\c@ s2dZddlmZmZddlZddlmZddlmZddlm Z m Z ddl m Z m Z dd lmZd Zd Zd Zd fdYZeZdZde fdYZddZdZdZdZedZeedZdZ dZ!dS(s" Generating and counting primes. i(tprint_functiontdivisionN(tbisect(tarray(tFunctiontS(tas_inttrangei(tisprimecC stddg|S(Ntli(t_array(tn((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt_azerosscG s td|S(NR (R (tv((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt_asetscC stdt||S(NR (R R(tatb((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt_arangestSievecB szeZdZdZdZd d d dZdZdZdZ dZ dZ d Z d Z d ZRS( sAn 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 sd_tdddddd_tdd d ddd _tdd d d dd _tfd jjjfDstdS( Niiiiii i iiiic3 s$|]}t|jkVqdS(N(tlent_n(t.0ti(tself(s5lib/python2.7/site-packages/sympy/ntheory/generate.pys 5s(RRt_listt_tlistt_mlisttalltAssertionError(R((Rs5lib/python2.7/site-packages/sympy/ntheory/generate.pyt__init__0s  cC sddt|j|jd|jd|jd|jd|jddt|j|jd|jd|jd|jd|jdd t|j|jd|jd|jd|jd|jdfS( Nss<%s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i %s sieve (%i): %i, %i, %i, ... %i, %i>tprimeiiiiittotienttmobius(RRRR(R((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt__repr__7scC std|||fDr0t}}}n|rL|j|j |_n|rh|j|j |_n|r|j|j |_ndS(s]Reset all caches (default). To reset one or more set the desired keyword to True.cs s|]}|dkVqdS(N(tNone(RR((s5lib/python2.7/site-packages/sympy/ntheory/generate.pys HsN(RtTrueRRRR(RRRR ((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt_resetEsc C st|}||jdkr#dSt|dd}|j||jdd}t||d}xR|jd|D]>}| |}x*t|t||D]}d||>> from sympy import sieve >>> sieve._reset() # this line for doctest only >>> sieve.extend(30) >>> sieve[10] == 29 True iNg?iiiR (tintRtextendRt primerangeRRR ( RR tmaxbasetbegintnewsievetpt startindexRtx((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyR&Qs   cC sJt|}x7t|j|krE|jt|jddqWdS(sExtend 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. ig?N(RRRR&R%(RR((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt extend_to_novs cc sddlm}tdt||}t||}||krMdS|j||j|d}t|jd}xC||kr|j|d}||kr|V|d7}qdSqWdS(sGenerate 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] i(tceilingiNi(t#sympy.functions.elementary.integersR/tmaxRR&tsearchRR(RRRR/RtmaxiR+((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyR's     c c sddlm}tdt||}t||}t|j}||kr\dS||krx.t||D]}|j|VqxWn|jt||7_x~td|D]m}|j|}||d||}x-t|||D]}|j|c|8>> from sympy import sieve >>> print([i for i in sieve.totientrange(7, 18)]) [6, 4, 6, 4, 10, 4, 12, 6, 8, 8, 16] i(R/iNi(R0R/R1RRRRR( RRRR/R RttiR,tj((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt totientranges.        c c sddlm}tdt||}t||}t|j}||kr\dS||krx/t||D]}|j|VqxWn |jt||7_x~td|D]m}|j|}||d||}x-t|||D]}|j|c|8>> from sympy import sieve >>> print([i for i in sieve.mobiusrange(7, 18)]) [-1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1] i(R/iNi(R0R/R1RRRRR ( RRRR/R RtmiR,R5((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt mobiusranges.       cC sddlm}t||}t|}|dkrMtd|n||jdkrp|j|nt|j|}|j|d|kr||fS||dfSdS(s~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) i(R/isn should be >= 2 but got: %siN(R0R/Rt ValueErrorRR&R(RR R/ttestR((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyR2s   cC svy"t|}|dks!tWnttfk r<tSX|ddkrW|dkS|j|\}}||kS(Nii(RRR9tFalseR2(RR RR((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt __contains__&s  cC spt|trD|j|j|j|jd|jd|jSt|}|j||j|dSdS(sReturn the nth prime numberiN(t isinstancetsliceR.tstopRtstarttstepR(RR ((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt __getitem__1s %  N(t__name__t __module__t__doc__RR!R"R$R&R.R'R6R8R2R<RB(((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyRs   %   ' 0 ! cC s$t|}|dkr'tdn|ttjkrDt|Sddlm}ddlm}d}t ||||||}xC||kr||d?}|||kr|}q|d}qWt |d}x3||krt |r|d7}n|d7}qW|dS(sD 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 is-nth must be a positive integer; prime(1) == 2i(tli(tlogi( RR9RtsieveRt'sympy.functions.special.error_functionsRFt&sympy.functions.elementary.exponentialRGR%tprimepiR(tnthR RFRGRRtmidtn_primes((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyR?s(1  &   RKcB seZdZedZRS(s: 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 >>> 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 c C s|tjkrtjS|tjkr,tjSyt|}Wn?tk r}|jtksj|tjkryt dndSX|dkrtjS|t j dkrtt j |dSt|d}|d8}t |d}x|||kr|d7}qW|d8}dg|d}dg|d}x;td|dD]&}|d||<||d||>> 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 iiiiiiiiN(R%Rt nextprimeRHRR2R(R tithRtprR5R tutnn((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyR^sN        +              cC sMddlm}t||}|dkr=tdn|dkrtidd6dd6dd6dd 6dd 6|S|tjdkrtj|\}}||krt|d St|Snd |d }||d kr|d }t|r|S|d8}n |d }x8t|r!|S|d8}t|r;|S|d8}qWd S( s 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 i(R/isno preceding primesiiiiiiiN(R0R/RR9RHRR2R(R R/R RaRb((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt prevprime@s2  +          cc sddlm}||kr dS|tjdkrYxtj||D] }|VqFWdSt||d}t||}x(t|}||kr|VqdSqWdS(s 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. 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] 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 ======== 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 i(R/Ni(R0R/RHRR'RR^(RRR/R((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyR'ss;    cC s||krdStt||f\}}tj|d|}t|}||krht|}n||krtdn|S(s$ 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 Nis&no primes exist in the specified range(tmapR%trandomtrandintR^RcR9(RRR R+((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt randprimes    cC s|rt|}n t|}|dkr<tdnd}|ryxVtd|dD]}|t|9}q\Wn(x%td|dD]}||9}qW|S(sE 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, 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 isprimorial argument must be >= 1i(RR%R9RRR'(R RLR+R((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt primorials2  c c sot|p d}d}}|||}}d}xv||kr| sW||kr|d7}||kr|}|d9}d}n|r|Vn||}|d7}q8W|r||kr|rdS|dfVdSn|skd} |}}x t|D]}||}qWx2||krI||}||}| d7} qW| r]| d8} n|| fVndS(sFor 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. iiiN(R%R"R( tftx0tnmaxtvaluestpowertlamttortoisethareRtmu((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt cycle_length-s>4 "          c C st|}|dkr'tdnddddddd d d d g }|dkrc||dSdtjd }}||t|dkrxK||dkr||d?}|t|d|kr|}q|}qWt|r|d8}n|Sd dlm}d dlm }d}t ||||||}xK||kr||d?}|||d|kr|}qK|d}qKW|t|d}x3||krt|s|d8}n|d8}qWt|r|d8}n|S(s 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 is1nth must be a positive integer; composite(1) == 4iiii i i iiiii(RF(RG( RR9RHRRKRRIRFRJRGR%( RLR t composite_arrRRRMRFRGt n_composites((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt compositesB  $      &     cC s.t|}|dkrdS|t|dS(sk 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 iii(R%RK(R ((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyt compositepis  ("REt __future__RRReRRR tsympyRRtsympy.core.compatibilityRRt primetestRR RRRRHRRKR^RcR'RgR#RhR"R;RrRuRv(((s5lib/python2.7/site-packages/sympy/ntheory/generate.pyts.      Kr D 3 R & BY B