'QQQR$++b/ :C C CrNctd|||fDrdx}x}}|r|jd|j|_|r|jd|j|_|r|jd|j|_yy)z]Reset all caches (default). To reset one or more set the desired keyword to True.c3$K|]}|du ywr0)r3r4s rr6zSieve._reset..Ps;QqDy;sTN)r:r7r2r8r9)r5r>r@rAs r_resetz Sieve._resetMsx ;5'6":; ;'+ +E +Gf HTWW-DJ ++htww/DK ++htww/DK rc t|}||jdkryt|dzdz}|j||jddz}t||dz}|j |D](}| |z}t |t ||D]}d||< *|xjtd|Dcgc]}|s| c}z c_ycc}w)aGrow the sieve to cover all primes <= n (a real number). Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True r.N?r rr )intr7extendr primerangerr1r) r5rmaxbasebeginnewsievep startindexr4xs rrJz Sieve.extendYs F  2   af+/ G 2"5!a%() A!&AJ:s8}a8     fSh"<!1"<== ">> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend_to_no(9) >>> sieve._list array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23]) Notes ===== The list is extended by 50% if it is too short, so it is likely that it will be longer than requested. r.g?N)r r1r7rJrI)r5r4s r extend_to_nozSieve.extend_to_no~sM. 1I$**o! KKDJJrNS01 2$**o!rc#NK|t|}d}n tdt|}t|}||k\ry|j||j|d}t |j dz}||kr)|j |dz }||kr ||dz }ny||kr(yyw)a(Generate all prime numbers in the range [2, a) or [a, b). Examples ======== >>> from sympy import sieve, prime All primes less than 19: >>> print([i for i in sieve.primerange(19)]) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> print([i for i in sieve.primerange(7, 19)]) [7, 11, 13, 17] All primes through the 10th prime >>> list(sieve.primerange(prime(10) + 1)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] Nr'r )r"maxrJsearchr1r7)r5rrr4maxirOs rrKzSieve.primeranges2 9"AAAq)*A"A 6  A KKN1 4::"$h 1q5!A1uQ $hs B B%#B%c#~Ktdt|}t|}t|j}||k\ry||kr#t ||D]}|j|y|xjt ||z c_t d|D]R}|j|}||zdz |z|z}t |||D]}|j|xx|zcc<||k\sO|Tt ||D]G}|j|}t d|z||D]}|j|xx|zcc<||k\sD|Iyw)zGenerate all totient numbers for the range [a, b). Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.totientrange(7, 18)]) [6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16] r Nr')rUr"r1r8rr)r5rrrr4tirPjs r totientrangezSieve.totientrangesH ?1% & A    6  !V1a[ %kk!n$ % KK71a= (K1a[ [[^!eaiA-1 z1a0)AKKNb(N)6H  1a[ [[^q1ua+)AKKNb(N)6H  sCD= AD=6D=c#Ktdt|}t|}t|j}||k\ry||kr#t ||D]}|j|y|xjt ||z z c_t d|D]R}|j|}||zdz |z|z}t |||D]}|j|xx|zcc<||k\sO|Tt ||D]G}|j|}t d|z||D]}|j|xx|zcc<||k\sD|Iyw)aGenerate all mobius numbers for the range [a, b). Parameters ========== a : integer First number in range b : integer First number outside of range 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 Nr')rUr"r1r9rr)r5rrrr4mirPrZs r mobiusrangezSieve.mobiusrangesJ& ?1% & A    6  !V1a[ %kk!n$ % KK71q5> )K1a[ [[^!eaiA-1 z1a0)AKKNb(N)6H  1a[ [[^q1ua+)AKKNb(N)6H  sCD?"AD?8D?ct|}t|}|dkrtd|z||jdkDr|j |t |j|}|j|dz |k(r||fS||dzfS)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'zn should be >= 2 but got: %sr.r )r"r ValueErrorr7rJr)r5rtestrs rrVz Sieve.searchs"q! 1I q5;a?@ @ tzz"~  KKN 4::q ! ::a!e  $a4Ka!e8Orc t|}|dk\sJ |dzdk(r|dk(S|j|\}}||k(S#ttf$rYywxYw)Nr'Fr)r r`AssertionErrorrV)r5rrrs r __contains__zSieve.__contains__1sd q A6M6 q5A:6M{{1~1Av N+  s;A  A c#:KtdD] }|| yw)Nr r)r5rs r__iter__zSieve.__iter__<s"q Aq'M sct|trq|j|j|j |jnd}|dkr t d|j |dz |jdz |jS|dkr t dt|}|j||j |dz S)zReturn the nth prime numberrr zSieve indices start at 1.) isinstanceslicerSstopstart IndexErrorr7stepr )r5rrks r __getitem__zSieve.__getitem__@s a    aff % !ww2AGGEqy!!<==::eai 1669: :1u!!<==q A   a ::a!e$ $r)NNNr0)__name__ __module__ __qualname____doc__r<rBrFrJrSrKr[r^rVrdrfrnrErrr$r$&sI"V C 0#>J36*X!F*X: %rr$c t|}|dkr td|ttjkr t|Sddlm}ddlm}d}t||||||zz}||kr!||zdz }|||kDr|}n|dz}||kr!t|dz }||krt|r|dz }|dz }||kr|dz S)aK 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 $\le x$, 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. 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 References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number_theorem#Table_of_.CF.80.28x.29.2C_x_.2F_log_x.2C_and_li.28x.29 .. [2] https://en.wikipedia.org/wiki/Prime_number_theorem#Approximations_for_the_nth_prime_number .. [3] https://en.wikipedia.org/wiki/Skewes%27_number r z-nth must be a positive integer; prime(1) == 2rloglir') r r`r1siever7&sympy.functions.elementary.exponentialru'sympy.functions.special.error_functionsrwrIprimepir )nthrrurwrrmidn_primess rr>r>Ysb s A1uHIIC Qx:: A As1vCF # $%A a%1ul c7Q;AaA a% q1u~H Q, 1: MH Q Q, q5Lrc eZdZdZedZy)r{aw Represents 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 $x= j \times a$ be such a number, where $2 \le a \le i / j$ Now, after sieving from primes $\le j - 1$, a must remain (because x, and hence a has no prime factor $\le j - 1$) Clearly, there are phi(i / j, j - 1) such a which remain on sieving from primes $\le j - 1$ Now, if a is a prime less than equal to j - 1, $x= j \times a$ has smallest prime factor = a, and has already been removed(by sieving from a). So, we do not 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)) $\Rightarrow$ 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, prime, prevprime, isprime >>> primepi(25) 9 So there are 9 primes less than or equal to 25. Is 25 prime? >>> isprime(25) False It is not. So the first prime less than 25 must be the 9th prime: >>> prevprime(25) == prime(9) True See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime c ,|tjurtjS|tjurtjS t |}|dkrtjS|tjdkr!ttj|dSt |dz}|dz}t|d}||z|kr|dz }||z|kr|dz}dg|dzz}dg|dzz}td|dzD]}|dz ||<||zdz ||<td|dzD]}||||dz k(r||dz }tdt|||zz|dzD]6}||z}||kr||xx|||z zcc<!||xx|||z|z zcc<8t|||zdz } t|| dD]}||xx|||z|z zcc<t|dS#t $r/|j dk(s|tjur tdYywxYw)NFzn must be realr'r.rrHr )rInfinityNegativeInfinityZerorI TypeErroris_realNaNr`rxr7rVrUrmin) clsrlimarr1arr2r4rOrZstlim2s revalz primepi.evalsT  ?::  "" "66M AA q566M  B U\\!_Q'( (!s(m q#qkCi1n 1HCCi1n qscAgscAgq#'" !A!eDG1fqjDG !q#'" ,AAw$q1u+%QU A1c!A,4q89 1U9GtBx!|+GGtAG}q00G  1 sAEAI&D3b) ,Q4Q>> 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 r r'r*r(r))r'r(r-r)r&r?r&r-)rIr nextprimerxr7rVr )rithr4prrZr unns rrr sN0 AAs A1u  2B FA1u  1u1uqQ1-a00EKKO||A1 6Q< 8O AqDB Qw Q 1:H Q R1 Q 1:H Q F 1:H Q 1:H Q rct|}|dkr td|dkr dddddd|S|tjdkr2tj |\}}||k(r t|dz St|Sd |d zz}||z dkr|dz }t |r|S|d z}n|dz} t |r|S|dz}t |r|S|d z}%) 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(zno preceding primesr'r))r(r-r)r&r*r.r r&r-)r"r`rxr7rVr )rr rrs r prevprimerds& A1u.//1uqQ1-a00EKKO||A1 61: 8O AqDB2v{ F 1:H Q F 1:H Q 1:H Q rc#K|d|}}||k\ry|tjdkrtj||Ed{yt|dz }t|} t |}||kr|ny76w)a Generate a list of all prime numbers in the range [2, a), or [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. Examples ======== >>> from sympy import primerange, prime All primes less than 19: >>> list(primerange(19)) [2, 3, 5, 7, 11, 13, 17] All primes greater than or equal to 7 and less than 19: >>> list(primerange(7, 19)) [7, 11, 13, 17] All primes through the 10th prime >>> list(primerange(prime(10) + 1)) [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: >>> from sympy import sieve >>> list(sieve.primerange(1, 30)) [2, 3, 5, 7, 11, 13, 17, 19, 23, 29] 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. See Also ======== prime : Return the nth prime 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. References ========== .. [1] https://en.wikipedia.org/wiki/Prime_number .. [2] https://primes.utm.edu/notes/gaps.html Nr'r.r )rxr7rKr"rrs rrKrKsV y!1AvEKKO##Aq))) QAA aL q5G  *s 1. 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 References ========== .. [1] https://en.wikipedia.org/wiki/Bertrand's_postulate Nr z&no primes exist in the specified range)maprIrandomrandintrrr`)rrrrOs r randprimersg4 Av sQF DAqq1ua A! AAv aL1uABB Hrc|r t|}n t|}|dkr tdd}|r$td|dzD]}|t |z}|St d|dzD]}||z} |S)a: Returns the product of the first n primes (default) or the primes less than or equal to n (when ``nth=False``). Examples ======== >>> from sympy.ntheory.generate import primorial, 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 r zprimorial argument must be >= 1r')r rIr`rr>rK)rr|rOr4s r primorialrsd 1I F1u:;; A q!a% A qMA  HAq1u% A FA  Hrc#Kt|xsd}dx}}|||}}d}||k7r;|r||kr4|dz }||k(r |}|dz}d}|r|||}|dz }||k7r |s.||kr4|r||k(r |ry|dfy|sLd} |x}}t|D] }||} ||k7r||}||}| dz } ||k7r| r| dz} || fyyw)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: https://en.wikipedia.org/wiki/Cycle_detection. rr r'N)rIr) fx0nmaxvaluespowerlamtortoiseharer4mus r cycle_lengthrZs'h tyq>DOEC2dH A d DAH Q C<H QJEC Jw q d DAH T  *    4s AT7D ${HT7D !GB$  !GB2g  sACC"AC3Cc tt|}|dkr tdgd}|dkr||dz Sdtjd}}||t |z dz krD||dz kr*||zdz }|t |z dz |kDr|}n|}||dz kr*t |r|dz}|Sddlm}dd lm }d}t||||||zz}||kr'||zdz }|||z dz |kDr|}n|dz}||kr'|t |z dz }||kDrt |s|dz}|dz}||kDrt |r|dz}|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 r z1nth must be a positive integer; composite(1) == 4) r-r&r rr-r.rrtrv) r r`rxr7r{r ryrurzrwrI) r|r composite_arrrrr}rurw n_compositess r compositers0 s A1uLMM8MBwQU## ekk"oqAA NQ !a%iq5Q,CWS\!A%) !a%i 1: FA:: A As1vCF # $%A a%1ul C=1 q AaA a%wqz>A%L  qz A L Q  qz Q HrcFt|}|dkry|t|z dz S)ak Return the number of positive composite numbers less than or equal to n. The first positive composite is 4, i.e. compositepi(4) = 1. Examples ======== >>> from sympy import compositepi >>> compositepi(25) 15 >>> compositepi(1000) 831 See Also ======== sympy.ntheory.primetest.isprime : Test if n is prime primerange : Generate all primes in a given range prime : Return the nth prime primepi : Return the number of primes less than or equal to n composite : Return the nth composite number r-rr )rIr{rs r compositepirs*, AA1u wqz>A r)r r0)T)NF)rrrr itertoolsrrrsympy.core.functionrsympy.core.singletonr primetestr sympy.utilities.miscr rrrr"r$rxr>r{rrrKrrrrrrErrrs "("'$m%m%` GTzhzzAH,^\~# L? DVr> Br