a RG5dr@sdZddlZddlmZddlmZddlmZddlmZddl m Z dd l m Z dd l mZd d Zd dZddZddZGdddZeZddZGdddeZd+ddZddZd,ddZdd Zd-d"d#Zd.d%d&Zd'd(Zd)d*ZdS)/z" Generating and counting primes. N)bisectcount)array)Function)S)isprime)as_intcCstddg|S)Nlr_arraynrR/var/www/html/django/DPS/env/lib/python3.9/site-packages/sympy/ntheory/generate.py_azerossrcGs td|SNr r )vrrr_asetsrcCstdt||Sr)r rangeabrrr_arangesrcCsddlm}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.integersrr )rrrrr_as_int_ceilings rc@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 jjjfDshJdS)N rrc3s|]}t|jkVqdSN)len_n.0iselfrr =z!Sieve.__init__..)r*r_list_tlist_mlistallr.rr.r__init__8s 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>primerrr r'totientmobius)r)r2r3r4r.rrr__repr__?s   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|]}|duVqdSr(rr+rrrr0Pr1zSieve._reset..TN)r5r2r*r3r4)r/r7r9r:rrr_resetMs z Sieve._resetcCst|}||jdkrdSt|dd}|||jdd}t||d}||D],}| |}t|t||D] }d||<qxqZ|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 r'N?rrr cSsg|] }|r|qSrr)r,xrrr |r1z Sieve.extend..)intr2extendr primerangerr)r )r/rZmaxbasebeginZnewsievep startindexr-rrrrAYs    z Sieve.extendcCs4t|}t|j|kr0|t|jddqdS)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)r r)r2rAr@)r/r-rrr extend_to_no~szSieve.extend_to_noccs|durt|}d}ntdt|}t|}||kr8dS||||d}t|jd}||kr|j|d}||kr|V|d7}q^dSq^dS)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)rmaxrAsearchr)r2)r/rrr-ZmaxirDrrrrBs   zSieve.primerangeccstdt|}t|}t|j}||kr,dS||krRt||D]}|j|Vq>n|jt||7_td|D]T}|j|}||d||}t|||D]}|j||8<q||krp|Vqpt||D]D}|j|}td|||D]}|j||8<q||kr|VqdS)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] rNr )rGrr)r3rr)r/rrrr-tirEjrrr totientranges,    zSieve.totientrangeccstdt|}t|}t|j}||kr,dS||krRt||D]}|j|Vq>n|jt||7_td|D]T}|j|}||d||}t|||D]}|j||8<q||krr|Vqrt||D]D}|j|}td|||D]}|j||8<q||kr|VqdS)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] rNr )rGrr)r4rr)r/rrrr-mirErJrrr mobiusranges,   zSieve.mobiusrangecCsrt|}t|}|dkr$td|||jdkr<||t|j|}|j|d|krb||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 zn should be >= 2 but got: %sr'rN)rr ValueErrorr2rAr)r/rtestrrrrrHs   z Sieve.searchc Cs\zt|}|dksJWnttfy0YdS0|ddkrF|dkS||\}}||kS)Nr Fr)r rNAssertionErrorrH)r/rrrrrr __contains__1s zSieve.__contains__ccstdD]}||VqdS)Nrr)r/rrrr__iter__<s zSieve.__iter__cCst|trV||j|jdur&|jnd}|dkr:td|j|d|jd|jS|dkrftdt|}|||j|dSdS)zReturn the nth prime numberNrrzSieve indices start at 1.) isinstanceslicerFstopstart IndexErrorr2stepr )r/rrVrrr __getitem__@s   zSieve.__getitem__)NNN)N)__name__ __module__ __qualname____doc__r6r;r<rArFrBrKrMrHrQrRrYrrrrr&s % ,#, rcCst|}|dkrtd|ttjkr.t|Sddlm}ddlm}d}t ||||||}||kr||d?}|||kr|}qf|d}qft |d}||krt |r|d7}|d7}q|dS)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 rz-nth must be a positive integer; prime(1) == 2rloglir ) r rNr)siever2&sympy.functions.elementary.exponentialr_'sympy.functions.special.error_functionsrar@primepir )nthrr_rarrmidZn_primesrrrr7Ys(1       r7c@seZdZdZeddZdS)reaw 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 Cs |tjurtjS|tjur tjSz t|}Wn0ty\|jdksN|tjurVtdYdS0|dkrltjS|t j dkrtt |dSt|d}|d8}t |d}|||kr|d7}q|d8}dg|d}dg|d}t d|dD] }|d||<||d||<qt d|dD]}||||dkr@q"||d}t dt||||dD]N}||}||kr|||||8<n||||||8<qht|||d} t || dD]"}||||||8<q֐q"t|dS)NFzn must be realr r'rr=r)rInfinityNegativeInfinityZeror@ TypeErroris_realNaNrNrbr2rHrGrmin) clsrlimarr1arr2r-rDrJstZlim2rrrevalsL             $z primepi.evalN)rZr[r\r] classmethodrtrrrrresMrecCs8t|}t|}|dkr@|}d}t|}|d7}||kr q>> 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 rr r#r!r")r r!r&r"rr8rr&N)r@r nextprimerbr2rHr )rZithr-prrJr unnrrrrv sJ       rvcCst|}|dkrtd|dkr4dddddd|S|tjdkrlt|\}}||krdt|dSt|Sd |d }||dkr|d}t|r|S|d 8}n|d}t|r|S|d8}t|r|S|d 8}qd 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!zno preceding primesr r")r!r&r"rr#r'rrr&N)rrNrbr2rHr )rr rxryrrr prevprimeds.    r{ccsx|durd|}}||krdS|tjdkrBt||EdHdSt|d}t|}t|}||krn|VqVdSqVdS)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)rbr2rBrrvrrrrrBsK  rBcCsZ||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. 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 Nrz&no primes exist in the specified range)mapr@randomrandintrvr{rN)rrrrDrrr randprimesrTcCsp|rt|}nt|}|dkr&tdd}|rPtd|dD]}|t|9}q>> 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 rzprimorial argument must be >= 1r )r r@rNrr7rB)rrfrDr-rrr primorials2  rFc cst|pd}d}}|||}}d}||krv|r:||krv|d7}||krZ|}|d9}d}|rd|V||}|d7}q&|r||kr|rdS|dfVdS|sd} |}}t|D] }||}q||kr||}||}| d7} q| 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: https://en.wikipedia.org/wiki/Cycle_detection. rrr N)r@r) fx0nmaxvaluespowerlamZtortoiseZharer-murrr cycle_lengthZs>4       rc Csft|}|dkrtdgd}|dkr4||dSdtjd}}||t|dkr||dkr||d?}|t|d|kr|}qX|}qXt|r|d8}|Sddlm}dd lm }d}t ||||||}||kr||d?}|||d|kr|}q|d}q|t|d}||krPt|sD|d8}|d8}q(t|rb|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) == 4) r&rrz rr&r'rr^r`) r rNrbr2rer rcr_rdrar@) rfrZ composite_arrrrrgr_raZ n_compositesrrr compositesB            rcCs$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 r&rr)r@rerrrr compositepisr)r)N)T)NF)r]r}r itertoolsrrr sympy.core.functionrsympy.core.singletonr primetestr sympy.utilities.miscr rrrrrrbr7rervr{rBrrrrrrrrrs4       2J} D/ _& B YA