o 8Vav@sdZddlZddlmZddlmZddlmZddlmZm Z ddl m Z dd l m Z d d Zd d ZddZGdddZeZddZGdddeZd(ddZddZd)ddZddZd*dd Zd+d"d#Zd$d%Zd&d'ZdS),z" Generating and counting primes. N)bisectcount)array)FunctionS)as_int)isprimecCstddg|S)Nlr_arraynr8/usr/lib/python3/dist-packages/sympy/ntheory/generate.py_azerosrcGs td|SNr r )vrrr_asets rcCstdt||Sr)r range)abrrr_arangerrc@steZdZdZddZddZdddZd d Zd d Zdd dZ ddZ ddZ ddZ ddZ ddZddZdS)SieveaAn infinite list of prime numbers, implemented as a dynamically growing sieve of Eratosthenes. When a lookup is requested involving an odd number that has not been sieved, the sieve is automatically extended up to that number. Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> 25 in sieve False >>> sieve._list array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23]) csld_tdddddd_tdd d ddd _tdd d d dd _tfd d jjjfDs4JdS)N rr c3s|] }t|jkVqdSN)len_n.0iselfrr 5sz!Sieve.__init__..)r'r_list_tlist_mlistallr+rr+r__init__0s *zSieve.__init__cCsddt|j|jd|jd|jd|jd|jddt|j|jd|jd|jd|jd|jdd t|j|jd|jd|jd|jd|jdfS) Nzs<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>primerr rr$totientmobius)r&r.r/r0r+rrr__repr__7s   zSieve.__repr__NcCsntdd|||fDrd}}}|r|jd|j|_|r(|jd|j|_|r5|jd|j|_dSdS)z]Reset all caches (default). To reset one or more set the desired keyword to True.css|]}|duVqdSr%rr(rrrr-HszSieve._reset..TN)r1r.r'r/r0)r,r3r5r6rrr_resetEs z Sieve._resetcCst|}||jdkr dSt|dd}|||jdd}t||d}||D]}| |}t|t||D]}d||<q>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True r$N?r rr cSsg|]}|r|qSrr)r)xrrr tsz Sieve.extend..)intr.extendr primerangerr&r )r,rZmaxbaseZbeginZnewsievep startindexr*rrrr=Qs    "z Sieve.extendcCsDt|}t|j|kr |t|jddt|j|ks dSdS)aExtend to include the ith prime number. Parameters ========== i : integer Examples ======== >>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend_to_no(9) >>> sieve._list array('l', [2, 3, 5, 7, 11, 13, 17, 19, 23]) Notes ===== The list is extended by 50% if it is too short, so it is likely that it will be longer than requested. r$g?N)rr&r.r=r<)r,r*rrr extend_to_novszSieve.extend_to_noccsddlm}|durt||}d}ntdt||}t||}||kr)dS||||d}t|jd}||kr[|j|d}||krS|V|d7}ndS||ks@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] rceilingNrr )#sympy.functions.elementary.integersrCrmaxr=searchr&r.)r,rrrCr*Zmaxir?rrrr>s&     zSieve.primerangec cs2ddlm}tdt||}t||}t|j}||kr!dS||kr5t||D]}|j|Vq*dS|jt||7_td|D]*}|j|}||d||}t|||D] }|j||8<q[||krn|VqDt||D]"}|j|}td|||D] }|j||8<q||kr|VqtdS)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] rrBr Nr)rDrCrErr&r/rr) r,rrrCrr*Ztir@jrrr totientranges8     zSieve.totientrangec cs4ddlm}tdt||}t||}t|j}||kr!dS||kr5t||D]}|j|Vq*dS|jt||7_td|D]*}|j|}||d||}t|||D] }|j||8<q\||kro|VqEt||D]"}|j|}td|||D] }|j||8<q||kr|VqudS)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] rrBr Nr)rDrCrErr&r0rr) r,rrrCrr*mir@rGrrr mobiusranges8     zSieve.mobiusrangecCs~ddlm}t||}t|}|dkrtd|||jdkr&||t|j|}|j|d|kr9||fS||dfS)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) rrBrzn should be >= 2 but got: %sr$r )rDrCr ValueErrorr.r=r)r,rrCtestrrrrrFs      z Sieve.searchc Cs\z t|}|dks JWn ttfyYdSw|ddkr#|dkS||\}}||kS)NrFr)rrKAssertionErrorrF)r,rrrrrr __contains__8s zSieve.__contains__ccstdD]}||VqdS)Nr r)r,rrrr__iter__Cs  zSieve.__iter__cCst|tr+||j|jdur|jnd}|dkrtd|j|d|jd|jS|dkr3tdt|}|||j|dS)zReturn the nth prime numberNrr zSieve indices start at 1.) isinstanceslicerAstopstart IndexErrorr.stepr)r,rrSrrr __getitem__Gs   zSieve.__getitem__)NNNr%)__name__ __module__ __qualname____doc__r2r7r8r=rAr>rHrJrFrNrOrVrrrrrs  % /'0! rcCst|}|dkr td|ttjkrt|Sddlm}ddlm}d}t ||||||}||krN||d?}|||krF|}n|d}||ks7t |d}||krht |r`|d7}|d7}||ksX|dS)aD 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. 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) == 2rlilogr) rrKr&siever.'sympy.functions.special.error_functionsr\&sympy.functions.elementary.exponentialr^r<primepir )nthrr\r^rrmidZn_primesrrrr3`s,1     r3c@seZdZdZeddZdS)rbaH 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*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, prime, prevprime, isprime >>> primepi(25) 9 So there are 9 primes less than or equal to 25. Is 25 prime? >>> isprime(25) False It isn't. 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 Cs|tjurtjS|tjurtjSzt|}Wnty.|jdks'|tjur+tdYdSw|dkr6tjS|t j dkrFtt |dSt|d}|d8}t |d}|||kre|d7}|||ks[|d8}dg|d}dg|d}t d|dD]}|d||<||d||<q~t d|dD]g}||||dkrq||d}t dt||||dD]%}||}||kr|||||8<q||||||8<qt|||d} t || dD]}||||||8<qqt|dS)NFzn must be realrr$rr9r )rZInfinityZNegativeInfinityZZeror< TypeErrorZis_realZNaNrKr_r.rFrErmin) clsrZlimZarr1Zarr2r*r?rGstZlim2rrrevalsR            z primepi.evalN)rWrXrYrZ classmethodrirrrrrbsMrbcCs2t|}t|}|dkr!|}d} t|}|d7}||kr |Sq|dkr'dS|dkr5dddddd|S|tjdkrQt|\}}||krMt|dSt|Sd|d}||krj|d7}t|re|S|d 7}n||dkr|d7}t|rz|S|d 7}n|d} t|r|S|d7}t|r|S|d 7}q) 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 r rr rr)rrr#rrr4rr#)r<r nextprimer_r.rFr )rZithr*prrGr unnrrrrk(sR     rkcCsddlm}t||}|dkrtd|dkr"dddddd|S|tjd kr>t|\}}||kr:t|d St|Sd |d }||d krY|d }t|rT|S|d 8}n|d } t|rd|S|d8}t|rn|S|d 8}q^) 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 rrBrzno preceding primesrr)rr#rrr r$r rr#)rDrCrrKr_r.rFr )rrCr rmrnrrr prevprimels4      rpccsddlm}|durd|}}||krdS|tjdkr(t||EdHdSt||d}t||} t|}||krC|VndSq7)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 rrBNrr$r )rDrCr_r.r>rrk)rrrCrrrr>s" K  r>cCsZ||krdStt||f\}}t|d|}t|}||kr#t|}||kr+td|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)mapr<randomZrandintrkrprK)rrrr?rrr randprimesrsTcCsr|rt|}nt|}|dkrtdd}|r)td|dD]}|t|9}q|Std|dD]}||9}q0|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)rr<rKrr3r>)rrcr?r*rrr primorial(s2  rtFc cst|pd}d}}|||}}d}||krE|r||krE|d7}||kr.|}|d9}d}|r3|V||}|d7}||krE|r||ks|rV||krV|rOdS|dfVdS|sd} |}}t|D]}||}qb||kr}||}||}| d7} ||ksm| r| d8} || 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 rN)r<r) fZx0ZnmaxvaluesZpowerZlamZtortoiseZharer*Zmurrr cycle_lengthjsF 4    rwc Cspt|}|dkr tdgd}|dkr||dSdtjd}}||t|dkrW||dkrM||d?}|t|d|krE|}n|}||dks2t|rU|d8}|Sddlm}dd lm }d}t ||||||}||kr||d?}|||d|kr|}n|d}||ksw|t|d}||krt|s|d8}|d8}||kst|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 r z1nth must be a positive integer; composite(1) == 4) r#rro ryr#r$rr[r]) rrKr_r.rbr r`r\rar^r<) rcrZ composite_arrrrrdr\r^Z n_compositesrrr compositesH       rcCs$t|}|dkr dS|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 r#rr )r<rbrrrr compositepisr)r r%)T)NF)rZrrr itertoolsrrr ZsympyrrZsympy.core.compatibilityrZ primetestr rrrrr_r3rbrkrpr>rsrtrwrrrrrrs2     AK }D 3c & BY B