\h7SrSSKrSSKrSSKJr SSKJr SSKJr SSKJ r J r J r JrJrJr SSKJrJr SS jrS rS r\"S 5S 5r\RrSrSrSrSrSrSrg)ad The routines here were removed from numbers.py, power.py, digits.py and factor_.py so they could be imported into core without raising circular import errors. Although the name 'intfunc' was chosen to represent functions that work with integers, it can also be thought of as containing internal/core functions that are needed by the classes of the core. N) lru_cache)sympify)S)gcdlcmsqrtiroot bit_scan1gcdext)as_int filldedentclUS:a [S5eU(dg[[U5U5up#SU-$)aReturn the number of digits needed to express n in give base. Examples ======== >>> from sympy.core.intfunc import num_digits >>> num_digits(10) 2 >>> num_digits(10, 2) # 1010 -> 4 digits 4 >>> num_digits(-100, 16) # -64 -> 2 digits 2 Parameters ========== n: integer The number whose digits are counted. b: integer The base in which digits are computed. See Also ======== sympy.ntheory.digits.digits, sympy.ntheory.digits.count_digits rzbase must be int greater than 1r) ValueError integer_logabs)nbaseets J/var/www/auris/envauris/lib/python3.13/site-packages/sympy/core/intfunc.py num_digitsrs88 ax:;;  s1vt $DA q5Lc[U5n[U5nUS:a/[[U5U*5up#U=(a US-US::HnX#4$US::a [S5eUS:aUS:wag[U*U5up#US4$US:Xa [S5eX:aSUS:H4$US:Xa#UR 5S- nU[ U5U:H4$[ U5nSU-U:Xa,[ UR 5S- 5U-nSX2--nX$U:H4$[R"[R"U5[R"U5- 5nX-nX@::aUS- nXA-nX@::aMXTU:- X@:H=(d XA-U:H4$)aG Returns ``(e, bool)`` where e is the largest nonnegative integer such that :math:`|n| \geq |b^e|` and ``bool`` is True if $n = b^e$. Examples ======== >>> from sympy import integer_log >>> integer_log(125, 5) (3, True) >>> integer_log(17, 9) (1, False) If the base is positive and the number negative the return value will always be the same except for 2: >>> integer_log(-4, 2) (2, False) >>> integer_log(-16, 4) (0, False) When the base is negative, the returned value will only be True if the parity of the exponent is correct for the sign of the base: >>> integer_log(4, -2) (2, True) >>> integer_log(8, -2) (3, False) >>> integer_log(-8, -2) (3, True) >>> integer_log(-4, -2) (2, False) See Also ======== integer_nthroot sympy.ntheory.primetest.is_square sympy.ntheory.factor_.multiplicity sympy.ntheory.factor_.perfect_power rrzbase must be 2 or more)rFFz n cannot be 0) r rrr bit_lengthtrailingintmathfloorlog10)rbrrn_ds rrr:sT q Aq A1u3q6A2&  "!a%AE"t Av1221u 6A2q!%xAv))u!q&yAv LLNQ (1+""" A!tqy  " #Q & 13Z'z 4::a=4::a=01A B ' Q  ' Q<"'/RUaZ 00rc:U(dg[[U55$)aCount the number of trailing zero digits in the binary representation of n, i.e. determine the largest power of 2 that divides n. Examples ======== >>> from sympy import trailing >>> trailing(128) 7 >>> trailing(63) 0 See Also ======== sympy.ntheory.factor_.multiplicity r)r rrs rrrs&  SV ric[U5S:a[S[U5-5e[[[ [ U565$)aComputes nonnegative integer greatest common divisor. Explanation =========== The algorithm is based on the well known Euclid's algorithm [1]_. To improve speed, ``igcd()`` has its own caching mechanism. If you do not need the cache mechanism, using ``sympy.external.gmpy.gcd``. Examples ======== >>> from sympy import igcd >>> igcd(2, 4) 2 >>> igcd(5, 10, 15) 5 References ========== .. [1] https://en.wikipedia.org/wiki/Euclidean_algorithm rz,igcd() takes at least 2 arguments (%s given))len TypeErrorr number_gcdmapr argss rigcdr.s;4 4y1}FTRSS z3vt,- ..rc[[U55[[U55pX:aXpS[RR-nUR 5U:aUS:waUR 5U- n[ X- 5[ X- 5pTSupgpXX-S::aOfXF-XX--n XJU-- XzU -- pX-S:aOGX[pTXXjU-- U 4upgpXY-S::aO/XG-XY--n XJU-- XjU-- pX-S:aOX[pTXXX-- 4upgpMoUS:XaXU-pMX`-Xq--X-X--pUR 5U:aUS:waMU(aXU-pU(aMU$)acComputes greatest common divisor of two integers. Explanation =========== Euclid's algorithm for the computation of the greatest common divisor ``gcd(a, b)`` of two (positive) integers $a$ and $b$ is based on the division identity $$ a = q \times b + r$$, where the quotient $q$ and the remainder $r$ are integers and $0 \le r < b$. Then each common divisor of $a$ and $b$ divides $r$, and it follows that ``gcd(a, b) == gcd(b, r)``. The algorithm works by constructing the sequence r0, r1, r2, ..., where r0 = a, r1 = b, and each rn is the remainder from the division of the two preceding elements. In Python, ``q = a // b`` and ``r = a % b`` are obtained by the floor division and the remainder operations, respectively. These are the most expensive arithmetic operations, especially for large a and b. Lehmer's algorithm [1]_ is based on the observation that the quotients ``qn = r(n-1) // rn`` are in general small integers even when a and b are very large. Hence the quotients can be usually determined from a relatively small number of most significant bits. The efficiency of the algorithm is further enhanced by not computing each long remainder in Euclid's sequence. The remainders are linear combinations of a and b with integer coefficients derived from the quotients. The coefficients can be computed as far as the quotients can be determined from the chosen most significant parts of a and b. Only then a new pair of consecutive remainders is computed and the algorithm starts anew with this pair. References ========== .. [1] https://en.wikipedia.org/wiki/Lehmer%27s_GCD_algorithm rr)rrrr)rr sysint_infobits_per_digitrr)ar"nbitsrxyABCDqx_qyB_qDA_qCs r igcd_lehmerr?sX vay>3vay>qu1  ++ +E ,,.5 Q!V LLNU "16{CK1 a6uzAE"A UAAI${Qqqq5y$.JA!uzAE"AUAAI${QqtY.JA!qz 6!eq uqu}aeaem1e ,,.5 Q!Vj a%1 ! Hrc[U5S:a[S[U5-5e[[[ [ U565$)zComputes integer least common multiple. Examples ======== >>> from sympy import ilcm >>> ilcm(5, 10) 10 >>> ilcm(7, 3) 21 >>> ilcm(5, 10, 15) 30 rz,ilcm() takes at least 2 arguments (%s given))r(r)r number_lcmr+r r,s rilcmrBVs; 4y1}FTRSS z3vt,- ..rcL[[U5[U55up#nX4U4$)zReturns x, y, g such that g = x*a + y*b = gcd(a, b). Examples ======== >>> from sympy.core.intfunc import igcdex >>> igcdex(2, 3) (-1, 1, 1) >>> igcdex(10, 12) (-1, 1, 2) >>> x, y, g = igcdex(100, 2004) >>> x, y, g (-20, 1, 4) >>> x*100 + y*2004 4 )r r)r3r"gr5r6s rigcdexrEjs%&SVSV$GA! 7NrcSn[U5[U5pUS:waUS:wa[X5up4nUS:XaX1-nUc[SU<SU<S35eU$![a [U5[U5pUR(aUR(d[ [ S55eUS:nU[R[R4;a[SU-5eU(aSU- nNf=f) a Return the number $c$ such that, $a \times c = 1 \pmod{m}$ where $c$ has the same sign as $m$. If no such value exists, a ValueError is raised. Examples ======== >>> from sympy import mod_inverse, S Suppose we wish to find multiplicative inverse $x$ of 3 modulo 11. This is the same as finding $x$ such that $3x = 1 \pmod{11}$. One value of x that satisfies this congruence is 4. Because $3 \times 4 = 12$ and $12 = 1 \pmod{11}$. This is the value returned by ``mod_inverse``: >>> mod_inverse(3, 11) 4 >>> mod_inverse(-3, 11) 7 When there is a common factor between the numerators of `a` and `m` the inverse does not exist: >>> mod_inverse(2, 4) Traceback (most recent call last): ... ValueError: inverse of 2 mod 4 does not exist >>> mod_inverse(S(2)/7, S(5)/2) 7/2 References ========== .. [1] https://en.wikipedia.org/wiki/Modular_multiplicative_inverse .. [2] https://en.wikipedia.org/wiki/Extended_Euclidean_algorithm Nrz Expected numbers for arguments; symbolic `mod_inverse` is not implemented but symbolic expressions can be handled with the similar function, sympy.polys.polytools.invertz*m > 1 did not evaluate; try to simplify %sz inverse of z (mod z) does not exist) r rErr is_numberr)rrtruefalse)r3mcr5_rDbigs r mod_inverserOsN Aay&)1 6a2gQlGA!AvE& yAqIJJ H) qz71:1  0  !e qvvqww' 'IAMN N AA#s9ABC10C1c^US:a [S5e[[[U555$)aReturn the largest integer less than or equal to `\sqrt{n}`. Parameters ========== n : non-negative integer Returns ======= int : `\left\lfloor\sqrt{n}\right\rfloor` Raises ====== ValueError If n is negative. TypeError If n is of a type that cannot be compared to ``int``. Therefore, a TypeError is raised for ``str``, but not for ``float``. Examples ======== >>> from sympy.core.intfunc import isqrt >>> isqrt(0) 0 >>> isqrt(9) 3 >>> isqrt(10) 3 >>> isqrt("30") Traceback (most recent call last): ... TypeError: '<' not supported between instances of 'str' and 'int' >>> from sympy.core.numbers import Rational >>> isqrt(Rational(-1, 2)) Traceback (most recent call last): ... ValueError: n must be nonnegative rzn must be nonnegative)rrr r&s risqrtrQs+V 1u011 tCF| rc\[[U5[U55up#[U5U4$)a Return a tuple containing x = floor(y**(1/n)) and a boolean indicating whether the result is exact (that is, whether x**n == y). Examples ======== >>> from sympy import integer_nthroot >>> integer_nthroot(16, 2) (4, True) >>> integer_nthroot(26, 2) (5, False) To simply determine if a number is a perfect square, the is_square function should be used: >>> from sympy.ntheory.primetest import is_square >>> is_square(26) False See Also ======== sympy.ntheory.primetest.is_square integer_log )r r r)r6rr5r"s rinteger_nthrootrSs(6 F1I &DA q619r) )__doc__rr0 functoolsrr singletonrsympy.external.gmpyrr*rrAr r r r sympy.utilities.miscr rrrrr.igcd2r?rBrErOrQrSrrr\s ;;3!HM1`0 4//< O d/(.B J-`r