g0ZdZddlmZddlZddlmZddlmZddlm Z m Z ddl m Z ddl mZdd lmZdd lmZdd lmZmZmZmZmZmZmZd d lmZmZmZd dl m!Z!m"Z"m#Z#d dl$m$Z$ddl%m&Z&ddl'm(Z(ddl)m*Z*m+Z+d dl,m-Z-dZ.d[dZ/dZ0dZ1d\dZ2d]dZ3d^dZ4d_d!Z5d`d#Z6d$Z7d%Z8d&Z9d'Z:d(Z;d)Zd,Z?d\d-Z@ dad.ZA dbd/ZBdcd0ZCd`d1ZDddd2ZEded3ZFd`d4ZGdfd5ZHd6ZId`d7ZJd8ZKd9ZLd`d:ZMd;ZNe&d?d@ZOe&dAd=d>?dBZPe&dCd=d>?dfdDZQdfdEeRdFeRdGeRfdHZSd\dIZTe&dJd=d>?dfdKZUe&dLd=d>?dMZVe&dNd=d>?dOZWdPZXdQZYdRZZdSZ[dTZ\dUZ]dVZ^dWZ_dXZ`dYZadZZbdS)gz Integer factorization ) defaultdictN)Dict)Mul)RationalInteger) num_digitsPow)_randint)S) SYMPY_INTSgcdsqrtsqrtremiroot bit_scan1remove)isprimeMERSENNE_PRIME_EXPONENTSis_mersenne_prime)sieve primerange nextprime)digits) deprecated)flatten)as_int filldedent)_ecm_one_factorc|dkrdSt|ttfdDfS)a Return the B-smooth and B-power smooth values of n. The smoothness of n is the largest prime factor of n; the power- smoothness is the largest divisor raised to its multiplicity. Examples ======== >>> from sympy.ntheory.factor_ import smoothness >>> smoothness(2**7*3**2) (3, 128) >>> smoothness(2**4*13) (13, 16) >>> smoothness(2) (2, 2) See Also ======== factorint, smoothness_p r)rrc3.K|]}||zVdSN).0mfacss Q/var/www/html/ai-engine/env/lib/python3.11/site-packages/sympy/ntheory/factor_.py zsmoothness..5s+33!T!W*333333) factorintmax)nr's @r( smoothnessr.sL0 Avvv Q<>> from sympy.ntheory.factor_ import smoothness_p, factorint >>> smoothness_p(10431, m=1) (1, [(3, (2, 2, 4)), (19, (1, 5, 5)), (61, (1, 31, 31))]) >>> smoothness_p(10431) (-1, [(3, (2, 2, 2)), (19, (1, 3, 9)), (61, (1, 5, 5))]) >>> smoothness_p(10431, power=1) (-1, [(3, (2, 2, 2)), (61, (1, 5, 5)), (19, (1, 3, 9))]) If visual=True then an annotated string will be returned: >>> print(smoothness_p(21477639576571, visual=1)) p**i=4410317**1 has p-1 B=1787, B-pow=1787 p**i=4869863**1 has p-1 B=2434931, B-pow=2434931 This string can also be generated directly from a factorization dictionary and vice versa: >>> factorint(17*9) {3: 2, 17: 1} >>> smoothness_p(_) 'p**i=3**2 has p-1 B=2, B-pow=2\np**i=17**1 has p-1 B=2, B-pow=16' >>> smoothness_p(_) {3: 2, 17: 1} The table of the output logic is: ====== ====== ======= ======= | Visual ------ ---------------------- Input True False other ====== ====== ======= ======= dict str tuple str str str tuple dict tuple str tuple str n str tuple tuple mul str tuple tuple ====== ====== ======= ======= See Also ======== factorint, smoothness )rr)TFNc,g|]}t|Sr$intr%is r( z smoothness_p..s0BBBqCFFBBBr*hasr=rz**TF)visualr/c zg|]7\}}|t|gtt|zzf8Sr$)tuplelistr.)r%fMr&s r(r6z smoothness_p..sY999!Qd:a!e+<+<&=&= =>>@999r*c0|d|dfSNrrr$)xks r(zsmoothness_p..sqtAw!or*keyz#p**i=%i**%i has p%+i B=%i, B-pow=%i )bool isinstancestr splitlinessplit smoothness_pr;r+sortedr<itemstyper3rrinsertappendjoin) r-r&powerr9dlivr'rvlinesdatrBs ` @r(rMrM8sCzf } $ $!S *  H ,,..  BBBHHUOOA&,,S11!4::4@@BBBDAqAaDD   &"5"5HAe,,,, 5 ! !*5)))   !U8 9999%)$**,,%7%799965557778 6--DGGSz4I4I E!uIIcll 1a :U3ZZGHHHH 99U  r*c t|t|}}n#t$rddlm}t d||fDr6t |}t |}|jdkr_|jdkrt|j|j cYSt|j|jt|j|jz cYS|jdkrt|j|jcYStt|j|jt|j|j}tt|j|jt|j|j}||z cYSt|ttfr^t||rNt|j dtr.|j ddkrt||j dcYStd|d|wxYw|dkrtd|zt||dS)a* Find the greatest integer m such that p**m divides n. Examples ======== >>> from sympy import multiplicity, Rational >>> [multiplicity(5, n) for n in [8, 5, 25, 125, 250]] [0, 1, 2, 3, 3] >>> multiplicity(3, Rational(1, 9)) -2 Note: when checking for the multiplicity of a number in a large factorial it is most efficient to send it as an unevaluated factorial or to call ``multiplicity_in_factorial`` directly: >>> from sympy.ntheory import multiplicity_in_factorial >>> from sympy import factorial >>> p = factorial(25) >>> n = 2**100 >>> nfac = factorial(n, evaluate=False) >>> multiplicity(p, nfac) 52818775009509558395695966887 >>> _ == multiplicity_in_factorial(p, n) True See Also ======== trailing r factorialc3NK|] }t|ttfV!dSr#)rIr rr4s r(r)zmultiplicity..s1EEz!j(344EEEEEEr*rz!expecting ints or fractions, got z and z9no such integer exists: multiplicity of %s is not-defined)r ValueError(sympy.functions.combinatorial.factorialsr]allrqp multiplicityminrIr rargsmultiplicity_in_factorialr)rcr-r]likecrosss r(rdrds@BPayy&))1 PPPFFFFFF EEq!fEEE E E ; A Asaxx3!88(ac222222#AC-- QS!#0F0FFFFF#AC----- ac** ac**,, ac** ac**,,e|###Z122 ;1i(( ;16!9g.. ;q Q,Qq :: : : :jAANOOO1P4 AvvTWXYZZZ !Q<<?s*"ABHA8H?Hct|tc}|dkrtd|zdkrtdztt}t |D]\}}t |||||<tfd|DS)areturn the largest integer ``m`` such that ``p**m`` divides ``n!`` without calculating the factorial of ``n``. Parameters ========== p : Integer positive integer n : Integer non-negative integer Examples ======== >>> from sympy.ntheory import multiplicity_in_factorial >>> from sympy import factorial >>> multiplicity_in_factorial(2, 3) 1 An instructive use of this is to tell how many trailing zeros a given factorial has. For example, there are 6 in 25!: >>> factorial(25) 15511210043330985984000000 >>> multiplicity_in_factorial(10, 25) 6 For large factorials, it is much faster/feasible to use this function rather than computing the actual factorial: >>> multiplicity_in_factorial(factorial(25), 2**100) 52818775009509558395695966887 See Also ======== multiplicity rz!expecting positive integer got %sz%expecting non-negative integer got %sc3vK|]3\}}|ztt|z |dz z|zV4dSrN)sumr)r%rWrBr-s r(r)z,multiplicity_in_factorial..sNMM41aAF1aLL)))QU3Q6MMMMMMr*)rr_rr3r+rOr,re)rcr-r=rBrWs ` r(rgrgsT !99fQiiDAqAvv 1``, else ``False`` (e.g. 1 is not a perfect power). Explanation =========== This is a low-level helper for ``perfect_power``, for internal use. Parameters ========== n : int assume that n is a nonnegative integer next_p : int Assume that n has no factor less than next_p. i.e., all(n % p for p in range(2, next_p)) is True Examples ======== >>> from sympy.ntheory.factor_ import _perfect_power >>> _perfect_power(16) (2, 4) >>> _perfect_power(17) False Frrct|dkrdS|||<tjfd|DfS)NrFc3.K|]\}}||zzVdSr#r$r%rcegs r(r)z/_perfect_power..done..F/??tq!QT??????r*)rmathprodrO)r-factorsrtmultis ` r(donez_perfect_power..doneAsY 5MM 665 y????w}}?????BBr*i@Bic3.K|]\}}||zzVdSr#r$rrs r(r)z!_perfect_power..Prur*rFc3.K|]\}}||zzVdSr#r$)r%rcrs_gs r(r)z!_perfect_power..sK%E%E,0Aq&'BZ%E%E%E%E%E%Er*c3.K|]\}}||zzVdSr#r$rrs r(r)z!_perfect_power..sK-I-I041./AY-I-I-I-I-I-Ir*(T?{Gz?)_factorint_smallrvaluesrvrwrOrrr bit_lengthrrrNr+keysr3loglog2powabs)r-next_prxryrzr&_exacttf_maxrct prime_iterlogn thresholdbrbrrts @@r(_perfect_powerrs6 AvvuG A ECCC I~~ Wav > >q A q555 !! " 665y????w}}?????BBzz aLL  !Avvu !GAGAJAvv!t "!QKK 6!Q36MQZZ! 5 a%1**!QKK 6  A aKEE  a%1** 619}}tAw5))) \\^^R " $F FF++ % %A!Q< 1``, else ``False`` (e.g. 1 is not a perfect power). A ValueError is raised if ``n`` is not Rational. By default, the base is recursively decomposed and the exponents collected so the largest possible ``e`` is sought. If ``big=False`` then the smallest possible ``e`` (thus prime) will be chosen. If ``factor=True`` then simultaneous factorization of ``n`` is attempted since finding a factor indicates the only possible root for ``n``. This is True by default since only a few small factors will be tested in the course of searching for the perfect power. The use of ``candidates`` is primarily for internal use; if provided, False will be returned if ``n`` cannot be written as a power with one of the candidates as an exponent and factoring (beyond testing for a factor of 2) will not be attempted. Examples ======== >>> from sympy import perfect_power, Rational >>> perfect_power(16) (2, 4) >>> perfect_power(16, big=False) (4, 2) Negative numbers can only have odd perfect powers: >>> perfect_power(-4) False >>> perfect_power(-8) (-2, 3) Rationals are also recognized: >>> perfect_power(Rational(1, 2)**3) (1/2, 3) >>> perfect_power(Rational(-3, 2)**3) (-3/2, 3) Notes ===== To know whether an integer is a perfect power of 2 use >>> is2pow = lambda n: bool(n and not n & (n - 1)) >>> [(i, is2pow(i)) for i in range(5)] [(0, False), (1, True), (2, True), (3, False), (4, True)] It is not necessary to provide ``candidates``. When provided it will be assumed that they are ints. The first one that is larger than the computed maximum possible exponent will signal failure for the routine. >>> perfect_power(3**8, [9]) False >>> perfect_power(3**8, [2, 4, 8]) (3, 8) >>> perfect_power(3**8, [4, 8], big=False) (9, 4) See Also ======== sympy.core.intfunc.integer_nthroot sympy.ntheory.primetest.is_square rrrFFNro )rFror|c4g|]}|cxkrknn|Sr$r$)r%r5 max_possible min_possibles r(r6z!perfect_power.. sD1111q////<////////r*c&g|] }|zdk |Srr$)r%r5rss r(r6z!perfect_power..$s"<<<1Q3!88!888r*c3BKddzz} |Vt|}NrF)r)rXr-s r(_factorszperfect_power.._factors-s2 QY HHH2B r*T) generator) candidatesrg@rr)bigfactor)rIr is_Integeras_numer_denomr One perfect_powerfuncrrrvrr3rrNrreversedrziprdivisors primefactorsr)r-rrrrcrbppnumpqden_rr not_squarerokrfacexactr&rEEe0rsrrs` @@@r(rrsYJ!Xq|!!1 ::q!!B /ffQ1&&1.q!!B -Q"1qc**-FCS))1,B q A1uu A2    DAq1u r1u uca   Avvu 9Q<L  l;; 11111 11122 Q3!88! A<<< 4rc2t|dzzSr)r)rAar-s r(rCzpollard_rho..s3q!Q< 0 then if no factor is found after the first attempt, a new ``a`` will be generated randomly (using the ``seed``) and the process repeated. Note: the value of M is lcm(1..B) = reduce(ilcm, range(2, B + 1)). A search is made for factors next to even numbers having a power smoothness less than ``B``. Choosing a larger B increases the likelihood of finding a larger factor but takes longer. Whether a factor of n is found or not depends on ``a`` and the power smoothness of the even number just less than the factor p (hence the name p - 1). Although some discussion of what constitutes a good ``a`` some descriptions are hard to interpret. At the modular.math site referenced below it is stated that if gcd(a**M - 1, n) = N then a**M % q**r is 1 for every prime power divisor of N. But consider the following: >>> from sympy.ntheory.factor_ import smoothness_p, pollard_pm1 >>> n=257*1009 >>> smoothness_p(n) (-1, [(257, (1, 2, 256)), (1009, (1, 7, 16))]) So we should (and can) find a root with B=16: >>> pollard_pm1(n, B=16, a=3) 1009 If we attempt to increase B to 256 we find that it does not work: >>> pollard_pm1(n, B=256) >>> But if the value of ``a`` is changed we find that only multiples of 257 work, e.g.: >>> pollard_pm1(n, B=256, a=257) 1009 Checking different ``a`` values shows that all the ones that did not work had a gcd value not equal to ``n`` but equal to one of the factors: >>> from sympy import ilcm, igcd, factorint, Pow >>> M = 1 >>> for i in range(2, 256): ... M = ilcm(M, i) ... >>> set([igcd(pow(a, M, n) - 1, n) for a in range(2, 256) if ... igcd(pow(a, M, n) - 1, n) != n]) {1009} But does aM % d for every divisor of n give 1? >>> aM = pow(255, M, n) >>> [(d, aM%Pow(*d.args)) for d in factorint(n, visual=True).args] [(257**1, 1), (1009**1, 1)] No, only one of them. So perhaps the principle is that a root will be found for a given value of B provided that: 1) the power smoothness of the p - 1 value next to the root does not exceed B 2) a**M % p != 1 for any of the divisors of n. By trying more than one ``a`` it is possible that one of them will yield a factor. Examples ======== With the default smoothness bound, this number cannot be cracked: >>> from sympy.ntheory import pollard_pm1 >>> pollard_pm1(21477639576571) Increasing the smoothness bound helps: >>> pollard_pm1(21477639576571, B=2000) 4410317 Looking at the smoothness of the factors of this number we find: >>> from sympy.ntheory.factor_ import smoothness_p, factorint >>> print(smoothness_p(21477639576571, visual=1)) p**i=4410317**1 has p-1 B=1787, B-pow=1787 p**i=4869863**1 has p-1 B=2434931, B-pow=2434931 The B and B-pow are the same for the p - 1 factorizations of the divisors because those factorizations had a very large prime factor: >>> factorint(4410317 - 1) {2: 2, 617: 1, 1787: 1} >>> factorint(4869863-1) {2: 1, 2434931: 1} Note that until B reaches the B-pow value of 1787, the number is not cracked; >>> pollard_pm1(21477639576571, B=1786) >>> pollard_pm1(21477639576571, B=1787) 4410317 The B value has to do with the factors of the number next to the divisor, not the divisors themselves. A worst case scenario is that the number next to the factor p has a large prime divisisor or is a perfect power. If these conditions apply then the power-smoothness will be about p/2 or p. The more realistic is that there will be a large prime factor next to p requiring a B value on the order of p/2. Although primes may have been searched for up to this level, the p/2 is a factor of p - 1, something that we do not know. The modular.math reference below states that 15% of numbers in the range of 10**15 to 15**15 + 10**4 are 10**6 power smooth so a B of 10**6 will fail 85% of the time in that range. From 10**8 to 10**8 + 10**3 the percentages are nearly reversed...but in that range the simple trial division is quite fast. References ========== .. [1] Richard Crandall & Carl Pomerance (2005), "Prime Numbers: A Computational Perspective", Springer, 2nd edition, 236-238 .. [2] https://web.archive.org/web/20150716201437/http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node81.html .. [3] https://www.cs.toronto.edu/~yuvalf/Factorization.pdf roz*pollard_pm1 should receive n > 3 and B > 2rrFN) r3r_r rrrrvrrr) r-Brrrrr5aMrcrsrts r( pollard_pm1rsB AA1uuAEFFFtax  G 7Q;   !!QU++ ' 'ADHQNN##ARQA&&BB QNN q9999199999q66MMM GAq1u  r*Fc|r!t|}t|}|D])}||zdkrt||z|\}}|dz||<*|r^t t |t |D]!}tt|||fz"t|t||kfS)a Helper function for integer factorization. Trial factors ``n` against all integers given in the sequence ``candidates`` and updates the dict ``factors`` in-place. Returns the reduced value of ``n`` and a flag indicating whether any factors were found. rr) r<rlenrrNset differenceprint factor_msgr3) rxr-rverbosefactors0nfactorsrUr&rBs r(_trialrps( ''7||H  q5A::!q&!$$DAqQGAJ0G //H >>?? 0 0A *71:. / / / / q663w<<8+ ++r*c@|rtd|dkr|rttdS||dzkst|r*d|t|<|rttdSt ||}|sdS|\} } | |dzkst| r| || <n_t | ||||d} | D]5\} } |rtt| | fzt| | z|| <6|rttdS)z Helper function for integer factorization. Checks if ``n`` is a prime or a perfect power, and in those cases updates the factorization. zCheck for terminationrTrFFr)r complete_msgrr3rr+rOr)rxr-limit use_trialuse_rhouse_pm1rrrcbaseexpr'rrss r(_check_terminationrsW ' %&&&Avv  ,   t619}} }A  ,   t q&!!A uID# fai74== ui'!&(((JJLL $ $DAq +jAq6)***SUGAJJ l 4r*z4Trial division with ints [%i ... %i] and fail_max=%iz&Trial division with primes [%i ... %i]z7Pollard's rho with retries %i, max_steps %i and seed %iz2Pollard's p-1 with smoothness bound %i and seed %iz;Elliptic Curve with B1 bound %i, B2 bound %i, num_curves %iz %i ** %iz/Close factors satisying Fermat condition found.zFactorization is complete.cd}|dz}t||}|dkrB|dzs)t|}||d<||z}t||}d}|dkr |||S|dkrh|dzsO|dz}d}|dzs.|dz}|dz }|dkrt|d\}} || z }n|dz.||d<t||}d}|dkr |||S|d z} || dkrd nd| z z }d } | |kr||zr| dz } nQ||z}d}||zs.||z}|dz }|dkrt||\}} || z }n||z.|||<d } t||}|dz }||dzkr |||S||zr| dz } nQ||z}d}||zs.||z}|dz }|dkrt||\}} || z }n||z.|||<d } t||}|d z }||dzkr |||S| |k|||S) a Return the value of n and either a 0 (indicating that factorization up to the limit was complete) or else the next near-prime that would have been tested. Factoring stops if there are fail_max unsuccessful tests in a row. If factors of n were found they will be in the factors dictionary as {factor: multiplicity} and the returned value of n will have had those factors removed. The factors dictionary is modified in-place. c$||z|kr||fS|dfS)zpreturn n, d if the sqrt(n) was not reached yet, else n, 0 indicating that factoring is done. rr$)r-rUs r(rzz_factorint_small..dones# Q3!88a4K!t r*rFror rr/rr)rerr) rxr-rfail_maxrrzlimit2 threshold2r&mmp6failss r(rrsAXFQJ zz1u (! AGAJ !GAQJ >>46?? " zz1u ( !GAA!e aQ77"1aLLEArGA !e GAJQJ ??46?? " !B R!VVrrb((F E (   v: ( QJEE &LAA&j f Q77"1f--EArGA &j  GFOEQJ!   ! !46?? " v: ( QJEE &LAA&j f Q77"1f--EArGA &j  GFOEQJ!   ! !46?? "O (  P 46??r*c T0t|trt|}|rBt||||||dd0t 0fdt 0Dg} | Si} |r3t|t tfst||||||d} n]t|t r1d|D} nt|tr|} | rt|t tfrt| D]w} t| r| | } t| |||||d} | D]%\}}|| vr| |xx|| zz cc<|| z| |<&x|st|tur|dur|dur| ikr tjSd| vr#| dtjg}ng}|dt | Dt |d diSt|tt fr| S|s|s|s|sJd d lm}t||rt)|jd }|d kri}d }t-jd |dzD]*}|dkrd ||z}}|d kr||z }||z}|d k|||<+|r3|r1t |D]!}t1t2|||fz"|rt1t4|S||}t)|}|rt9|}d}|d krt| |||||d}d|d<|S|r|d kr |dkriS|diS|dkr%d diid diddid d iddidddddid didd ig |Si}|rnt;|}t=|dkr>> from sympy.ntheory import factorint >>> factorint(2000) # 2000 = (2**4) * (5**3) {2: 4, 5: 3} >>> factorint(65537) # This number is prime {65537: 1} For input less than 2, factorint behaves as follows: - ``factorint(1)`` returns the empty factorization, ``{}`` - ``factorint(0)`` returns ``{0:1}`` - ``factorint(-n)`` adds ``-1:1`` to the factors and then factors ``n`` Partial Factorization: If ``limit`` (> 3) is specified, the search is stopped after performing trial division up to (and including) the limit (or taking a corresponding number of rho/p-1 steps). This is useful if one has a large number and only is interested in finding small factors (if any). Note that setting a limit does not prevent larger factors from being found early; it simply means that the largest factor may be composite. Since checking for perfect power is relatively cheap, it is done regardless of the limit setting. This number, for example, has two small factors and a huge semi-prime factor that cannot be reduced easily: >>> from sympy.ntheory import isprime >>> a = 1407633717262338957430697921446883 >>> f = factorint(a, limit=10000) >>> f == {991: 1, int(202916782076162456022877024859): 1, 7: 1} True >>> isprime(max(f)) False This number has a small factor and a residual perfect power whose base is greater than the limit: >>> factorint(3*101**7, limit=5) {3: 1, 101: 7} List of Factors: If ``multiple`` is set to ``True`` then a list containing the prime factors including multiplicities is returned. >>> factorint(24, multiple=True) [2, 2, 2, 3] Visual Factorization: If ``visual`` is set to ``True``, then it will return a visual factorization of the integer. For example: >>> from sympy import pprint >>> pprint(factorint(4200, visual=True)) 3 1 2 1 2 *3 *5 *7 Note that this is achieved by using the evaluate=False flag in Mul and Pow. If you do other manipulations with an expression where evaluate=False, it may evaluate. Therefore, you should use the visual option only for visualization, and use the normal dictionary returned by visual=False if you want to perform operations on the factors. You can easily switch between the two forms by sending them back to factorint: >>> from sympy import Mul >>> regular = factorint(1764); regular {2: 2, 3: 2, 7: 2} >>> pprint(factorint(regular)) 2 2 2 2 *3 *7 >>> visual = factorint(1764, visual=True); pprint(visual) 2 2 2 2 *3 *7 >>> print(factorint(visual)) {2: 2, 3: 2, 7: 2} If you want to send a number to be factored in a partially factored form you can do so with a dictionary or unevaluated expression: >>> factorint(factorint({4: 2, 12: 3})) # twice to toggle to dict form {2: 10, 3: 3} >>> factorint(Mul(4, 12, evaluate=False)) {2: 4, 3: 1} The table of the output logic is: ====== ====== ======= ======= Visual ------ ---------------------- Input True False other ====== ====== ======= ======= dict mul dict mul n mul dict dict mul mul dict dict ====== ====== ======= ======= Notes ===== Algorithm: The function switches between multiple algorithms. Trial division quickly finds small factors (of the order 1-5 digits), and finds all large factors if given enough time. The Pollard rho and p-1 algorithms are used to find large factors ahead of time; they will often find factors of the order of 10 digits within a few seconds: >>> factors = factorint(12345678910111213141516) >>> for base, exp in sorted(factors.items()): ... print('%s %s' % (base, exp)) ... 2 2 2507191691 1 1231026625769 1 Any of these methods can optionally be disabled with the following boolean parameters: - ``use_trial``: Toggle use of trial division - ``use_rho``: Toggle use of Pollard's rho method - ``use_pm1``: Toggle use of Pollard's p-1 method ``factorint`` also periodically checks if the remaining part is a prime number or a perfect power, and in those cases stops. For unevaluated factorial, it uses Legendre's formula(theorem). If ``verbose`` is set to ``True``, detailed progress is printed. See Also ======== smoothness, smoothness_p, divisors Frrrrrr9multiplec3|K|]6}|dkr |g|zntj|z g| zV7dSrNr r)r%rcrs r(r)zfactorint..sd55#$+.a&1**1#A,,157)c!fW:M555555r*)rrrrrr9cNi|]"\}}t|t|#Sr$r2r%rBrWs r( zfactorint..s6(((Ac!ffc!ff(((r*Tr/c&g|]}t|ddiSevaluateFr r4s r(r6zfactorint..s5:::!,e,,:::r*rrr\rrFrrror)rFror|2z Factoring %sNz..(%i other digits).. FactoringiiXzExceeded limit:rrrrrr)rr)rrrruse_ecmrr)rrrr~i'd)r)1rIrdictr+rmrNras_powers_dictrOr<rrpoprPr r NegativeOneextendr`r]rrfrrrrrrr3rJrre trial_int_msgrrisqrtrr fermat_msgget trial_msgrpm1_msgrrho_msgrrecm_msgr )1r-rrrrrrr9r factorlist factordictrErsrUrBrWrfr]rArxr&rcrbsnsmallrrsqrt_nra2b2rrfermatrr'lowhigh_limit iterationhigh_ps found_trialcB1B2 num_curvesrrs1 @r(r+r+s j!T GG)#*G#*55JJJ5555(.s 55568:: J jS$K00q'.'.u>>>  As  ((     $ $ & &((( At    (jS$K00 ( ))** ( (Cs|| s##A#Ui")75JJJA  ( (1 ??qMMMQqS(MMMM$%aCJqMM  ( $q''T//$$%%   5L    NN2   M?DDD ::$Z%5%5%7%788::: ; ; ;D)5))) Ac{ # # 5575g555BBBBBB!Y 16!9   77GA%aQ//  q55a1fqAq&&Qaq&&  87 888A*71:67777 $l###Nq Aq A E  1uu Bey'WU<<<  9 66I1v RAQFQFQFQF! q!fq!fq!f6679 9G" VV r77R<< .2bqb6))SWWr\:;=?WE F F F F +q ! ! ! EH u~ & &E4 mq%22333 !UH==IAv070 0 0A *71:. / / / / {{ q55GCFFO  ,    %  , #U + + + gq%%w A A N q55GCFFO'1eY!7GV==1XXF A A q1u Q AB aB 1XXBKK 6  "j!!!!eQU^ 7 7 %9)0')0222!JJLL77DAq!(Q!2!2Q!6GAJJ7 $l###NNN q1ul Q&Cov "FIH!D&!!   0i3,.///!#u--B#GQG<>> from sympy import factorrat, S >>> factorrat(S(8)/9) # 8/9 = (2**3) * (3**-2) {2: 3, 3: -2} >>> factorrat(S(-1)/987) # -1/789 = -1 * (3**-1) * (7**-1) * (47**-1) {-1: 1, 3: -1, 7: -1, 47: -1} Please see the docstring for ``factorint`` for detailed explanations and examples of the following keywords: - ``limit``: Integer limit up to which trial division is done - ``use_trial``: Toggle use of trial division - ``use_rho``: Toggle use of Pollard's rho method - ``use_pm1``: Toggle use of Pollard's p-1 method - ``verbose``: Toggle detailed printing of progress - ``multiple``: Toggle returning a list of factors or dict - ``visual``: Toggle product form of output Frc3K|]9\}}|dkr |g|zntj|z g| zV:dSrr)r%rcrrs r(r)zfactorrat..spII#'1a+.a&1**1#A,,157)c!fW:MIIIIIIr*c@|ddkr|dn d|dz Sr@r$)elems r(rCzfactorrat..s,;?7Q;;JNa=>tAwYr*rDrrr/c&g|]}t|ddiSrr r4s r(r6zfactorrat..s5111!,e,,111r*r) factorratrmrNrOr+rccopyrr3rbrrrr rrr)ratrrrrrr9rrr=rcrsrfrs @r(r(r(s0 5I!7!%%AAAIIII+1#))++=G=G,H,H,HIIIKM NN #%u !7! # # ##'466 CA#%u$-")")") +++,1577 1 !  1vvzza1ff aD  *Aww 77 EE"IIIM?DDD 11$QWWYY//111 2 2 2D)5)))r*c t|}|dd||dttdd|i|}d|ddD}|r!t |dr ||dgz }|S)aMReturn a sorted list of n's prime factors, ignoring multiplicity and any composite factor that remains if the limit was set too low for complete factorization. Unlike factorint(), primefactors() does not return -1 or 0. Parameters ========== n : integer limit, verbose, **kwargs : Additional keyword arguments to be passed to ``factorint``. Since ``kwargs`` is new in version 1.13, ``limit`` and ``verbose`` are retained for compatibility purposes. Returns ======= list(int) : List of prime numbers dividing ``n`` Examples ======== >>> from sympy.ntheory import primefactors, factorint, isprime >>> primefactors(6) [2, 3] >>> primefactors(-5) [5] >>> sorted(factorint(123456).items()) [(2, 6), (3, 1), (643, 1)] >>> primefactors(123456) [2, 3, 643] >>> sorted(factorint(10000000001, limit=200).items()) [(101, 1), (99009901, 1)] >>> isprime(99009901) False >>> primefactors(10000000001, limit=300) [101] See Also ======== factorint, divisors NF)r9rrrr-cg|]}|dv| S))r/rrr$)r%r=s r(r6z primefactors..:s"999qQj%8%8%8%8%8r*r/r$)r3updaterNr+rr)r-rrkwargsrxrs r(rrs^ AA MMTu!g77888Y---f--224455G :9GDRDM999A772;'' gbk] Hr*c#Kdkr |srdVdSttdfd |rfdDEd{VdSEd{VdS)zHelper function for divisors which generates the divisors. Parameters ========== n : int a nonnegative integer proper: bool If `True`, returns the generator that outputs only the proper divisor (i.e., excluding n). rNrc3 K|tkrdVdSdgt|D]&}d|z'fd|dzDEd{VdS)Nrr/c3,K|]}D] }||zV dSr#r$)r%rbrcpowss r(r)z-_divisors..rec_gen..[s4DD!tDD!ADDDDDDDr*)rrrR)r-rr2rrrec_gens @r(r3z_divisors..rec_genTs B<<GGGGG3D:be,-- . . DHr!u,----DDDDwwq1u~~DDD D D D D D D D D Dr*c3(K|] }|k|V dSr#r$)r%rcr-s r(r)z_divisors..^s'33!AFFAFFFF33r*r)r+rNr)r-properrrr3s` @@@r( _divisorsr6@s Avv ! GGG1J  !! " "BEEEEEEEE3333wwyy333333333333799r*c|ttt||}|r|nt|S)a- Return all divisors of n sorted from 1..n by default. If generator is ``True`` an unordered generator is returned. The number of divisors of n can be quite large if there are many prime factors (counting repeated factors). If only the number of factors is desired use divisor_count(n). Examples ======== >>> from sympy import divisors, divisor_count >>> divisors(24) [1, 2, 3, 4, 6, 8, 12, 24] >>> divisor_count(24) 8 >>> list(divisors(120, generator=True)) [1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60, 120] Notes ===== This is a slightly modified version of Tim Peters referenced at: https://stackoverflow.com/questions/1010381/python-factorization See Also ======== primefactors, factorint, divisor_count )r6rrrN)r-rr5rXs r(rrcs5@ 6#a&&>>6 * *B *22r *r*c|sdS|dkrt||\}}|rdS|dkrdStdt|D}|r|r|dz}|S)a Return the number of divisors of ``n``. If ``modulus`` is not 1 then only those that are divisible by ``modulus`` are counted. If ``proper`` is True then the divisor of ``n`` will not be counted. Examples ======== >>> from sympy import divisor_count >>> divisor_count(6) 4 >>> divisor_count(6, 2) 2 >>> divisor_count(6, proper=True) 3 See Also ======== factorint, divisors, totient, proper_divisor_count rrc*g|]\}}|dk |dzSrr$rs r(r6z!divisor_count..s% > > >1Aa!er*)divmodrr+rO)r-modulusr5rs r( divisor_countr=s0 q Aa!!1  1Avvq > > ! 2 2 4 4 > > >?AV Q Hr*c&t||dS)a Return all divisors of n except n, sorted by default. If generator is ``True`` an unordered generator is returned. Examples ======== >>> from sympy import proper_divisors, proper_divisor_count >>> proper_divisors(24) [1, 2, 3, 4, 6, 8, 12] >>> proper_divisor_count(24) 7 >>> list(proper_divisors(120, generator=True)) [1, 2, 4, 8, 3, 6, 12, 24, 5, 10, 20, 40, 15, 30, 60] See Also ======== factorint, divisors, proper_divisor_count T)rr5)r)r-rs r(proper_divisorsr?s, A4 8 8 88r*c&t||dS)a$ Return the number of proper divisors of ``n``. Examples ======== >>> from sympy import proper_divisor_count >>> proper_divisor_count(6) 3 >>> proper_divisor_count(6, modulus=2) 1 See Also ======== divisors, proper_divisors, divisor_count T)r<r5)r=)r-r<s r(proper_divisor_countrAs& GD 9 9 99r*c#FK|dkr |dkrdVdSdt|D}tdt|zD]A}d}t|D]}|dzr |||z}|dz}|VBdS)zHelper function for udivisors which generates the unitary divisors. Parameters ========== n : int a nonnegative integer rNcg|] \}}||z Sr$r$)r%rcrss r(r6z_udivisors..s 88841a!Q$888r*rF)r+rOrrr)r- factorpowsr5rUrBs r( _udivisorsrEs Avv 66GGG889Q<<#5#5#7#7888J1c*oo% & & q||~~&&  A1u #Z]" !GAA r*czttt|}|r|nt|S)a) Return all unitary divisors of n sorted from 1..n by default. If generator is ``True`` an unordered generator is returned. The number of unitary divisors of n can be quite large if there are many prime factors. If only the number of unitary divisors is desired use udivisor_count(n). Examples ======== >>> from sympy.ntheory.factor_ import udivisors, udivisor_count >>> udivisors(15) [1, 3, 5, 15] >>> udivisor_count(15) 4 >>> sorted(udivisors(120, generator=True)) [1, 3, 5, 8, 15, 24, 40, 120] See Also ======== primefactors, factorint, divisors, divisor_count, udivisor_count References ========== .. [1] https://en.wikipedia.org/wiki/Unitary_divisor .. [2] https://mathworld.wolfram.com/UnitaryDivisor.html )rErrrNr-rrXs r( udivisorsrHs3B F3q66NN # #B *22r *r*cd|dkrdSdtdt|DzS)a Return the number of unitary divisors of ``n``. Parameters ========== n : integer Examples ======== >>> from sympy.ntheory.factor_ import udivisor_count >>> udivisor_count(120) 8 See Also ======== factorint, divisors, udivisors, divisor_count, totient References ========== .. [1] https://mathworld.wolfram.com/UnitaryDivisorFunction.html rrFcg|] }|dk| Sr:r$)r%rcs r(r6z"udivisor_count..:s444a!ee1eeer*)rr+r-s r(udivisor_countrLs:8 Avvq c44ill44455 55r*c#*K|dkrdSt|D]}d|z}||kr ||zr|Vtd|zdz D]}||cxkrdkr n||zr|Vtd|zdzD]}||cxkrdkr n||zr|VdS)zHelper function for antidivisors which generates the antidivisors. Parameters ========== n : int a nonnegative integer rFNr)r6)r-rUys r( _antidivisorsrO=s Avv q\\ aC q55QU5GGG qs1u   q::::A:::::!a%:GGG qs1u   q::::A:::::!a%:GGGr*czttt|}|r|nt|S)av Return all antidivisors of n sorted from 1..n by default. Antidivisors [1]_ of n are numbers that do not divide n by the largest possible margin. If generator is True an unordered generator is returned. Examples ======== >>> from sympy.ntheory.factor_ import antidivisors >>> antidivisors(24) [7, 16] >>> sorted(antidivisors(128, generator=True)) [3, 5, 15, 17, 51, 85] See Also ======== primefactors, factorint, divisors, divisor_count, antidivisor_count References ========== .. [1] definition is described in https://oeis.org/A066272/a066272a.html )rOrrrNrGs r( antidivisorsrQUs28 vc!ff~~ & &B *22r *r*ctt|}|dkrdStd|zdz td|zdzzt|zt|dz dz S)a Return the number of antidivisors [1]_ of ``n``. Parameters ========== n : integer Examples ======== >>> from sympy.ntheory.factor_ import antidivisor_count >>> antidivisor_count(13) 4 >>> antidivisor_count(27) 5 See Also ======== factorint, divisors, antidivisors, divisor_count, totient References ========== .. [1] formula from https://oeis.org/A066272 rFrrr)rrr=rKs r(antidivisor_countrSusz< s1vvAAvvq 1q ! !M!A#'$:$: :a (A.. /12 33r*zfThe `sympy.ntheory.factor_.totient` has been moved to `sympy.functions.combinatorial.numbers.totient`.z1.13z%deprecated-ntheory-symbolic-functions)deprecated_since_versionactive_deprecations_targetc$ddlm}||S)ax Calculate the Euler totient function phi(n) .. deprecated:: 1.13 The ``totient`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.totient` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. ``totient(n)`` or `\phi(n)` is the number of positive integers `\leq` n that are relatively prime to n. Parameters ========== n : integer Examples ======== >>> from sympy.functions.combinatorial.numbers import totient >>> totient(1) 1 >>> totient(25) 20 >>> totient(45) == totient(5)*totient(9) True See Also ======== divisor_count References ========== .. [1] https://en.wikipedia.org/wiki/Euler%27s_totient_function .. [2] https://mathworld.wolfram.com/TotientFunction.html r)totient)%sympy.functions.combinatorial.numbersrW)r-_totients r(rWrWs&ZJIIIII 8A;;r*zvThe `sympy.ntheory.factor_.reduced_totient` has been moved to `sympy.functions.combinatorial.numbers.reduced_totient`.c$ddlm}||S)av Calculate the Carmichael reduced totient function lambda(n) .. deprecated:: 1.13 The ``reduced_totient`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.reduced_totient` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. ``reduced_totient(n)`` or `\lambda(n)` is the smallest m > 0 such that `k^m \equiv 1 \mod n` for all k relatively prime to n. Examples ======== >>> from sympy.functions.combinatorial.numbers import reduced_totient >>> reduced_totient(1) 1 >>> reduced_totient(8) 2 >>> reduced_totient(30) 4 See Also ======== totient References ========== .. [1] https://en.wikipedia.org/wiki/Carmichael_function .. [2] https://mathworld.wolfram.com/CarmichaelFunction.html r)reduced_totient)rXr[)r-_reduced_totients r(r[r[s)PZYYYYY  A  r*zrThe `sympy.ntheory.factor_.divisor_sigma` has been moved to `sympy.functions.combinatorial.numbers.divisor_sigma`.c&ddlm}|||S)a? Calculate the divisor function `\sigma_k(n)` for positive integer n .. deprecated:: 1.13 The ``divisor_sigma`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.divisor_sigma` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. ``divisor_sigma(n, k)`` is equal to ``sum([x**k for x in divisors(n)])`` If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: \sigma_k(n) = \prod_{i=1}^\omega (1+p_i^k+p_i^{2k}+\cdots + p_i^{m_ik}). Parameters ========== n : integer k : integer, optional power of divisors in the sum for k = 0, 1: ``divisor_sigma(n, 0)`` is equal to ``divisor_count(n)`` ``divisor_sigma(n, 1)`` is equal to ``sum(divisors(n))`` Default for k is 1. Examples ======== >>> from sympy.functions.combinatorial.numbers import divisor_sigma >>> divisor_sigma(18, 0) 6 >>> divisor_sigma(39, 1) 56 >>> divisor_sigma(12, 2) 210 >>> divisor_sigma(37) 38 See Also ======== divisor_count, totient, divisors, factorint References ========== .. [1] https://en.wikipedia.org/wiki/Divisor_function r) divisor_sigma)rXr^)r-rBfunc_divisor_sigmas r(r^r^s+BZYYYYY  a # ##r*r-rBreturncdkr=tjdt|DStjfdt|DS)a  Calculate the divisor function `\sigma_k(n)` for positive integer n Parameters ========== n : int positive integer k : int nonnegative integer See Also ======== sympy.functions.combinatorial.numbers.divisor_sigma rc3 K|] }|dzV dSrlr$)r%rss r(r)z!_divisor_sigma..Ms&>>1Q>>>>>>r*c3LK|]\}}||dzzzdz |zdz zVdSrlr$)r%rcrsrBs r(r)z!_divisor_sigma..NsEWW$!Qa!QU)nq(AqD1H5WWWWWWr*)rvrwr+rrO)r-rBs `r(_divisor_sigmard;sv" Avvy>> ! (;(;(=(=>>>>>> 9WWWW)A,,BTBTBVBVWWW W WWr*ct|}t|}|dkrtd|dkrtdd}t|D]\}}||||zzz}|S)a Calculate core(n, t) = `core_t(n)` of a positive integer n ``core_2(n)`` is equal to the squarefree part of n If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: core_t(n) = \prod_{i=1}^\omega p_i^{m_i \mod t}. Parameters ========== n : integer t : integer core(n, t) calculates the t-th power free part of n ``core(n, 2)`` is the squarefree part of ``n`` ``core(n, 3)`` is the cubefree part of ``n`` Default for t is 2. Examples ======== >>> from sympy.ntheory.factor_ import core >>> core(24, 2) 6 >>> core(9424, 3) 1178 >>> core(379238) 379238 >>> core(15**11, 10) 15 See Also ======== factorint, sympy.solvers.diophantine.diophantine.square_factor References ========== .. [1] https://en.wikipedia.org/wiki/Square-free_integer#Squarefree_core rzn must be a positive integerrzt must be >= 2)rr_r+rO)r-rrNrcrss r(corerfQsl q Aq AAvv7888 a)*** aLL&&((  DAq QUOAAr*ztThe `sympy.ntheory.factor_.udivisor_sigma` has been moved to `sympy.functions.combinatorial.numbers.udivisor_sigma`.c&ddlm}|||S)aL Calculate the unitary divisor function `\sigma_k^*(n)` for positive integer n .. deprecated:: 1.13 The ``udivisor_sigma`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.udivisor_sigma` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. ``udivisor_sigma(n, k)`` is equal to ``sum([x**k for x in udivisors(n)])`` If n's prime factorization is: .. math :: n = \prod_{i=1}^\omega p_i^{m_i}, then .. math :: \sigma_k^*(n) = \prod_{i=1}^\omega (1+ p_i^{m_ik}). Parameters ========== k : power of divisors in the sum for k = 0, 1: ``udivisor_sigma(n, 0)`` is equal to ``udivisor_count(n)`` ``udivisor_sigma(n, 1)`` is equal to ``sum(udivisors(n))`` Default for k is 1. Examples ======== >>> from sympy.functions.combinatorial.numbers import udivisor_sigma >>> udivisor_sigma(18, 0) 4 >>> udivisor_sigma(74, 1) 114 >>> udivisor_sigma(36, 3) 47450 >>> udivisor_sigma(111) 152 See Also ======== divisor_count, totient, divisors, udivisors, udivisor_count, divisor_sigma, factorint References ========== .. [1] https://mathworld.wolfram.com/UnitaryDivisorFunction.html r)udivisor_sigma)rXrh)r-rB_udivisor_sigmas r(rhrhs*|XWWWWW ?1a  r*zfThe `sympy.ntheory.factor_.primenu` has been moved to `sympy.functions.combinatorial.numbers.primenu`.c$ddlm}||S)a Calculate the number of distinct prime factors for a positive integer n. .. deprecated:: 1.13 The ``primenu`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.primenu` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. If n's prime factorization is: .. math :: n = \prod_{i=1}^k p_i^{m_i}, then ``primenu(n)`` or `\nu(n)` is: .. math :: \nu(n) = k. Examples ======== >>> from sympy.functions.combinatorial.numbers import primenu >>> primenu(1) 0 >>> primenu(30) 3 See Also ======== factorint References ========== .. [1] https://mathworld.wolfram.com/PrimeFactor.html r)primenu)rXrk)r-_primenus r(rkrks&XJIIIII 8A;;r*zlThe `sympy.ntheory.factor_.primeomega` has been moved to `sympy.functions.combinatorial.numbers.primeomega`.c$ddlm}||S)aR Calculate the number of prime factors counting multiplicities for a positive integer n. .. deprecated:: 1.13 The ``primeomega`` function is deprecated. Use :class:`sympy.functions.combinatorial.numbers.primeomega` instead. See its documentation for more information. See :ref:`deprecated-ntheory-symbolic-functions` for details. If n's prime factorization is: .. math :: n = \prod_{i=1}^k p_i^{m_i}, then ``primeomega(n)`` or `\Omega(n)` is: .. math :: \Omega(n) = \sum_{i=1}^k m_i. Examples ======== >>> from sympy.functions.combinatorial.numbers import primeomega >>> primeomega(1) 0 >>> primeomega(20) 3 See Also ======== factorint References ========== .. [1] https://mathworld.wolfram.com/PrimeFactor.html r) primeomega)rXrn)r- _primeomegas r(rnrn s&ZPOOOOO ;q>>r*ct|}|dkrtd|dkrtdt|dz S)aReturns the exponent ``i`` for the nth Mersenne prime (which has the form `2^i - 1`). Examples ======== >>> from sympy.ntheory.factor_ import mersenne_prime_exponent >>> mersenne_prime_exponent(1) 2 >>> mersenne_prime_exponent(20) 4423 rz?nth must be a positive integer; mersenne_prime_exponent(1) == 23zGThere are only 51 perfect numbers; nth must be less than or equal to 51)rr_r)nthr-s r(mersenne_prime_exponentrs7 sN s A1uuZ[[[2vvbccc #AE **r*ctdkrdSdzdkrOdzdz }d|dz zd|zdz zkrdS|tvptd|zdz SddzkrdSdzdkrdSt fdd DrdSt dk}|r,t td tz|S) aCReturns True if ``n`` is a perfect number, else False. A perfect number is equal to the sum of its positive, proper divisors. Examples ======== >>> from sympy.functions.combinatorial.numbers import divisor_sigma >>> from sympy.ntheory.factor_ import is_perfect, divisors >>> is_perfect(20) False >>> is_perfect(6) True >>> 6 == divisor_sigma(6) - 6 == sum(divisors(6)[:-1]) True References ========== .. [1] https://mathworld.wolfram.com/PerfectNumber.html .. [2] https://en.wikipedia.org/wiki/Perfect_number rFrFrriic30K|]\}}|z|kVdSr#r$)r%r&rr-s r(r)zis_perfect..s s/ C C$!Q1q5A: C C C C C Cr*)) r)iu)iDQaIn 1888, Sylvester stated: " ...a prolonged meditation on the subject has satisfied me that the existence of any one such [odd perfect number] -- its escape, so to say, from the complex web of conditions which hem it in on all sides -- would be little short of a miracle." I guess SymPy just found that miracle and it factors like this: %s) rrrrra abundancer_rr+)r-r&results` r( is_perfectr|L s.0 q A1uuu1uzz \\^^a A % !a%La1f\ *a / /5,,K0A!Q$(0K0KK 2t8||u3w!||u C C C C"B C C CCCuq\\Q F 6%% (1|| %45566 6 Mr*c,t|d|zz S)aEReturns the difference between the sum of the positive proper divisors of a number and the number. Examples ======== >>> from sympy.ntheory import abundance, is_perfect, is_abundant >>> abundance(6) 0 >>> is_perfect(6) True >>> abundance(10) -2 >>> is_abundant(10) False rFrdrKs r(rzrz s" !  q1u $$r*ct|}t|rdS|dzdkptt|dkS)aReturns True if ``n`` is an abundant number, else False. A abundant number is smaller than the sum of its positive proper divisors. Examples ======== >>> from sympy.ntheory.factor_ import is_abundant >>> is_abundant(20) True >>> is_abundant(15) False References ========== .. [1] https://mathworld.wolfram.com/AbundantNumber.html Frrrr|rHrzrKs r( is_abundantr sH( q A!}}u q5A: /illQ.///r*ct|}t|rdStt|dkS)aReturns True if ``n`` is a deficient number, else False. A deficient number is greater than the sum of its positive proper divisors. Examples ======== >>> from sympy.ntheory.factor_ import is_deficient >>> is_deficient(20) False >>> is_deficient(15) True References ========== .. [1] https://mathworld.wolfram.com/DeficientNumber.html FrrrKs r( is_deficientr s;( q A!}}u  ! q ! !!r*ch||ko,||zt|cxkot|kncS)a(Returns True if the numbers `m` and `n` are "amicable", else False. Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to that of the other. Examples ======== >>> from sympy.functions.combinatorial.numbers import divisor_sigma >>> from sympy.ntheory.factor_ import is_amicable >>> is_amicable(220, 284) True >>> divisor_sigma(220) == divisor_sigma(284) True References ========== .. [1] https://en.wikipedia.org/wiki/Amicable_numbers r~)r&r-s r( is_amicabler sC, 6 Ea!e~a00EEEEN14E4EEEEEEr*cdkrdSdzoIt o9tfdtDS)z Returns True if the numbers `n` is Carmichael number, else False. Parameters ========== n : Integer References ========== .. [1] https://en.wikipedia.org/wiki/Carmichael_number .. [2] https://oeis.org/A002997 1FrFc3HK|]\}}|dkodz |dz zdkVdS)rrNr$)r%rcrsr-s r(r)z is_carmichael.. sBRRTQqAv01q5QU+q0RRRRRRr*)rrar+rOrKs`r( is_carmichaelr sh 3wwu q5 S^ S RRRRYq\\=O=O=Q=QRRR R RSr*cd|cxkr|krEnnB|dzdkrdt|dz|dDSdt||dDStd)zl Returns a list of the number of Carmichael in the range See Also ======== is_carmichael rrFc0g|]}t||Sr$rr4s r(r6z4find_carmichael_numbers_in_range.. s%FFF!]15E5EFAFFFr*rc0g|]}t||Sr$rr4s r(r6z4find_carmichael_numbers_in_range.. s%BBB!q1A1ABABBBr*zQThe provided range is not valid. x and y must be non-negative integers and x <= y)rr_)rArNs r( find_carmichael_numbers_in_ranger s{ A{{{{{{{{{ q5A::FFuQUAq11FFF FBBuQ1~~BBB Blmmmr*cd}g}t||kr>> from sympy.ntheory.factor_ import dra >>> dra(3110, 12) 8 References ========== .. [1] https://en.wikipedia.org/wiki/Digital_root r(Base should be an integer greater than 1r)rrr_)r-rrs r(drar( s^* fQii..Cq AAvvCDDD axxq qQU# #$r*ctt|}t|}|dkrtd||kr6d}|dkr&t||\}}|dkrdS||z}|dk&|}||k6|S)a Returns the multiplicative digital root of a natural number ``n`` in a given base ``b`` which is a single digit value obtained by an iterative process of multiplying digits, on each iteration using the result from the previous iteration to compute the digit multiplication. Examples ======== >>> from sympy.ntheory.factor_ import drm >>> drm(9876, 10) 0 >>> drm(49, 10) 8 References ========== .. 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