EdrdZddlZddlmZddlmZddlmZddlmZddl m Z dd l m Z dd l mZd Zd Zd ZdZGddZeZdZGddeZddZdZddZdZd dZd!dZdZdZdS)"z" Generating and counting primes. N)bisectcount)array)Function)S)isprime)as_intc*tddg|zS)Nlr_arrayns 6lib/python3.11/site-packages/sympy/ntheory/generate.py_azerosrs #s1u  c"td|SNr r)vs r_asetrs #q>>rc>tdt||Sr)rrangeabs r_arangers #uQ{{ # ##rc>ddlm}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 ZdZdS)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]) c d_tdddddd_tdd d ddd _tdd d d dd _t fd jjjfDsJdS) N rr c3HK|]}t|jkVdSN)len_n).0iselfs r z!Sieve.__init__..=s0UU3q66TW$UUUUUUr)r2r_list_tlist_mlistallr5s`r__init__zSieve.__init__8s1aAr2.. Aq!Q1-- Aq"b!R00 UUUUtz4; .TUUUUUUUUUUrcddt|j|jd|jd|jd|jd|jddt|j|jd|jd|jd|jd|jdd t|j|jd|jd|jd|jd|jdfzS) Nzs<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>primerr r'r.totientmobius)r1r7r8r9r;s r__repr__zSieve.__repr__?s6c$*ooA 1 tz!}BB DK((QQQR$+b/ s4;''QQQR$+b/ :C C CrNctd|||fDrdx}x}}|r|jd|j|_|r|jd|j|_|r|jd|j|_dSdS)z]Reset all caches (default). To reset one or more set the desired keyword to True.c3K|]}|duV dSr0)r3r4s rr6zSieve._reset..Ps&;;QqDy;;;;;;rTN)r:r7r2r8r9)r5r>r@rAs r_resetz Sieve._resetMs ;;5'6":;;; ; ; ,'+ +E +Gf  .HTWH-DJ  0+htwh/DK  0+htwh/DKKK 0 0rct|}||jdkrdSt|dzdz}|||jddz}t||dz}||D].}| |z}t |t ||D]}d||</|xjtdd|Dz c_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 r.N?r rr cg|]}||SrErE)r3xs r z Sieve.extend..|s"<"<"<!"<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 r1r7rMrL)r5r4s r extend_to_nozSieve.extend_to_no~st. 1II$*oo! 3 KKDJrNS011 2 2 2$*oo! 3 3 3 3 3rc#K|t|}d}n,tdt|}t|}||krdS||||d}t |jdz}||kr*|j|dz }||kr |V|dz }ndS||k(dSdS)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"maxrMsearchr1r7)r5rrr4maxirRs rrNzSieve.primeranges2  #""AAAAq))**A""A 6  F A KKNN1 4:"$h  1q5!A1u Q $h     rc#Ktdt|}t|}t|j}||krdS||kr$t ||D]}|j|VdS|xjt ||z c_t d|D]P}|j|}||zdz |z|z}t |||D]}|j|xx|zcc<||kr|VQt ||D]E}|j|}t d|z||D]}|j|xx|zcc<||kr|VFdS)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')rWr"r1r8rr)r5rrrr4tirSjs r totientrangezSieve.totientranges ?1%% & & A      6  F !V 1a[[ % %k!n$$$$ % % KK71a== (KK1a[[  [^!eaiA-1 z1a00))AKNNNb(NNNN6HHH1a[[  [^q1ua++))AKNNNb(NNNN6HHH   rc#Ktdt|}t|}t|j}||krdS||kr$t ||D]}|j|VdS|xjt ||z z c_t d|D]P}|j|}||zdz |z|z}t |||D]}|j|xx|zcc<||kr|VQt ||D]E}|j|}t d|z||D]}|j|xx|zcc<||kr|VFdS)aGenerate all mobius numbers for the range [a, b). Parameters ========== a : integer First number in range b : integer First number outside of range Examples ======== >>> from sympy import sieve >>> print([i for i in sieve.mobiusrange(7, 18)]) [-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1] r Nr')rWr"r1r9rr)r5rrrr4mirSr\s r mobiusrangezSieve.mobiusranges& ?1%% & & A      6  F !V 1a[[ % %k!n$$$$ % % KK71q5>> )KK1a[[  [^!eaiA-1 z1a00))AKNNNb(NNNN6HHH1a[[  [^q1ua++))AKNNNb(NNNN6HHH   rc"t|}t|}|dkrtd|z||jdkr||t |j|}|j|dz |kr||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 ValueErrorr7rMr)r5rtestrs rrXz Sieve.searchs"q!! 1II q5 A;a?@@ @ tz"~   KKNNN 4:q ! ! :a!e  $ a4Ka!e8Orc t|}|dksJn#ttf$rYdSwxYw|dzdkr|dkS||\}}||kS)Nr'Fr)r rbAssertionErrorrX)r5rrrs r __contains__zSieve.__contains__1s} q A6MMMMN+   55  q5A: 6M{{1~~1Av s //c#BKtdD] }||V dS)Nr r)r5rs r__iter__zSieve.__iter__<s4q  Aq'MMMM  rc|t|tr_||j|j|jnd}|dkrt d|j|dz |jdz |jS|dkrt dt|}|||j|dz S)zReturn the nth prime numberNrr zSieve indices start at 1.) isinstanceslicerUstopstart IndexErrorr7stepr )r5rrms r __getitem__zSieve.__getitem__@s a   %   af % % % !w9AGGEqy >!!<===:eai 169: :1u >!!<===q A   a :a!e$ $r)NNNr0)__name__ __module__ __qualname____doc__r<rBrFrMrUrNr]r`rXrfrhrprErrr$r$&s"VVV C C C 0 0 0 0#>#>#>J3336****X!!!F***X:   %%%%%rr$c t|}|dkrtd|ttjkr t|Sddlm}ddlm}d}t||||||zz}||kr%||zdz }|||kr|}n|dz}||k%t|dz }||krt|r|dz }|dz }||k|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 rbr1siever7&sympy.functions.elementary.exponentialrw'sympy.functions.special.error_functionsryrLprimepir )nthrrwryrrmidn_primess rr>r>YsEb s A1uJHIIIC  Qx:::::::::::: A Ass1vvCCFF # $%%A a%1ul 2c77Q; AAaA a% q1u~~H Q, 1::  MH Q Q, q5Lrc(eZdZdZedZdS)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 |tjur tjS|tjur tjS t |}n9#t $r,|jdks|tjurtdYdSwxYw|dkr tjS|tj dkr-tt |dSt |dz}|dz}t|d}||z|kr|dz }||z|k|dz}dg|dzz}dg|dzz}td|dzD]}|dz ||<||zdz ||<td|dzD]}||||dz kr||dz }tdt|||zz|dzD]C}||z}||kr||xx|||z zcc<'||xx|||z|z zcc<Dt|||zdz } t|| dD]}||xx|||z|z zcc<t|dS)NFzn must be realr'r.rrHr )rInfinityNegativeInfinityZerorL TypeErroris_realNaNrbrzr7rXrWrmin) clsrlimarr1arr2r4rRr\stlim2s revalz primepi.evals  ? :  " " 6M AAA   yE! 3Q!%Z 3 !1222 FF  q5 6M  B  )U\\!__Q'(( (!s(mm q#qkkCi1n  1HCCi1n  qscAgscAgq#'"" ! !A!eDG1fqjDGGq#'"" , ,AAw$q1u+% QU A1c!A,44q899 1 1U91GGGtBx!|+GGGGGGGtAG}q00GGGGsAEAI&&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-)rLr nextprimerzr7rXr )rithr4prr\r unns rrr s0 AAs A1u   2B FA1u     1uq1u1qQ1--a00EKO||A1 6 Q< 8O AqDB Qw  Q 1:: H Q R1 Q 1:: H Q F 1:: H Q 1:: H Q rct|}|dkrtd|dkrdddddd|S|tjdkr@t|\}}||krt|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"rbrzr7rXr )rr rrs r prevprimerds)& A1u0.///1u1qQ1--a00EKO||A1 6 1: 8O AqDB2v{ F 1:: H Q F 1:: H Q 1:: H Q rc#K|d|}}||krdS|tjdkr#t||Ed{VdSt|dz }t|} t |}||kr|VndS)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] http://primes.utm.edu/notes/gaps.html Nr'r.r )rzr7rNr"rrs rrNrNsV !1AvEKO##Aq))))))))) QAA aLL q5 GGGG F rc||krdStt||f\}}tj|dz |}t |}||krt |}||krt d|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. 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)maprLrandomrandintrrrb)rrrrRs r randprimers4 Av sQF  DAqq1ua  A! AAv aLL1uCABBB HrTc|rt|}nt|}|dkrtdd}|r)td|dzD]}|t |z}nt 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 rLrbrr>rN)rr~rRr4s r primorialrsd  1II FF1u<:;;; A q!a%  A qMAA Aq1u%%  A FAA HrFc#Kt|pd}dx}}|||}}d}||kr@|r||kr8|dz }||kr |}|dz}d}|r|V||}|dz }||kr|2||k8|r||kr |rdS|dfVdS|sYd} |x}}t|D] }||}||kr!||}||}| dz } ||k!| r| dz} || fVdSdS)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)rLr) fx0nmaxvaluespowerlamtortoiseharer4mus r cycle_lengthrZsh tyq>>DOEC2dH A d  D AH  Q C< H QJEC  JJJqww q d  D AH  T    F*    F   4s  A1T77DD$ q{{H1T77D !GB$    !GB2g   rc t|}|dkrtdgd}|dkr ||dz Sdtjd}}||t |z dz krN||dz kr/||zdz }|t |z dz |kr|}n|}||dz k/t |r|dz}|Sddlm}dd lm }d}t||||||zz}||kr+||zdz }|||z dz |kr|}n|dz}||k+|t |z dz }||krt |s|dz}|dz}||kt |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.rrvrx) r rbrzr7r}r r{rwr|ryrL) r~r composite_arrrrrrwry n_compositess r compositers0 s A1uNLMMM888MBw$QU## ek"oqAA NQ  !a%i q5Q,CWS\\!A%)  !a%i  1::  FA:::::::::::: A Ass1vvCCFF # $%%A a%1ul C=1 q  AAaA a%wqzz>A%L  qzz  A L Q  qzz Q HrcZt|}|dkrdS|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 )rLr}rs r compositepirs5, AA1uq wqzz>A r)r r0)T)NF)rtrr itertoolsrrrsympy.core.functionrsympy.core.singletonr primetestr sympy.utilities.miscr rrrr"r$rzr>r}rrrNrrrrrrErrrs  "!!!!!((((((""""""''''''$$$m%m%m%m%m%m%m%m%` GGGTzzzzzhzzzzAAAAH,,,^\\\\~# # # L? ? ? ? DVVVVr> > > Br