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Lk&HH5 LL&HeH5LH"1H;H5fLH1HH5LH0HH5;LH0HH53LH0HH5LHx0fInfHnH1flHH sH\)UfHnfHnH[flfHnH)=fInfHnH|flfHnH|)%"fHnH)fInfHnHi!fHnH)fHnH H5flfHnHLxfHnH H=flfHnHӏfHnH /flH<ŏfHnH QRflfHnH JYflfHnH IflfHnH lRflfHnH 0flfHnfl\/HHtH5L.L%L+HHe-LHHH@#LH5vH-#H)H5H#E1HHھH=i+IHZH菟H臟LL%Lg*HH,LHHH"LH5H"HY)H5 Hy"E1HHھH=jv*IHHHLH=,ԞE1HL[]A\A]A^A_H=`賞H=J襞}'u'H=!脞H= vH=hH=Z2'HJHBl'H/H'HH2aPa`apaa0accccxccp`H0DddDDDDDDDDDDD$4DUUUUUUUUUUUUUUUUUUUUUUUUȲUȲe\$ context manager Return a new context for controlling MPFR and MPC arithmetic. To load the new context, use set_context(). Options can only be specified as keyword arguments. Options precision: precision, in bits, of an MPFR result real_prec: precision, in bits, of Re(MPC) -1 implies use mpfr_prec imag_prec: precision, in bits, of Im(MPC) -1 implies use real_prec round: rounding mode for MPFR real_round: rounding mode for Re(MPC) -1 implies use mpfr_round imag_round: rounding mode for Im(MPC) -1 implies use real_round e_max: maximum allowed exponent e_min: minimum allowed exponent subnormalize: if True, subnormalized results can be returned trap_underflow: if True, raise exception for underflow if False, set underflow flag trap_overflow: if True, raise exception for overflow if False, set overflow flag and return Inf or -Inf trap_inexact: if True, raise exception for inexact result if False, set inexact flag trap_invalid: if True, raise exception for invalid operation if False, set invalid flag and return NaN trap_erange: if True, raise exception for range error if False, set erange flag trap_divzero: if True, raise exception for division by zero if False, set divzero flag and return Inf or -Inf allow_complex: if True, allow mpfr functions to return mpc if False, mpfr functions cannot return an mpc rational_division: if True, mpz/mpz returns an mpq if False, mpz/mpz follows default behavior allow_release_gil: if True, mpq operations may release the GIL if False, mpq operations may not release the GIL local_context([context[,keywords]]) -> context manager Create a context manager object that will restore the current context when the 'with ...' block terminates. The temporary context for the 'with ...' block is based on the current context if no context is specified. Keyword arguments are supported and will modify the temporary new context.context.copy() -> gmpy2 context Return a copy of a context.get_context() -> gmpy2 context Return a reference to the current context.ieee(size[,subnormalize=True]) -> context Return a new context corresponding to a standard IEEE floating point format. The supported sizes are 16, 32, 64, 128, and multiples of 32 greater than 128.set_context(context) Activate a context object controlling gmpy2 arithmetic. mpc() -> mpc(0.0+0.0j) If no argument is given, return mpc(0.0+0.0j). mpc(c [, precision=0]) -> mpc Return a new 'mpc' object from an existing complex number (either a Python complex object or another 'mpc' object). mpc(real [,imag=0 [, precision=0]]) -> mpc Return a new 'mpc' object by converting two non-complex numbers into the real and imaginary components of an 'mpc' object. mpc(s [, precision=0 [, base=10]]) -> mpc Return a new 'mpc' object by converting a string s into a complex number. If base is omitted, then a base-10 representation is assumed otherwise the base must be in the interval [2,36]. Note: The precision can be specified either a single number that is used for both the real and imaginary components, or as a tuple that can specify different precisions for the real and imaginary components. If a precision greater than or equal to 2 is specified, then it is used. A precision of 0 (the default) implies the precision of the current context is used. A precision of 1 minimizes the loss of precision by following these rules: 1) If n is a radix-2 floating point number, then the full precision of n is retained. 2) If n is an integer, then the precision is the bit length of the integer. mpfr() -> mpfr(0.0) If no argument is given, return mpfr(0.0). mpfr(n [, precision=0 [, context]]) -> mpfr Return an 'mpfr' object after converting a numeric value. See below for the interpretation of precision. mpfr(s [, precision=0 [, base=0 [, context]]]) -> mpfr Return a new 'mpfr' object by converting a string s made of digits in the given base, possibly with fraction-part (with a period as a separator) and/or exponent-part (with an exponent marker 'e' for base<=10, else '@'). The base of the string representation must be 0 or in the interval [2,62]. If the base is 0, the leading digits of the string are used to identify the base: 0b implies base=2, 0x implies base=16, otherwise base=10 is assumed. Note: If a precision greater than or equal to 2 is specified, then it is used. A precision of 0 (the default) implies the precision of either the specified context or the current context is used. A precision of 1 minimizes the loss of precision by following these rules: 1) If n is a radix-2 floating point number, then the full precision of n is retained. 2) If n is an integer, then the precision is the bit length of the integer. mpq() -> mpq(0,1) If no argument is given, return mpq(0,1). mpq(n) -> mpq Return an 'mpq' object with a numeric value n. Fraction values are converted exactly. mpq(n,m) -> mpq Return an 'mpq' object with a numeric value n/m. mpq(s[, base=10]) -> mpq Return an 'mpq' object from a string s made up of digits in the given base. s may be made up of two numbers in the same base separated by a '/' character. xmpz() -> xmpz(0) If no argument is given, return xmpz(0). xmpz(n) -> xmpz Return an 'xmpz' object with a numeric value 'n' (truncating n to its integer part if it's a Fraction, 'mpq', float or 'mpfr'). xmpz(s[, base=0]): Return an 'xmpz' object from a string 's' made of digits in the given base. If base=0, binary, octal, or hex Python strings are recognized by leading 0b, 0o, or 0x characters, otherwise the string is assumed to be decimal. Values for base can range between 2 and 62. Note: 'xmpz' is a mutable integer. It can be faster when used for augmented assignment (+=, *=, etc.). 'xmpz' objects cannot be used as dictionary keys. The use of 'mpz' objects is recommended in most cases.mpz() -> mpz(0) If no argument is given, return mpz(0). mpz(n) -> mpz Return an 'mpz' object with a numeric value 'n' (truncating n to its integer part if it's a Fraction, 'mpq', float or 'mpfr'). mpz(s[, base=0]): Return an 'mpz' object from a string 's' made of digits in the given base. If base=0, binary, octal, or hex Python strings are recognized by leading 0b, 0o, or 0x characters, otherwise the string is assumed to be decimal. Values for base can range between 2 and 62.xmpz.limbs_finish(n) Must be called after writing to the address returned by 'xmpz.limbs_write(n)' or 'xmpz.limbs_modify(n)' to update the limbs of 'xpmz'.xmpz.limbs_modify(n) -> int Returns the address of a mutable buffer representing the limbs of 'xmpz', resized so that it may hold at least 'n' limbs. Must be followed by a call to 'xmpz.limbs_finish(n)' after writing to the returned address in order for the changes to take effect.xmpz.limbs_write(n) -> int Returns the address of a mutable buffer representing the limbs of 'xmpz', resized so that it may hold at least 'n' limbs. Must be followed by a call to 'xmpz.limbs_finish(n)' after writing to the returned address in order for the changes to take effect. WARNING: this operation is destructive and may destroy the old value of 'xmpz'xmpz.limbs_read() -> int Returns the address of the immutable buffer representing the limbs of 'xmpz'.xmpz.num_limbs() -> int Return the number of limbs of 'xmpz'.x.__sizeof__() Returns the amount of memory consumed by x. Note: deleted xmpz objects are reused and may or may not be resized when a new value is assigned.xmpz.iter_clear(start=0, stop=-1) -> iterator Return every bit position that is clear in 'xmpz', beginning at 'start'. If a positive value is specified for 'stop', iteration is continued until 'stop' is reached. If a negative value is specified, iteration is continued until the last 1-bit. Note: the value of the underlying xmpz object can change during iteration.xmpz.iter_set(start=0, stop=-1) -> iterator Return an iterator yielding the bit position for every bit that is set in 'xmpz', beginning at 'start'. If a positive value is specified for 'stop', iteration is continued until 'stop' is reached. To match the behavior of slicing, 'stop' is not included. If a negative value is specified, iteration is continued until the last 1-bit. Note: the value of the underlying xmpz object can change during iteration.xmpz.iter_bits(start=0, stop=-1) -> iterator Return True or False for each bit position in 'xmpz' beginning at 'start'. If a positive value is specified for 'stop', iteration is continued until 'stop' is reached. If a negative value is specified, iteration is continued until the last 1-bit. Note: the value of the underlying xmpz object can change during iteration.xmpz.copy() -> xmpz Return a copy of an xmpz.xmpz.make_mpz() -> mpz Return an mpz by converting an 'xmpz' to an 'mpz' as quickly as possible. NOTE: Optimized for speed so the original xmpz is set to 0!xbit_mask(n) -> xmpz Return an 'xmpz' exactly n bits in length with all bits set. x.conjugate() -> number Return the conjugate of x (which is just a new reference to x since x is not a complex number).x.__sizeof__() Returns the amount of memory consumed by x. Note: deleted mpz objects are reused and may or may not be resized when a new value is assigned.x.is_odd() -> bool Return True if x is odd, False otherwise.is_odd(x) -> bool Return True if x is odd, False otherwise.x.is_even() -> bool Return True if x is even, False otherwise.is_even(x) -> bool Return True if x is even, False otherwise.kronecker(x, y) -> mpz Return the Kronecker-Jacobi symbol (x|y).legendre(x, y) -> mpz Return the Legendre symbol (x|y). y is assumed to be an odd prime.jacobi(x, y) -> mpz Return the Jacobi symbol (x|y). y must be odd and >0.next_prime(x) -> mpz Return the next _probable_ prime number > x.x.is_prime([n=25]) -> bool Return True if x is _probably_ prime, else False if x is definitely composite. x is checked for small divisors and up to n Miller-Rabin tests are performed.is_prime(x[, n=25]) -> bool Return True if x is _probably_ prime, else False if x is definitely composite. x is checked for small divisors and up to n Miller-Rabin tests are performed.x.is_power() -> bool Return True if x is a perfect power (there exists a y and an n > 1, such that x=y**n), else return False.is_power(x) -> bool Return True if x is a perfect power (there exists a y and an n > 1, such that x=y**n), else return False.x.is_congruent(y, m) -> bool Returns True if x is congruent to y modulo m, else return False.is_congruent(x, y, m) -> bool Returns True if x is congruent to y modulo m, else return False.x.is_divisible(d) -> bool Returns True if x is divisible by d, else return False.is_divisible(x, d) -> bool Returns True if x is divisible by d, else return False.x.is_square() -> bool Returns True if x is a perfect square, else return False.is_square(x) -> bool Returns True if x is a perfect square, else return False.divexact(x, y) -> mpz Return the quotient of x divided by y. Faster than standard division but requires the remainder is zero!invert(x, m) -> mpz Return y such that x*y == 1 modulo m. Raises ZeroDivisionError if no inverse exists.remove(x, f) -> tuple Return a 2-element tuple (y,m) such that x=y*(f**m) and f does not divide y. Remove the factor f from x as many times as possible. m is the multiplicity f in x. f > 1.isqrt_rem(x) -> tuple Return a 2-element tuple (s,t) such that s=isqrt(x) and t=x-s*s. x >=0.isqrt(x) -> mpz Return the integer square root of an integer x. x >= 0.comb(n, k) -> mpz Return the number of combinations of 'n things, taking k at a time'. k >= 0. Same as bincoef(n, k)bincoef(n, k) -> mpz Return the binomial coefficient ('n choose k'). k >= 0.lucas2(n) -> tuple Return a 2-tuple with the (n-1)-th and n-th Lucas numbers.lucas(n) -> mpz Return the n-th Lucas number.fib2(n) -> tuple Return a 2-tuple with the (n-1)-th and n-th Fibonacci numbers.fib(n) -> mpz Return the n-th Fibonacci number.multi_fac(n,m) -> mpz Return the exact m-multi factorial of n. The m-multifactorial is defined as n*(n-m)*(n-2m)...primorial(n) -> mpz Return the product of all positive prime numbers less than or equal to n.double_fac(n) -> mpz Return the exact double factorial (n!!) of n. The double factorial is defined as n*(n-2)*(n-4)...fac(n) -> mpz Return the exact factorial of n. See factorial(n) to get the floating-point approximation.divm(a, b, m) -> mpz Return x such that b*x == a mod m. Raises a ZeroDivisionError exception if no such value x exists.gcdext(a, b) - > tuple Return a 3-element tuple (g,s,t) such that g == gcd(a,b) and g == a*s + b*tlcm(*integers) -> mpz Return the lowest common multiple of integers.gcd(*integers) -> mpz Return the greatest common divisor of integers.Round an mpz to power of 10.Truncating an mpz returns itself.Floor of an mpz returns itself.Ceiling of an mpz returns itself.iroot_rem(x,n) -> (number, number) Return a 2-element tuple (y,r), such that y is the integer n-th root of x and x=y**n + r. x >= 0. n > 0.iroot(x,n) -> (number, boolean) Return the integer n-th root of x and boolean value that is True iff the root is exact. x >= 0. n > 0.num_digits(x[, base]) -> int Return length of string representing the absolute value of x in the given base. Values for base can range between 2 and 62. The value returned may be 1 too large.x.num_digits([base]) -> int Return length of string representing the absolute value of x in the given base. Values for base can range between 2 and 62. The value returned may be 1 too large.x.__sizeof__() Returns the amount of memory consumed by x. Note: deleted mpq objects are reused and may or may not be resized when a new value is assigned.Round an mpq to power of 10.Return integer portion of an mpq.Return least integer greater than or equal to an mpq.Return greatest integer less than or equal to an mpq.qdiv(x[, y=1]) -> number Return x/y as 'mpz' if possible, or as 'mpq' if x is not exactly divisible by y.denom(x) -> mpz Return the denominator of x.numer(x) -> mpz Return the numerator of x.__round__(x[, n = 0]) -> mpfr Return x rounded to n decimal digits before (n < 0) or after (n > 0) the decimal point. Rounds to an integer if n is not specified.x.__sizeof__() Returns the amount of memory consumed by x.context.check_range(x) -> mpfr Return a new 'mpfr' with exponent that lies within the range of emin and emax specified by context.check_range(x) -> mpfr Return a new 'mpfr' with exponent that lies within the current range of emin and emax.x.as_simple_fraction([precision=0]) -> mpq Return a simple rational approximation to x. The result will be accurate to 'precision' bits. If 'precision' is 0, the precision of 'x' will be used.x.as_mantissa_exp() -> (mantissa,exponent) Return the mantissa and exponent of an mpfr.x.as_integer_ratio() -> (num, den) Return the exact rational equivalent of an mpfr. Value is a tuple for compatibility with Python's float.as_integer_ratio().zero(n) -> mpfr Return an 'mpfr' initialized to 0.0 with the same sign as n. If n is not given, +0.0 is returned.inf(n) -> mpfr Return an 'mpfr' initialized to Infinity with the same sign as n. If n is not given, +Infinity is returned.nan() -> mpfr Return an 'mpfr' initialized to NaN (Not-A-Number).copy_sign(mpfr, mpfr) -> mpfr Return an 'mpfr' composed of the first argument with the sign of the second argument.set_sign(mpfr, bool) -> mpfr If 'bool' is True, then return an 'mpfr' with the sign bit set.set_exp(mpfr, n) -> mpfr Set the exponent of an mpfr to n. If n is outside the range of valid exponents, set_exp() will set the erange flag and either return the original value or raise an exception if trap_erange is set.get_exp(mpfr) -> integer Return the exponent of an mpfr. Returns 0 for NaN or Infinity and sets the erange flag and will raise an exception if trap_erange is set.get_max_precision() -> integer Return the maximum bits of precision that can be used for calculations. Note: to allow extra precision for intermediate calculations, avoid setting precision close the maximum precision.get_emax_max() -> integer Return the maximum possible exponent that can be set for 'mpfr'.get_emin_min() -> integer Return the minimum possible exponent that can be set for 'mpfr'.can_round(b, err, rnd1, rnd2, prec) Let b be an approximation to an unknown number x that is rounded according to rnd1. Assume the b has an error at most two to the power of E(b)-err where E(b) is the exponent of b. Then return true if x can be rounded correctly to prec bits with rounding mode rnd2.free_cache() Free the internal cache of constants maintained by MPFR.f2q(x,[err]) -> mpq Return the 'best' mpq approximating x to within relative error 'err'. Default is the precision of x. Uses Stern-Brocot tree to find the 'best' approximation. An 'mpz' is returned if the denominator is 1. If 'err'<0, relative error is 2.0 ** err.x.__sizeof__() Returns the amount of memory consumed by x.x.conjugate() -> mpc Returns the conjugate of x.proj(x) -> mpc Returns the projection of a complex x on to the Riemann sphere.context.proj(x) -> mpc Returns the projection of a complex x on to the Riemann sphere.rect(r, phi) -> mpc Return the rectangular coordinate form of a complex number that is given in polar form.context.rect(r, phi) -> mpc Return the rectangular coordinate form of a complex number that is given in polar form.polar(x) -> (abs(x), phase(x)) Return the polar coordinate form of a complex x that is in rectangular form.context.polar(x) -> (abs(x), phase(x)) Return the polar coordinate form of a complex x that is in rectangular form.norm(x) -> mpfr Return the norm of a complex x. The norm(x) is defined as x.real**2 + x.imag**2. abs(x) is the square root of norm(x). context.norm(x) -> mpfr Return the norm of a complex x. The norm(x) is defined as x.real**2 + x.imag**2. abs(x) is the square root of norm(x). root_of_unity(n, k) -> mpc Return the n-th root of mpc(1) raised to the k-th power..context.root_of_unity(n, k) -> mpc Return the n-th root of mpc(1) raised to the k-th power..phase(x) -> mpfr Return the phase angle, also known as argument, of a complex x.context.phase(x) -> mpfr Return the phase angle, also known as argument, of a complex x.cmp_abs(x, y) -> integer Return -1 if |x| < |y|; 0 if |x| = |y|; or 1 if |x| > |y|. Both x and y can be integer, rational, real, or complex.cmp(x, y) -> integer Return -1 if x < y; 0 if x = y; or 1 if x > y. Both x and y must be integer, rational or real. Note: 0 is returned (and exception flag set) if either argument is NaN.sign(x) -> number Return -1 if x < 0, 0 if x == 0, or +1 if x >0.is_unordered(x,y) -> boolean Return True if either x and/or y is NaN.is_lessgreater(x,y) -> boolean Return True if x > y or x < y. Return False if x == y or either x and/or y is NaN.x.is_integer() -> boolean Return True if x is an integer; False otherwise.context.is_integer(x) -> boolean Return True if x is an integer; False otherwise.is_integer(x) -> boolean Return True if x is an integer; False otherwise.x.is_regular() -> boolean Return True if x is not zero, NaN, or Infinity; False otherwise.context.is_regular(x) -> boolean Return True if x is not zero, NaN, or Infinity; False otherwise.is_regular(x) -> boolean Return True if x is not zero, NaN, or Infinity; False otherwise.x.is_signed() -> boolean Return True if the sign bit of x is set.context.is_signed(x) -> boolean Return True if the sign bit of x is set.is_signed(x) -> boolean Return True if the sign bit of x is set.x.is_zero() -> boolean Return True if x is equal to 0. If x is an mpc, return True if both x.real and x.imag are equal to 0.context.is_zero(x) -> boolean Return True if x is equal to 0. If x is an mpc, return True if both x.real and x.imag are equal to 0.is_zero(x) -> boolean Return True if x is equal to 0. If x is an mpc, return True if both x.real and x.imag are equal to 0.x.is_finite() -> boolean Return True if x is an actual number (i.e. non NaN or Infinity). If x is an mpc, return True if both x.real and x.imag are finite.context.is_finite(x) -> boolean Return True if x is an actual number (i.e. non NaN or Infinity). If x is an mpc, return True if both x.real and x.imag are finite.is_finite(x) -> boolean Return True if x is an actual number (i.e. non NaN or Infinity). If x is an mpc, return True if both x.real and x.imag are finite.x.is_infinite() -> boolean Return True if x is +Infinity or -Infinity. If x is an mpc, return True if either x.real or x.imag is infinite. Otherwise return False.context.is_infinite(x) -> boolean Return True if x is +Infinity or -Infinity. If x is an mpc, return True if either x.real or x.imag is infinite. Otherwise return False.is_infinite(x) -> boolean Return True if x is +Infinity or -Infinity. If x is an mpc, return True if either x.real or x.imag is infinite. Otherwise return False.x.is_nan() -> boolean Return True if x is NaN (Not-A-Number) else False.context.is_nan(x) -> boolean Return True if x is NaN (Not-A-Number) else False.is_nan(x) -> boolean Return True if x is NaN (Not-A-Number) else False.div_2exp(x, n) -> number Return 'mpfr' or 'mpc' divided by 2**n.context.div_2exp(x, n) -> number Return 'mpfr' or 'mpc' divided by 2**n.mul_2exp(x, n) -> number Return 'mpfr' or 'mpc' multiplied by 2**n.context.mul_2exp(x, n) -> number Return 'mpfr' or 'mpc' multiplied by 2**n.fmms(x, y, z, t) -> number Return correctly rounded result of (x * y) - (z + t).context.fmms(x, y, z, t) -> number Return correctly rounded result of (x * y) - (z * t).fmma(x, y, z, t) -> number Return correctly rounded result of (x * y) + (z + t).context.fmma(x, y, z, t) -> number Return correctly rounded result of (x * y) + (z * t).fms(x, y, z) -> number Return correctly rounded result of (x * y) - z.context.fms(x, y, z) -> number Return correctly rounded result of (x * y) - z.fma(x, y, z) -> number Return correctly rounded result of (x * y) + z.context.fma(x, y, z) -> number Return correctly rounded result of (x * y) + z.digits(x[, base[, prec]]) -> string Return string representing x. Calls mpz.digits, mpq.digits, mpfr.digits, or mpc.digits as appropriate.c.digits(base=10, prec=0) -> ((mant, exp, prec), (mant, exp, prec)) Returns up to 'prec' digits in the given base. If 'prec' is 0, as many digits that are available given c's precision are returned. 'base' must be between 2 and 62. The result consists of 2 three-element tuples that contain the mantissa, exponent, and number of bits of precision of the real and imaginary components.x.digits([base=10[, prec=0]]) -> (mantissa, exponent, bits) Returns up to 'prec' digits in the given base. If 'prec' is 0, as many digits that are available are returned. No more digits than available given x's precision are returned. 'base' must be between 2 and 62, inclusive. The result is a three element tuple containing the mantissa, the exponent, and the number of bits of precision.x.digits([base=10]) -> string Return a Python string representing x in the given base (2 to 62, default is 10). A leading '-' is present if x<0, but no leading '+' is present if x>=0. x.digits([base=10]) -> string Return Python string representing x in the given base. Values for base can range between 2 to 62. A leading '-' is present if x<0 but no leading '+' is present if x>=0.x.__format__(fmt) -> string Return a Python string by formatting 'x' using the format string 'fmt'. A valid format string consists of: optional alignment code: '<' -> left shifted in field '>' -> right shifted in field '^' -> centered in field optional leading sign code '+' -> always display leading sign '-' -> only display minus for negative values ' ' -> minus for negative values, space for positive values optional width.real_precision.imag_precision optional rounding mode: 'U' -> round toward plus infinity 'D' -> round toward minus infinity 'Z' -> round toward zero 'N' -> round to nearest optional output style: 'P' -> Python style, 1+2j, (default) 'M' -> MPC style, (1 2) optional conversion code: 'a','A' -> hex format 'b' -> binary format 'e','E' -> scientific format 'f','F' -> fixed point format 'g','G' -> fixed or scientific format The default format is 'f'.x.__format__(fmt) -> string Return a Python string by formatting 'x' using the format string 'fmt'. A valid format string consists of: optional alignment code: '<' -> left shifted in field '>' -> right shifted in field '^' -> centered in field optional leading sign code '+' -> always display leading sign '-' -> only display minus for negative values ' ' -> minus for negative values, space for positive values optional width.precision optional rounding mode: 'U' -> round toward plus Infinity 'D' -> round toward minus Infinity 'Y' -> round away from zero 'Z' -> round toward zero 'N' -> round to nearest optional conversion code: 'a','A' -> hex format 'b' -> binary format 'e','E' -> scientific format 'f','F' -> fixed point format 'g','G' -> fixed or float format The default format is '.6f'.x.__format__(fmt) -> string Return a Python string by formatting mpz 'x' using the format string 'fmt'. A valid format string consists of: optional alignment code: '<' -> left shifted in field '>' -> right shifted in field '^' -> centered in field optional leading sign code: '+' -> always display leading sign '-' -> only display minus sign ' ' -> minus for negative values, space for positive values optional base indicator '#' -> precede binary, octal, or hex with 0b, 0o or 0x optional width optional conversion code: 'd' -> decimal format 'b' -> binary format 'o' -> octal format 'x' -> hex format 'X' -> upper-case hex format The default format is 'd'.context.square(x) -> number Return x * x. If x is an integer, then the result is an 'mpz'. If x is a rational, then the result is an 'mpq'. If x is a float, then the result is an 'mpfr'. If x is a complex number, then the result is an 'mpc'.square(x) -> number Return x * x. If x is an integer, then the result is an 'mpz'. If x is a rational, then the result is an 'mpq'. If x is a float, then the result is an 'mpfr'. If x is a complex number, then the result is an 'mpc'.context.const_catalan() -> number Return the catalan constant using the context's precision.const_catalan([precision=0]) -> number Return the catalan constant using the specified precision. If no precision is specified, the default precision is used.context.const_log2() -> number Return the log2 constant using the context's precision.const_log2([precision=0]) -> number Return the log2 constant using the specified precision. If no precision is specified, the default precision is used.context.const_euler() -> number Return the euler constant using the context's precision.const_euler([precision=0]) -> number Return the euler constant using the specified precision. If no precision is specified, the default precision is used.context.const_pi() -> number Return the constant pi using the context's precision.const_pi([precision=0]) -> number Return the constant pi using the specified precision. If no precision is specified, the default precision is used.fsum(iterable) -> mpfr Return an accurate sum of the values in the iterable.fsum(iterable) -> mpfr Return an accurate sum of the values in the iterable.context.factorial(n) -> mpfr Return the floating-point approximation to the factorial of n. See fac(n) to get the exact integer result.factorial(n) -> mpfr Return the floating-point approximation to the factorial of n. See fac(n) to get the exact integer result.context.next_below(x) -> mpfr Return the next 'mpfr' from x toward -Infinity.next_below(x) -> mpfr Return the next 'mpfr' from x toward -Infinity.context.next_above(x) -> mpfr Return the next 'mpfr' from x toward +Infinity.next_above(x) -> mpfr Return the next 'mpfr' from x toward +Infinity.context.next_toward(x, y) -> mpfr Return the next 'mpfr' from x in the direction of y. The result has the same precision as x.next_toward(x, y) -> mpfr Return the next 'mpfr' from x in the direction of y. The result has the same precision as x.context.frexp(x) -> (int, mpfr) Return a tuple containing the exponent and mantissa of x.frexp(x) -> (int, mpfr) Return a tuple containing the exponent and mantissa of x.context.remquo(x, y) -> (mpfr, int) Return a tuple containing the remainder(x,y) and the low bits of the quotient.remquo(x, y) -> (mpfr, int) Return a tuple containing the remainder(x,y) and the low bits of the quotient.context.lgamma(x) -> (mpfr, int) Return a tuple containing the logarithm of the absolute value of gamma(x) and the sign of gamma(x)lgamma(x) -> (mpfr, int) Return a tuple containing the logarithm of the absolute value of gamma(x) and the sign of gamma(x)context.modf(x) -> (mpfr, mpfr) Return a tuple containing the integer and fractional portions of x.modf(x) -> (mpfr, mpfr) Return a tuple containing the integer and fractional portions of x.context.round_away(x) -> mpfr Return an 'mpfr' that is x rounded to the nearest integer, with ties rounded away from 0.round_away(x) -> mpfr Return an 'mpfr' that is x rounded to the nearest integer, with ties rounded away from 0.context.trunc(x) -> mpfr Return an 'mpfr' that is x truncated towards 0. Same as x.floor() if x>=0 or x.ceil() if x<0.trunc(x) -> mpfr Return an 'mpfr' that is x truncated towards 0. Same as x.floor() if x>=0 or x.ceil() if x<0.x.__trunc__() -> mpfr Return an 'mpfr' that is truncated towards 0. Same as x.floor() if x>=0 or x.ceil() if x<0.context.floor(x) -> mpfr Return an 'mpfr' that is the largest integer <= x.floor(x) -> mpfr Return an 'mpfr' that is the largest integer <= x.x.__floor__() -> mpfr Return an 'mpfr' that is the largest integer <= x.context.ceil(x) ->mpfr Return an 'mpfr' that is the smallest integer >= x.ceil(x) ->mpfr Return an 'mpfr' that is the smallest integer >= x.x.__ceil__() -> mpfr Return an 'mpfr' that is the smallest integer >= x.context.reldiff(x, y) -> mpfr Return the relative difference between x and y. Result is equal to abs(x-y)/x.reldiff(x, y) -> mpfr Return the relative difference between x and y. Result is equal to abs(x-y)/x.context.round2(x[, n]) -> mpfr Return x rounded to n bits. Uses default precision if n is not specified. See round_away() to access the mpfr_round() function.round2(x[, n]) -> mpfr Return x rounded to n bits. Uses default precision if n is not specified. See round_away() to access the mpfr_round() function.context.fmod(x, y) -> mpfr Return x - n*y where n is the integer quotient of x/y, rounded to 0.fmod(x, y) -> mpfr Return x - n*y where n is the integer quotient of x/y, rounded to 0.context.remainder(x, y) -> mpfr Return x - n*y where n is the integer quotient of x/y, rounded to the nearest integer and ties rounded to even.remainder(x, y) -> mpfr Return x - n*y where n is the integer quotient of x/y, rounded to the nearest integer and ties rounded to even.context.minnum(x, y) -> mpfr Return the minimum number of x and y. If x and y are not 'mpfr', they are converted to 'mpfr'. The result is rounded to match the specified context. If only one of x or y is a number, then that number is returned.minnum(x, y) -> mpfr Return the minimum number of x and y. If x and y are not 'mpfr', they are converted to 'mpfr'. The result is rounded to match the current context. If only one of x or y is a number, then that number is returned.context.maxnum(x, y) -> mpfr Return the maximum number of x and y. If x and y are not 'mpfr', they are converted to 'mpfr'. The result is rounded to match the specified context. If only one of x or y is a number, then that number is returned.maxnum(x, y) -> mpfr Return the maximum number of x and y. If x and y are not 'mpfr', they are converted to 'mpfr'. The result is rounded to match the current context. If only one of x or y is a number, then that number is returned.context.agm(x, y) -> mpfr Return arithmetic-geometric mean of x and y.agm(x, y) -> mpfr Return arithmetic-geometric mean of x and y.context.yn(x,n) -> mpfr Return the second kind Bessel function of order n of x.yn(x,n) -> mpfr Return the second kind Bessel function of order n of x.context.jn(x,n) -> mpfr Return the first kind Bessel function of order n of x.jn(x,n) -> mpfr Return the first kind Bessel function of order n of x.context.rootn(x, n) -> mpfr Return n-th root of x. The result always an 'mpfr'. Note: this is IEEE 754-2008 compliant version of root().rootn(x, n) -> mpfr Return n-th root of x. The result always an 'mpfr'. Note: this is IEEE 754-2008 compliant version of root().context.root(x, n) -> mpfr Return n-th root of x. The result always an 'mpfr'. Note: not IEEE 754-2008 compliant; result differs when x = -0 and n is even. See rootn().root(x, n) -> mpfr Return n-th root of x. The result always an 'mpfr'. Note: not IEEE 754-2008 compliant; result differs when x = -0 and n is even. See rootn().sqrt(x) -> number Return the square root of x.context.sqrt(x) -> number Return the square root of x.exp(x) -> number Return the exponential of x.context.exp(x) -> number Return the exponential of x.log(x) -> number Return the natural logarithm of x.context.log(x) -> number Return the natural logarithm of x.log10(x) -> number Return the base-10 logarithm of x.context.log10(x) -> number Return the base-10 logarithm of x.context.radians(x) -> mpfr Convert angle x from degrees to radians. Note: In rare cases the result may not be correctly rounded.radians(x) -> mpfr Convert angle x from degrees to radians. Note: In rare cases the result may not be correctly rounded.context.degrees(x) -> mpfr Convert angle x from radians to degrees. Note: In rare cases the result may not be correctly rounded.degrees(x) -> mpfr Convert angle x from radians to degrees. Note: In rare cases the result may not be correctly rounded.sinh_cosh(x) -> (number, number) Return a tuple containing the hyperbolic sine and cosine of x.context.sinh_cosh(x) -> (number, number) Return a tuple containing the hyperbolic sine and cosine of x.sin_cos(x) -> (number, number) Return a tuple containing the sine and cosine of x; x in radians.context.sin_cos(x) -> (number, number) Return a tuple containing the sine and cosine of x; x in radians.context.hypot(x, y) -> number Return square root of (x**2 + y**2).hypot(x, y) -> number Return square root of (x**2 + y**2).context.atan2(y, x) -> number Return arc-tangent of (y/x); result in radians.atan2(y, x) -> number Return arc-tangent of (y/x); result in radians.atanh(x) -> number Return inverse hyperbolic tangent of x.context.atanh(x) -> number Return inverse hyperbolic tanget of x.asin(x) -> number Return inverse sine of x; result in radians.context.asin(x) -> number Return inverse sine of x; result in radians.acos(x) -> number Return inverse cosine of x; result in radians.context.acos(x) -> number Return inverse cosine of x; result in radians.ai(x) -> number Return Airy function of x.context.ai(x) -> number Return Airy function of x.y1(x) -> number Return second kind Bessel function of order 1 of x.context.y1(x) -> number Return second kind Bessel function of order 1 of x.y0(x) -> number Return second kind Bessel function of order 0 of x.context.y0(x) -> number Return second kind Bessel function of order 0 of x.j1(x) -> number Return first kind Bessel function of order 1 of x.context.j1(x) -> number Return first kind Bessel function of order 1 of x.j0(x) -> number Return first kind Bessel function of order 0 of x.context.j0(x) -> number Return first kind Bessel function of order 0 of x.erfc(x) -> number Return complementary error function of x.context.erfc(x) -> number Return complementary error function of x.erf(x) -> number Return error function of x.context.erf(x) -> number Return error function of x.zeta(x) -> number Return Riemann zeta of x.context.zeta(x) -> number Return Riemann zeta of x.digamma(x) -> number Return digamma of x.context.digamma(x) -> number Return digamma of x.lngamma(x) -> number Return natural logarithm of gamma(x).context.lngamma(x) -> number Return natural logarithm of gamma(x).gamma(x) -> number Return gamma of x.context.gamma(x) -> number Return gamma of x.li2(x) -> number Return real part of dilogarithm of x.context.li2(x) -> number Return real part of dilogarithm of x.eint(x) -> number Return exponential integral of x.context.eint(x) -> number Return exponential integral of x.expm1(x) -> number Return exp(x) - 1.context.expm1(x) -> number Return exp(x) - 1.log1p(x) -> number Return natural logarithm of (1+x).context.log1p(x) -> number Return natural logarithm of (1+x).exp10(x) -> number Return 10**x.context.exp10(x) -> number Return 10**x.exp2(x) -> number Return 2**x.context.exp2(x) -> number Return 2**x.log2(x) -> number Return base-2 logarithm of x.context.log2(x) -> number Return base-2 logarithm of x.cbrt(x) -> number Return the cube root of x.context.cbrt(x) -> number Return the cube root of x.frac(x) -> number Return fractional part of x.context.frac(x) -> number Return fractional part of x.rint_trunc(x) -> number Return x rounded to the nearest integer by first rounding towards zero and then, if needed, using the current rounding mode.context.rint_trunc(x) -> number Return x rounded to the nearest integer by first rounding towards zero and then, if needed, using the context rounding mode.rint_round(x) -> number Return x rounded to the nearest integer by first rounding to the nearest integer (ties away from 0) and then, if needed, using the current rounding mode.context.rint_round(x) -> number Return x rounded to the nearest integer by first rounding to the nearest integer (ties away from 0) and then, if needed, using the context rounding mode.rint_floor(x) -> number Return x rounded to the nearest integer by first rounding to the next lower or equal integer and then, if needed, using the current rounding mode.context.rint_floor(x) -> number Return x rounded to the nearest integer by first rounding to the next lower or equal integer and then, if needed, using the context rounding mode.rint_ceil(x) -> number Return x rounded to the nearest integer by first rounding to the next higher or equal integer and then, if needed, using the current rounding mode.context.rint_ceil(x) -> number Return x rounded to the nearest integer by first rounding to the next higher or equal integer and then, if needed, using the context rounding mode.rint(x) -> number Return x rounded to the nearest integer using the current rounding mode.context.rint(x) -> number Return x rounded to the nearest integer using the context rounding mode.rec_sqrt(x) -> number Return the reciprocal of the square root of x.context.rec_sqrt(x) -> number Return the reciprocal of the square root of x.coth(x) -> number Return hyperbolic cotangent of x.context.coth(x) -> number Return hyperbolic cotangent of x.csch(x) -> number Return hyperbolic cosecant of x.context.csch(x) -> number Return hyperbolic cosecant of x.sech(x) -> number Return hyperbolic secant of x.context.sech(x) -> number Return hyperbolic secant of x.cot(x) -> number Return cotangent of x; x in radians.context.cot(x) -> number Return cotangent of x; x in radians.csc(x) -> number Return cosecant of x; x in radians.context.csc(x) -> number Return cosecant of x; x in radians.sec(x) -> number Return secant of x; x in radians.context.sec(x) -> number Return secant of x; x in radians.acosh(x) -> number Return inverse hyperbolic cosine of x.context.acosh(x) -> number Return inverse hyperbolic cosine of x.asinh(x) -> number Return inverse hyperbolic sine of x.context.asinh(x) -> number Return inverse hyperbolic sine of x.tanh(x) -> number Return hyperbolic tangent of x.context.tanh(x) -> number Return hyperbolic tangent of x.cosh(x) -> number Return hyperbolic cosine of x.context.cosh(x) -> number Return hyperbolic cosine of x.sinh(x) -> number Return hyperbolic sine of x.context.sinh(x) -> number Return hyperbolic sine of x.atan(x) -> number Return inverse tangent of x; result in radians.context.atan(x) -> number Return inverse tangent of x; result in radians.tan(x) -> number Return tangent of x; x in radians.context.tan(x) -> number Return tangent of x; x in radians.cos(x) -> number Return cosine of x; x in radians.context.cos(x) -> number Return cosine of x; x in radians.sin(x) -> number Return sine of x; x in radians.context.sin(x) -> number Return sine of x; x in radians.context.div(x, y) -> number Return x / y; uses true division.div(x, y) -> number Return x / y; uses true division.context.sub(x, y) -> number Return x - y.sub(x, y) -> number Return x - y.context.pow(x, y) -> number Return x ** y.powmod_sec(x, y, m) -> mpz Return (x**y) mod m. Calculates x ** y (mod m) but using a constant time algorithm to reduce the risk of side channel attacks. y must be an integer >0. m must be an odd integer.powmod(x, y, m) -> mpz Return (x**y) mod m. Same as the three argument version of Python's built-in pow(), but converts all three arguments to mpz.powmod_exp_list(base, exp_lst, mod) -> list Returns list(powmod(base, i, mod) for i in exp_lst). Will always release the GIL. (Experimental in gmpy2 2.1.x).powmod_base_list(base_lst, exp, mod) -> list Returns list(powmod(i, exp, mod) for i in base_lst). Will always release the GIL. (Experimental in gmpy2 2.1.x).context.plus(x) -> number Return +x, the context is applied to the result.context.mul(x, y) -> number Return x * y.mul(x, y) -> number Return x * y.context.mod(x, y) -> number Return mod(x, y). Note: overflow, underflow, and inexact exceptions are not supported for mpfr arguments to context.mod().mod(x, y) -> number Return mod(x, y). Note: overflow, underflow, and inexact exceptions are not supported for mpfr arguments to mod().context.minus(x) -> number Return -x. The context is applied to the result.context.floor_div(x, y) -> number Return x // y; uses floor division.floor_div(x, y) -> number Return x // y; uses floor division.context.div_mod(x, y) -> (quotient, remainder) Return div_mod(x, y); uses alternate spelling to avoid naming conflicts. Note: overflow, underflow, and inexact exceptions are not supported for mpfr arguments to context.div_mod().context.add(x, y) -> number Return x + y.add(x, y) -> number Return x + y.context.abs(x) -> number Return abs(x), the context is applied to the result.hamdist(x, y) -> int Return the Hamming distance (number of bit-positions where the bits differ) between integers x and y.bit_count(x) -> int Return the number of 1-bits set in abs(x).x.bit_count() -> int Return the number of 1-bits set in abs(x).popcount(x) -> int Return the number of 1-bits set in x. If x<0, the number of 1-bits is infinite so -1 is returned in that case.x.bit_flip(n) -> mpz Return a copy of x with the n-th bit inverted.bit_flip(x, n) -> mpz Return a copy of x with the n-th bit inverted.x.bit_set(n) -> mpz Return a copy of x with the n-th bit set.bit_set(x, n) -> mpz Return a copy of x with the n-th bit set.x.bit_clear(n) -> mpz Return a copy of x with the n-th bit cleared.bit_clear(x, n) -> mpz Return a copy of x with the n-th bit cleared.x.bit_test(n) -> bool Return the value of the n-th bit of x.bit_test(x, n) -> bool Return the value of the n-th bit of x.bit_scan1(x, n=0) -> int Return the index of the first 1-bit of x with index >= n. n >= 0. If there are no more 1-bits in x at or above index n (which can only happen for x>=0, assuming an infinitely long 2's complement format), then None is returned.x.bit_scan1(n=0) -> int Return the index of the first 1-bit of x with index >= n. n >= 0. If there are no more 1-bits in x at or above index n (which can only happen for x>=0, assuming an infinitely long 2's complement format), then None is returned.bit_scan0(x, n=0) -> int Return the index of the first 0-bit of x with index >= n. n >= 0. If there are no more 0-bits in x at or above index n (which can only happen for x<0, assuming an infinitely long 2's complement format), then None is returned.x.bit_scan0(n=0) -> int Return the index of the first 0-bit of x with index >= n. n >= 0. If there are no more 0-bits in x at or above index n (which can only happen for x<0, assuming an infinitely long 2's complement format), then None is returned.bit_mask(n) -> mpz Return an 'mpz' exactly n bits in length with all bits set. bit_length(x) -> int Return the number of significant bits in the radix-2 representation of x. Note: bit_length(0) returns 0.x.bit_length() -> int Return the number of significant bits in the radix-2 representation of x. Note: mpz(0).bit_length() returns 0.unpack(x, n) -> list Unpack an integer 'x' into a list of n-bit values. Equivalent to repeated division by 2**n. Raises error if 'x' is negative.pack(lst, n) -> mpz Pack a list of integers 'lst' into a single 'mpz' by concatenating each integer element of 'lst' after padding to length n bits. Raises an error if any integer is negative or greater than n bits in length.t_mod_2exp(x, n) -> remainder Return the remainder of x divided by 2**n. The remainder will have the same sign as x. x must be an integer. n must be >0.t_div_2exp(x, n) -> quotient Return the quotient of x divided by 2**n. The quotient is rounded towards zero (truncation). n must be >0.t_divmod_2exp(x, n) -> (quotient, remaidner) Return the quotient and remainder of x divided by 2**n. The quotient is rounded towards zero (truncation) and the remainder will have the same sign as x. x must be an integer. n must be >0.f_mod_2exp(x, n) -> remainder Return remainder of x divided by 2**n. The remainder will be positive. x must be an integer. n must be >0.f_div_2exp(x, n) -? quotient Return the quotient of x divided by 2**n. The quotient is rounded towards -Inf (floor rounding). x must be an integer. n must be >0.f_divmod_2exp(x, n) -> (quotient, remainder) Return quotient and remainder after dividing x by 2**n. The quotient is rounded towards -Inf (floor rounding) and the remainder will be positive. x must be an integer. n must be >0.c_mod_2exp(x, n) -> remainder Return the remainder of x divided by 2**n. The remainder will be negative. x must be an integer. n must be >0.c_div_2exp(x, n) -> quotient Returns the quotient of x divided by 2**n. The quotient is rounded towards +Inf (ceiling rounding). x must be an integer. n must be >0.c_divmod_2exp(x ,n) -> (quotient, remainder) Return the quotient and remainder of x divided by 2**n. The quotient is rounded towards +Inf (ceiling rounding) and the remainder will be negative. x must be an integer. n must be >0.t_mod(x, y) -> remainder Return the remainder of x divided by y. The remainder will have the same sign as x. x and y must be integers.t_div(x, y) -> quotient Return the quotient of x divided by y. The quotient is rounded towards 0. x and y must be integers.t_divmod(x, y) -> (quotient, remainder) Return the quotient and remainder of x divided by y. The quotient is rounded towards zero (truncation) and the remainder will have the same sign as x. x and y must be integers.f_mod(x, y) -> remainder Return the remainder of x divided by y. The remainder will have the same sign as y. x and y must be integers.f_div(x, y) -> quotient Return the quotient of x divided by y. The quotient is rounded towards -Inf (floor rounding). x and y must be integers.f_divmod(x, y) -> (quotient, remainder) Return the quotient and remainder of x divided by y. The quotient is rounded towards -Inf (floor rounding) and the remainder will have the same sign as y. x and y must be integers.c_mod(x, y) -> remainder Return the remainder of x divided by y. The remainder will have the opposite sign of y. x and y must be integers.c_div(x, y) -> quotient Return the quotient of x divided by y. The quotient is rounded towards +Inf (ceiling rounding). x and y must be integers.c_divmod(x, y) -> (quotient, remainder) Return the quotient and remainder of x divided by y. The quotient is rounded towards +Inf (ceiling rounding) and the remainder will have the opposite sign of y. x and y must be integers._mpmath_create(...): helper function for mpmath._mpmath_normalize(...): helper function for mpmath.is_strong_bpsw_prp(n) -> boolean Return True if n is a strong Baillie-Pomerance-Selfridge-Wagstaff probable prime. A strong BPSW probable prime passes the is_strong_prp() test with base and the is_strong_selfridge_prp() test. is_bpsw_prp(n) -> boolean Return True if n is a Baillie-Pomerance-Selfridge-Wagstaff probable prime. A BPSW probable prime passes the is_strong_prp() test with base 2 and the is_selfridge_prp() test. is_strong_selfridge_prp(n) -> boolean Return True if n is a strong Lucas probable prime with Selfidge parameters (p,q). The Selfridge parameters are chosen by finding the first element D in the sequence {5, -7, 9, -11, 13, ...} such that Jacobi(D,n) == -1. Then let p=1 and q = (1-D)/4. Then perform a strong Lucas probable prime test.is_selfridge_prp(n) -> boolean Return True if n is a Lucas probable prime with Selfidge parameters (p,q). The Selfridge parameters are chosen by finding the first element D in the sequence {5, -7, 9, -11, 13, ...} such that Jacobi(D,n) == -1. Then let p=1 and q = (1-D)/4. Then perform a Lucas probable prime test.is_extra_strong_lucas_prp(n,p) -> boolean Return True if n is an extra strong Lucas probable prime with parameters (p,1). Assuming: n is odd D = p*p - 4, D != 0 gcd(n, 2*D) == 1 n = s*(2**r) + Jacobi(D,n), s odd Then an extra strong Lucas probable prime requires: lucasu(p,1,s) == 0 (mod n) or lucasv(p,1,s) == +/-2 (mod n) or lucasv(p,1,s*(2**t)) == 0 (mod n) for some t, 0 <= t < ris_strong_lucas_prp(n,p,q) -> boolean Return True if n is a strong Lucas probable prime with parameters (p,q). Assuming: n is odd D = p*p - 4*q, D != 0 gcd(n, 2*q*D) == 1 n = s*(2**r) + Jacobi(D,n), s odd Then a strong Lucas probable prime requires: lucasu(p,q,s) == 0 (mod n) or lucasv(p,q,s*(2**t)) == 0 (mod n) for some t, 0 <= t < ris_lucas_prp(n,p,q) -> boolean Return True if n is a Lucas probable prime with parameters (p,q). Assuming: n is odd D = p*p - 4*q, D != 0 gcd(n, 2*q*D) == 1 Then a Lucas probable prime requires: lucasu(p,q,n - Jacobi(D,n)) == 0 (mod n)is_fibonacci_prp(n,p,q) -> boolean Return True if n is a Fibonacci probable prime with parameters (p,q). Assuming: n is odd p > 0, q = +/-1 p*p - 4*q != 0 Then a Fibonacci probable prime requires: lucasv(p,q,n) == p (mod n).is_strong_prp(n,a) -> boolean Return True if n is a strong (also known as Miller-Rabin) probable prime to the base a. Assuming: gcd(n,a) == 1 n is odd n = s*(2**r) + 1, with s odd Then a strong probable prime requires one of the following is true: a**s == 1 (mod n) or a**(s*(2**t)) == -1 (mod n) for some t, 0 <= t < r.is_euler_prp(n,a) -> boolean Return True if n is an Euler (also known as Solovay-Strassen) probable prime to the base a. Assuming: gcd(n,a) == 1 n is odd Then an Euler probable prime requires: a**((n-1)/2) == 1 (mod n)is_fermat_prp(n,a) -> boolean Return True if n is a Fermat probable prime to the base a. Assuming: gcd(n,a) == 1 Then a Fermat probable prime requires: a**(n-1) == 1 (mod n)lucasv_mod(p,q,k,n) -> mpz Return the k-th element of the Lucas V sequence defined by p,q (mod n). p*p - 4*q must not equal 0; k must be greater than or equal to 0; n must be greater than 0.lucasv(p,q,k) -> mpz Return the k-th element of the Lucas V sequence defined by p,q. p*p - 4*q must not equal 0; k must be greater than or equal to 0.lucasu_mod(p,q,k,n) -> mpz Return the k-th element of the Lucas U sequence defined by p,q (mod n). p*p - 4*q must not equal 0; k must be greater than or equal to 0; n must be greater than 0.lucasu(p,q,k) -> mpz Return the k-th element of the Lucas U sequence defined by p,q. p*p - 4*q must not equal 0; k must be greater than or equal to 0.mpc_random(random_state) -> mpc Return uniformly distributed number in the unit square [0,1]x[0,1].mpfr_grandom(random_state) -> (mpfr, mpfr) Return two random numbers with gaussian distribution.mpfr_nrandom(random_state) -> (mpfr) Return a random number with gaussian distribution.mpfr_random(random_state) -> mpfr Return uniformly distributed number between [0,1].mpz_random(random_state, int) -> mpz Return uniformly distributed random integer between 0 and n-1.mpz_rrandomb(random_state, bit_count) -> mpz Return a random integer between 0 and 2**bit_count-1 with long sequences of zeros and one in its binary representation.mpz_urandomb(random_state, bit_count) -> mpz Return uniformly distributed random integer between 0 and 2**bit_count-1.random_state([seed]) -> object Return new object containing state information for the random number generator. An optional integer can be specified as the seed value.Convert 'mpc' to 'complex'.to_binary(x) -> bytes Return a Python byte sequence that is a portable binary representation of a gmpy2 object x. The byte sequence can be passed to gmpy2.from_binary() to obtain an exact copy of x's value. Works with mpz, xmpz, mpq, mpfr, and mpc types. Raises TypeError if x is not a gmpy2 object.from_binary(bytes) -> gmpy2 object Return a Python object from a byte sequence created by gmpy2.to_binary().mpfr_from_old_binary(string) -> mpfr Return an 'mpfr' from a GMPY 1.x binary mpf format.mpq_from_old_binary(string) -> mpq Return an 'mpq' from a GMPY 1.x binary format.mpz_from_old_binary(string) -> mpz Return an 'mpz' from a GMPY 1.x binary format._printf(fmt, x) -> string Return a Python string by formatting 'x' using the format string 'fmt'. WARNING: Invalid format strings will cause a crash. Please see the GMP and MPFR manuals for details on the format code. 'mpc' objects are not supported.set_cache(cache_size, object_size) Set the current cache size (number of objects) and the maximum size per object (number of limbs). Raises ValueError if cache size exceeds 1000 or object size exceeds 16384.get_cache() -> (cache_size, object_size) Return the current cache size (number of objects) and maximum size per object (number of limbs) for all GMPY2 objects.mp_limbsize() -> integer Return the number of bits per limb.mpc_version() -> string Return string giving current MPC version.mpfr_version() -> string Return string giving current MPFR version.mp_version() -> string Return string giving the name and version of the multiple precision library used.version() -> string Return string giving current GMPY2 version.license() -> string Return string giving license information.RoundToNearestRoundToZeroRoundUpRoundDownRoundAwayZeroDefaultcan't covert 'mpc' to 'float'can't covert mpc to intcannot get thread stateinvalid value for round mode(ii)emin must be Python integeremax must be Python integerO!liilinvalid value for precision(nn)4.2.1MPFR%s %s1.3.1MPCGMP(sii)-inf-0Internal error in mpfr_asciisOinvalid value for imag_precinvalid value for real_precrequires 'mpc' object+- 0123456789(s)__format__divzero must be True or Falseerange must be True or Falseinvalid must be True or Falseinexact must be True or False{0:.%ldg}format{0:.%ld.%ldg}mpfr('{0:.%ldg}',%ld)mpfr('{0:.%ldg}')mpc('{0:.%ld.%ldg}')Fraction__mpc____mpfr____mpq____mpz__strictnumeratordenominatorcannot convert object to mpzinvalid digits|nnrequires mpz type|llliiilliiiiiiiiiiinvalid value for roundinvalid value for real_roundinvalid value for imag_roundinvalid value for emininvalid value for emax'mpq' does not support NaNzero denominator in mpq()c_div() division by 0c_divmod() division by 0c_mod() division by 0f_div() division by 0f_divmod() division by 0f_mod() division by 0t_div() division by 0t_divmod() division by 0t_mod() division by 0isqrt() of negative number|iiinvalid mpq binary (num len)seed must be an integer6 arguments requiredargument is not an mpzcannot convert object to mpqy must be odd, prime, and >0y must be odd and >0invert() division by 0invert() no inverse existsrequires mpfr typeOllOi(O)xmpz division by zeroxmpz modulo by zero'mpz' does not support NaNO|iinexact resultmask length must be >= 0(NN)bit value must be 0 or 1deleting bits not supporteddivision or modulo by zerodigits must be 0 or >= 2can not convert NaN to MPQn must be > 0exponent too largenew exponent is out-of-boundsiroot() of negative numberfactor must be > 1(Nk)qdiv() division by zero|l'mpc' division by zerodivexact() division by 0argument must be an iterableO|OiOinvalid string in mpc()O|OOOO|OO|liOO|lOpow() invalid operationpow() base not invertiblempq.pow() outrageous exponentpow() requires 2 arguments.minus() requires 1 argument.yn() requires 2 argumentsremquo() requires 2 argumentsmin() requires 2 argumentsmax() requires 2 argumentsjn() requires 2 argumentshypot() requires 2 argumentsfmod() requires 2 argumentsplus() requires 1 argument.fmms() requires 4 argumentsfmma() requires 4 argumentsfms() requires 3 argumentsfma() requires 3 argumentsatan2() requires 2 argumentsagm() requires 2 argumentssub() requires 2 argumentsmul() requires 2 argumentsmod() modulo by zeroadd() requires 2 argumentsdiv() requires 2 arguments.divmod() division by zerodivmod() invalid operationrect() requires 2 argumentsinvalid precisionrootn() requires 2 argumentsroot() requires 2 argumentssign() of invalid value (NaN)unpack() requires x >= 0cmp() requires 2 argumentsinvalid comparison with NaN'xmpz' does not support NaNgmpy2.gmpy2Errorgmpy2.RangeErrorgmpy2.InexactResultErrorgmpy2.OverflowResultErrorgmpy2.UnderflowResultErrorgmpy2.InvalidOperationErrorgmpy2.DivisionByZeroErrorxmpzlimb_sizempfrmpc__GMPY2_CTX__HAVE_THREADSgmpy2._C_APIcopyreggmpy2numbersstartstopsubnormalizetrap_underflowtrap_overflowtrap_inexacttrap_invalidtrap_erangetrap_divzeroallow_complexrational_divisionallow_release_gilbaserealimag_printfaddbit_clearbit_countbit_flipbit_lengthbit_scan0bit_scan1bit_setbit_testbincoefcmpcmp_abscombc_divc_div_2expc_divmodc_divmod_2expc_modc_mod_2expdenomdivexactdivmdouble_facfibfib2floor_divfrom_binaryf_divf_div_2expf_divmodf_divmod_2expf_modf_mod_2expgcdgcdextget_cachehamdistinvertirootiroot_remisqrtisqrt_remis_bpsw_prpis_congruentis_divisibleis_evenis_euler_prpis_extra_strong_lucas_prpis_fermat_prpis_fibonacci_prpis_lucas_prpis_oddis_poweris_primeis_selfridge_prpis_squareis_strong_prpis_strong_bpsw_prpis_strong_lucas_prpis_strong_selfridge_prpjacobikroneckerlcmlegendrelicenselucaslucasulucasu_modlucasvlucasv_modlucas2mp_versionmp_limbsizempc_versionmpfr_versionmpq_from_old_binarympz_from_old_binarympz_randommpz_rrandombmpz_urandombmulmulti_facnext_primenumernum_digitspopcountpowmodpowmod_base_listpowmod_exp_listpowmod_secprimorialqdivremoverandom_stateset_cachesubto_binaryt_divt_div_2expt_divmodt_divmod_2expt_modt_mod_2expunpackxbit_mask_mpmath_normalize_mpmath_createacosacoshaiagmasinasinhatanatanhatan2can_roundcbrtcheck_rangeconst_catalanconst_eulerconst_log2const_picopy_signcotcothcsccschdegreesdigammaeinterferfcexpm1exp10exp2f2qfactorialfmafmsfmmafmmsfmodfracfree_cachefrexpfsumget_contextget_emax_maxget_emin_minget_expget_max_precisionhypotieeeis_finiteis_infiniteis_integeris_lessgreateris_nanis_regularis_signedis_unorderedis_zerojnj0j1lgammali2lngammalocal_contextloglog1plog10maxnumminnummodfmpfr_from_old_binarympfr_randommpfr_grandommpfr_nrandommul_2expnext_abovenext_belownext_towardradiansrec_sqrtreldiffremainderremquorintrint_ceilrint_floorrint_roundrint_truncrootnround_awayround2sechset_contextset_expset_signsin_cossinh_coshyny0y1zetampc_randomnormpolarphaseprojroot_of_unityrect__enter____exit__clear_flagscopyminuspluspowprecision in bitsrcreturn codeimaginary componentreal component__complex____sizeof__conjugate__ceil____floor____round____trunc__as_integer_ratioas_mantissa_expas_simple_fractioniter_bitsiter_cleariter_setmake_mpznum_limbslimbs_readlimbs_writelimbs_modifylimbs_finishgmpy2 random stategmpy2 contextGMPY2 Context managerGMPY2 Context Objectgmpy2 iteratorGMPY2 Iterator Objectset_context() requires a context argumentround mode must be Python integerbase must be in the interval [2, 62]requested minimum exponent is invalidrequested maximum exponent is invalidinvalid value for rounding modeinternal error in Pympq_To_Binarycannot support current limb sizebase must be in the interval 2 ... 62_printf() could not format the 'mpz' or 'mpq' object_printf() could not format the 'mpfr' object_printf() does not support 'mpc'_printf() argument type not supportedimag_prec must be Python integerreal_prec must be Python integerprecision must be Python integerIllegal iter_type in gmpy2.Iterator.cache size must between 0 and 1000object size must between 0 and 16384Invalid conversion specificationInternal error in mpfr_asprintfInvalid conversion specification for imagallow_release_gil must be True or Falserational_division must be True or Falseallow_complex must be True or Falsetrap_divzero must be True or Falsetrap_erange must be True or Falsetrap_invalid must be True or Falsetrap_inexact must be True or Falsetrap_overflow must be True or Falsetrap_underflow must be True or Falseoverflow must be True or Falseunderflow must be True or Falsesubnormalize must be True or Falsenumber of limbs must be an int or longnumber of limbs must be an int or a long'mpq' too large to convert to float'mpz' too large to convert to floatmpz_from_old_binary() requires bytes argumentargument too large to convert to an indexbit positions must be integersmpc('{0:.%ld.%ldg}',(%ld,%ld))value could not be converted to C longcould not convert object to integerstring contains non-ASCII charactersobject is not string or UnicodeObject does not appear to be Fractioninvalid keyword arguments for contextcontext() only supports keyword arguments'mpq' does not support Infinityillegal string: both . and / foundillegal string: embedded . requires base=10c_div() requires 'mpz','mpz' argumentsc_divmod() requires 'mpz','mpz' argumentsc_mod() requires 'mpz','mpz' argumentsf_div() requires 'mpz','mpz' argumentsf_divmod() requires 'mpz','mpz' argumentsf_mod() requires 'mpz','mpz' argumentshamdist() requires 'mpz','mpz' argumentsisqrt_rem() requires 'mpz' argumentisqrt_rem() of negative numberis_congruent() requires 3 integer argumentst_div() requires 'mpz','mpz' argumentst_divmod() requires 'mpz','mpz' argumentst_mod() requires 'mpz','mpz' argumentsbit_count() requires 'mpz' argumentbit_length() requires 'mpz' argumentis_even() requires 'mpz' argumentis_odd() requires 'mpz' argumentis_power() requires 'mpz' argumentis_square() requires 'mpz' argumentnum_digits() requires 'mpz',['int'] argumentspopcount() requires 'mpz' argumentis_euler_prp() requires 2 integer argumentsis_euler_prp() requires 'a' greater than or equal to 2is_euler_prp() requires 'n' be greater than 0is_euler_prp() requires gcd(n,a) == 1is_fermat_prp() requires 2 integer argumentsis_fermat_prp() requires 'a' greater than or equal to 2is_fermat_prp() requires 'n' be greater than 0is_fermat_prp() requires gcd(n,a) == 1is_fibonacci_prp() requires 3 integer argumentsinvalid values for p,q in is_fibonacci_prp()is_fibonacci_prp() requires 'n' be greater than 0is_strong_prp() requires 2 integer argumentsis_strong_prp() requires 'a' greater than or equal to 2is_strong_prp() requires 'n' be greater than 0is_strong_prp() requires gcd(n,a) == 1lucasu() requires 3 integer argumentsinvalid values for p,q in lucasu()invalid value for k in lucasu()lucasu_mod() requires 4 integer argumentsinvalid values for p,q in lucasu_mod()invalid value for k in lucasu_mod()invalid value for n in lucasu_mod()lucasv() requires 3 integer argumentsinvalid values for p,q in lucasv()invalid value for k in lucasv()lucasv_mod() requires 4 integer argumentsinvalid values for p,q in lucasv_mod()invalid value for k in lucasv_mod()invalid value for n in lucasv_mod()next_prime() requires 'mpz' argumentkronecker() requires 'mpz','mpz' argumentsis_lucas_prp() requires 3 integer argumentsinvalid values for p,q in is_lucas_prp()is_lucas_prp() requires 'n' be greater than 0is_lucas_prp() requires gcd(n,2*q*D) == 1isqrt() requires 'mpz' argumentis_extra_strong_lucas_prp() requires 2 integer argumentsinvalid value for p in is_extra_strong_lucas_prp()is_extra_strong_lucas_prp() requires 'n' be greater than 0is_extra_strong_lucas_prp() requires gcd(n,2*D) == 1is_strong_lucas_prp() requires 3 integer argumentsinvalid values for p,q in is_strong_lucas_prp()is_strong_lucas_prp() requires 'n' be greater than 0is_strong_lucas_prp() requires gcd(n,2*q*D) == 1is_congruent() requires 2 integer argumentsis_prime() takes at most 1 argumentbit_scan1() requires 'mpz',['int'] argumentsbit_scan0() requires 'mpz',['int'] argumentspack() requires 'list','int' argumentsresult too large to store in an 'mpz'pack() requires list elements be positive integers < 2^n bitsargument can not be converted to 'mpz'ieee() requires 'int' argumentieee() requires positive value for sizeinvalid keyword arguments for ieee()bitwidth must be 16, 32, 64, 128; or must be greater than 128 and divisible by 32.mpq_from_old_binary() requires bytes argumentinvalid mpq binary (too short)random_state() requires 0 or 1 integer argumentsis_divisible() requires 2 integer argumentsarguments long, MPZ_Object*, PyObject*, long, long, char neededinvalid rounding mode specifiedcontext(precision=%s, real_prec=%s, imag_prec=%s, round=%s, real_round=%s, imag_round=%s, emax=%s, emin=%s, subnormalize=%s, trap_underflow=%s, underflow=%s, trap_overflow=%s, overflow=%s, trap_inexact=%s, inexact=%s, trap_invalid=%s, invalid=%s, trap_erange=%s, erange=%s, trap_divzero=%s, divzero=%s, allow_complex=%s, rational_division=%s, allow_release_gil=%s)internal error in GMPy_CTXT_Reprlegendre() requires 'mpz','mpz' argumentsjacobi() requires 'mpz','mpz' argumentsf_div_2exp() requires 'mpz','int' argumentst_mod_2exp() requires 'mpz','int' argumentsf_mod_2exp() requires 'mpz','int' argumentsc_mod_2exp() requires 'mpz','int' argumentst_div_2exp() requires 'mpz','int' argumentsc_div_2exp() requires 'mpz','int' argumentsmpz_urandomb() requires 2 argumentsmpz_urandomb() requires 'random_state' and 'bit_count' argumentsmpz_rrandomb() requires 2 argumentsmpz_rrandomb() requires 'random_state' and 'bit_count' argumentsmpz_random() requires 2 argumentsmpz_random() requires 'random_state' and 'int' argumentsc_divmod_2exp() requires 'mpz','int' argumentsinvert() requires 'mpz','mpz' argumentsf_divmod_2exp() requires 'mpz','int' argumentst_divmod_2exp() requires 'mpz','int' argumentsbit_test() requires 'mpz','int' argumentsmulti_fac() requires 2 integer argumentsis_selfridge_prp() requires 1 integer argumentis_selfridge_prp() requires 'n' be greater than 0appropriate value for D cannot be found in is_selfridge_prp()is_strong_bpsw_prp() requires 1 integer argumentis_strong_bpsw_prp() requires 'n' be greater than 0is_bpsw_prp() requires 1 integer argumentis_bpsw_prp() requires 'n' be greater than 0is_strong_selfridge_prp() requires 1 integer argumentis_strong_selfridge_prp() requires 'n' be greater than 0appropriate value for D cannot be found in is_strong_selfridge_prp()mul() argument type not supportedadd() argument type not supportedsub() argument type not supportedmpfc_random() requires 1 argumentmpc_random() requires 'random_state' argumentset_sign() requires 'mpfr', 'boolean' argumentsmpfr_nrandom() requires 1 argumentmpfr_nrandom() requires 'random_state' argumentmpfr_random() requires 1 argumentmpfr_random() requires 'random_state' argumentmpfr_from_old_binary() requires bytes argumentinvalid mpf binary encoding (too short)get_exp() requires 'mpfr' argumentCan not get exponent from NaN or Infinity.copy_sign() requires 'mpfr', 'mpfr' argumentsargument too large to be converted to an index'mpz' does not support Infinitympz.__new__() requires mpz typeobject of type '%.200s' can not be interpreted as mpzmpz() requires numeric or string argumentbase for mpz() must be 0 or in the interval [2, 62]mpz() with number argument only takes 1 argumentmpz() requires numeric or string (and optional base) argumentslocal_context() only supports [context[,keyword]] argumentsfrom_binary() requires bytes argumentbyte sequence too short for from_binary()byte sequence invalid for from_binary()from_binary() argument type not supported__round__() requires 0 or 1 argument__round__() requires 'int' argumentToo many arguments for __round__()xbit_mask() requires 'int' argument'mpz' to large to convert to 'mpfr' mpfr_grandom() requires 1 argumentmpfr_grandom() requires 'random_state' argumentmust specify bit sequence as an integermod() argument type not supportedlcm() requires 'mpz' argumentsgcd() requires 'mpz' argumentsCannot pass NaN to mpfr.as_mantissa_exp.Cannot pass Infinity to mpfr.as_mantissa_exp.Cannot pass NaN to mpfr.as_integer_ratio.Cannot pass Infinity to mpfr.as_integer_ratio.divmod() arguments not supportedbase must be in the interval [2,62]Internal error in Pympfr_To_PyStrcan not convert Infinity to MPQmpq.__new__() requires mpq typempq() takes at most 2 argumentsmpq() requires at least one non-keyword argumentbase for mpq() must be 0 or in the interval [2, 62]mpq() requires numeric or string argumentargument can not be converted to 'mpq'divm() requires 'mpz','mpz','mpz' argumentsiroot_rem() requires 'int','int' argumentsiroot_rem() of negative numberdiv() argument type not supportedbit_flip() requires 'mpz','int' argumentsbit_set() requires 'mpz','int' argumentsbit_clear() requires 'mpz','int' argumentsgcdext() requires 'mpz','mpz' argumentspowmod_sec() requires 3 arguments.powmod_sec() base must be an integer.powmod_sec() exponent must be an integer.powmod_sec() exponent must be > 0.powmod_sec() modulus must be an integer.powmod_sec() modulus must be odd.bincoef() requires two integer argumentsmpmath_create() expects 'mpz','int'[,'int','str'] argumentscould not convert prec to positive intset_exp() requires 'mpfr', 'integer' argumentsiroot() requires 'int','int' argumentsremove() requires 'mpz','mpz' argumentsfloor_div() argument type not supportedqdiv() requires 1 or 2 integer or rational argumentsCannot convert NaN to a number.Cannot convert Infinity to a number.Requested precision out-of-bounds.is_prime() requires 'mpz'[,'int'] argumentsobject could not be converted to 'mpc'can't convert argument to 'mpc'sqrt() argument type not supportedroot_of_unity() requires 2 argumentsroot_of_unity() requires positive integer arguments.root_of_unity() requires integer argumentsproj() argument type not supporteddivexact() requires 'mpz','mpz' argumentscheck_range() argument types not supportedphase() argument type not supportedpolar() argument type not supportednorm() argument type not supportedpowmod_base_list requires 3 argumentsthe first argument to powmod_base_list must be a sequencepowmod_base_list() 'mod' must be > 0all items in iterable must be integerspowmod_base_list() requires integer argumentspowmod_exp_list requires 3 argumentsthe second argument to powmod_exp_list must be a sequencepowmod_exp_list() 'mod' must be > 0powmod_exp_list() requires integer argumentsobject could not be converted to 'mpfr'mpc.__new__() requires mpc typempc() takes at most 4 argumentsmpc() requires at least one non-keyword argumentcontext argument is not a valid contextprecision for mpc() must be integer or tupleinvalid value for precision in mpc()precision for mpc() must be >= 0base for mpc() must be in the interval [2,36]object of type '%.200s' can not be interpreted as mpcinvalid type for imaginary component in mpc()mpc() requires string or numeric argument.mpc() requires numeric or string argumentsquare() argument type not supportedmpfr.__new__() requires mpfr typempfr() takes at most 4 argumentsmpfr() requires at least one non-keyword argumentprecision for mpfr() must be >= 0base for mpfr() must be 0 or in the interval [2, 62]object of type '%.200s' can not be interpreted as mpfrmpfr() requires numeric or string argumentargument can not be converted to 'mpfr'pow() 3rd argument not allowed unless all arguments are integerspow() argument types not supportedpow() modulus must be an integerpow() 3rd argument cannot be 0powmod() requires 3 arguments.powmod() argument types not supportedpow() 0 base to negative exponentpow() argument type not supportedzeta() argument type not supportedy1() argument type not supportedy0() argument type not supportedtanh() argument type not supportedtan() argument type not supportedsinh() argument type not supportedsin() argument type not supportedsech() argument type not supportedsec() argument type not supportedrint_trunc() argument type not supportedrint_round() argument type not supportedrint_floor() argument type not supportedrint_ceil() argument type not supportedrint() argument type not supportedrec_sqrt() argument type not supportedlog2() argument type not supportedlog10() argument type not supportedlog1p() argument type not supportedlog() argument type not supportedlngamma() argument type not supportedli2() argument type not supportedj1() argument type not supportedj0() argument type not supportedis_zero() argument type not supportedis_signed() argument type not supportedis_regular() argument type not supportedis_nan() argument type not supportedis_integer() argument type not supportedis_infinite() argument type not supportedis_finite() argument type not supportedgamma() argument type not supportedfrac() argument type not supportedexp2() argument type not supportedexp10() argument type not supportedexpm1() argument type not supportedexp() argument type not supportederfc() argument type not supportederf() argument type not supportedeint() argument type not supporteddigamma() argument type not supportedcsch() argument type not supportedcsc() argument type not supportedcoth() argument type not supportedcot() argument type not supportedcosh() argument type not supportedcos() argument type not supportedcbrt() argument type not supportedatanh() argument type not supportedatan() argument type not supportedasinh() argument type not supportedasin() argument type not supportedai() argument type not supportedacosh() argument type not supportedacos() argument type not supportedminus() argument type not supportedyn() argument type not supportedtrunc() argument type not supportedsinh_cosh() argument type not supportedsin_cos() argument type not supportedround() argument type not supportedremquo() argument type not supportedremainder() requires 2 argumentsremainder() argument type not supportedreldiff() requires 2 argumentsreldiff() argument type not supportedmodf() argument type not supportedmin() argument type not supportedmax() argument type not supportedlgamma() argument type not supportedjn() argument type not supportedis_unordered() requires 2 argumentsis_unordered() argument type not supportedis_lessgreater() requires 2 argumentsis_lessgreater() argument type not supportedhypot() argument type not supportedfrexp() argument type not supportedfmod() argument type not supportedplus() argument type not supportedfmms() argument type not supportedfmma() argument type not supportedfms() argument type not supportedfma() argument type not supportedfloor() argument type not supportedceil() argument type not supportedatan2() argument type not supportedagm() argument type not supportedcan't take mod of complex numberdigits() requires at least one argumentdigits() accepts at most three argumentsdigits() argument type not supportedno ordering relation is defined for complex numbersabs() argument type not supporteddiv_mod() requires 2 argumentscan't take floor or mod of complex number.divmod() argument type not supportedall items in iterable must be real numbersnext_above() argument type not supportednext_below() argument type not supportedrect() argument type not supportednext_toward() requires 2 argumentsnext_toward() argument type not supportedf2q() argument types not supportedf2q() requires 1 or 2 argumentsround2() argument type not supportedround2() requires 1 or 2 argumentsdiv_2exp() requires 2 argumentsdiv_2exp() argument type not supportedmul_2exp() requires 2 argumentsmul_2exp() argument type not supportedrootn() argument type not supportedroot() argument type not supportedsign() argument type not supportedfloor_div() requires 2 argumentscan't take floor of complex numberunpack() requires 'int','int' argumentsto_binary() argument type not supportedcmp_abs() requires integer, rational, real, or complex argumentsxmpz.__new__() requires xmpz type'xmpz' does not support Infinityxmpz() requires numeric or string argumentbase for xmpz() must be 0 or in the interval [2, 62]xmpz() with number argument only takes 1 argumentxmpz() requires numeric or string (and optional base) argumentscmp() requires integer, rational, or real argumentsdef gmpy2_reducer(x): return (gmpy2.from_binary, (gmpy2.to_binary(x),)) copyreg.pickle(type(gmpy2.mpz(0)), gmpy2_reducer) copyreg.pickle(type(gmpy2.xmpz(0)), gmpy2_reducer) copyreg.pickle(type(gmpy2.mpq(0)), gmpy2_reducer) copyreg.pickle(type(gmpy2.mpfr(0)), gmpy2_reducer) copyreg.pickle(type(gmpy2.mpc(0,0)), gmpy2_reducer) numbers.Integral.register(type(gmpy2.mpz())) numbers.Rational.register(type(gmpy2.mpq())) numbers.Real.register(type(gmpy2.mpfr())) numbers.Complex.register(type(gmpy2.mpc())) the numerator of a rational number in lowest termsthe denominator of a rational number in lowest termsthe real part of a complex numberthe imaginary part of a complex numberGMPY2 Random number generator state?0b0o0x0XU)yPD?@9B.??0C?d%U/%%U,%;thx_lsXh0D8XHlXh H h4H\pHxH  P p  8   < T hH    (8Hh0DXl(Hh(@X(lHXhxX@p!h""""" " "4#H#\(#p8#H#X#h#x#####$#8&h('H'h''(8(*H+d+8,,-H.4h.L.d//800X1@H2\h2p;X;;; X< <8 =P x=h = 8> > > (? x? ?!@(!h@@!@`!HA!xA!A!A!(B!(Dh"XD"D"F#F,#XGL#Gl#G#H#hH#XI$hI$I`$hJ|$K$K%HL4%hLP%Lx%L%M%M$&O&P&HQ 'xQ<'R'8R'HR'XR'hR'xR'R'R(R$(R8(RL(R`(S(HS(xS(S((T(T)T0)hUP)Ux)U)(V)hV)V)V*W(*xWT*W*X*Y*Z*xZ0+[l+\+^+_<,xax,(b,d-d0-Xed-xf-g-h.iP.j.m.hn/nL/r/s/s 0hxl0xy0z0|<1}x11؀102(t2238D33H3x4T44ȍ4x 5XD5X585x6(h6H67اX777H888xD9|99:h:::8@;;;x;h4<Xx<<=H\===x0>d>x>h>X ?HD?8|?(??$@\@@A8@A xA AA,BB8B(Cx\CCX#Dh%D%Dh&D)4Eh*pEx*E*E+E,F-8FH.Fx/F2G4XGX7G9G<0H8?xHAHCHhF0II|ILIxOJRdJUJ8ZJ\@K`KcKeLhgXL8jLlLm4MoMpMrMs N8tXNtNuNuNhvO(ylOyOyOhzOzPz$PH{LP{tP}PQ`Q8QXQ4R؇pRR8RxS8S`S؍ShSTȢTTبULUUUh V`VVVHWxW(WXx\XXX8YlYYhY(ZtZZZ [l[h[\XX\\ \X4]X]X]X`^^X^#H_)_H+`.h`(3`h5a8Ta8BAA j ABG m ABH <PIBBB A(A0 (A BBBF 8DK>BAA j ABG m ABH 8HL>BAA j ABG m ABH <LMBBB A(A0 (A BBBF 8HN>BAA j ABG m ABH @O AAD0V AAB ] AAA W AAG 0P[BDA D0  DABD \QnBBA A(D0 (A ABBE o (C ABBI N (A ABBD 8\R>BAA j ABG m ABH <SBBB A(A0 (A BBBF 8|U>BAA j ABG m ABH P V.BDA v DBM o DBK J DBP h CBC <h \WADD _ DAK d DAO [CA< WNDD0{ AAE NAAHx0< LX]DD0xAAAp0N AAH 4(!XAHD y DAE N DAE 4`!dYAKD C DAH P DAK @! ZAAD0Q AAG W AAG H AAF D!ZBDA C DBP I HBM x CBC H$"`[4BEB B(A0A8Gp  8D0A(B BBBB Hp"T^BEB B(A0A8Gp  8D0A(B BBBB L"`BEB E(A0A8G 8D0A(B BBBH L #g=BEB B(A0D8D 8D0A(B BBBA L\#xjOBHB B(A0A8Gu 8A0A(B BBBJ L#xpBHB B(A0A8G 8A0A(B BBBH L#HxBHB B(A0A8G 8A0A(B BBBC LL$}BEE B(A0A8G 8A0A(B BBBA P$XBKD X ABD h ABE f ABG Q ABD P$BKD X ABD h ABE f ABG Q ABD PD%BKD X ABD h ABE f ABG Q ABD 4%,AKD g DAD _ DAD D%BAA M DBI X CBC J DBP P& BAH G0  AABG f  AABF D  AABH Ll&BHB B(A0A8G8 8A0A(B BBBG 4&HAHG A DAJ f DAE L&5 BHB B(A0A8G 8D0A(B BBBF LD'5 BEB E(A0A8Gk 8A0A(B BBBD D'BAA ` ABI K ABB ^ CBE ,'A1 N I G I G I G 4 (AAD v AAB N AAH @D(AAD0W AAA D DAG ] AAA @(,AAD0W AAA D DAG ] AAA H(ئPBEE E(D0D8D` 8D0A(B BBBF H)ܧPBEE E(D0D8D` 8D0A(B BBBF Hd)PBEE E(D0D8D` 8D0A(B BBBF H)PBEE E(D0D8D` 8D0A(B BBBF L)BBB B(A0A8G 8D0A(B BBBF 4L*AKG B FAD m CAG 0*@ BAA D@  AABH 4*AKG U DAC _ CAE 4*ԴAKG U DAC _ CAE 4(+AKG U DAC _ CAE 4`+DAHG0^ AAH W CAE 4+AHG0^ AAH W CAE 4+AHG0^ AAH W CAE 4,lAHG0^ AAH W CAE 4@,$AHG0^ AAH W CAE 4x, AKG0Z AAA N AAH \,BBB B(A0A8DHCAY 8D0A(B BBBG D-D BHD Z ABE O ABF u ABH 8X- BHA  ABA a ABD 4-vAKG0Z DAF N DAE 8-8 BBD D(D0z (D ABBE 8. 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BBD D(D0z (D ABBE `.BBB H(D0 (D BBBD r (D BBBD B (D BBBD `.BBB H(D0 (D BBBD r (D BBBD B (D BBBD 0H/<BHA G0  DABD 0|/BHA G0  DABD @/nBEE I(A0G@f 0D(A BBBB `/@mBBB B(A0A8D`p 8A0A(B BBBH { 8C0A(B BBBI X0LBBA H(G@ (A ABBI D (D ABBK  (C ABBE G (D ABBH D (A ABBN 40}AAG t DAF M DAF 41 }AAG t DAF M DAF 8L1hBAI  ABD E ABH 81VBID H(Dp (D ABBA 11 1AG k AD D2TAD H AB p AG D AK  DE D AK 0X2lAAD z DAK VDAD2(BAA b DBL X CBK z DBP D2(BAA b DBL X CBK z DBP D3BAA  ABG ] ABH } ABH Dd3BAA  ABG ] ABH } ABH D3XBAA  ABG ] ABH } ABH D3BAA  ABG ] ABH } ABH D<4BAA  ABG ] ABH } ABH D4`BAA  ABG ] ABH } ABH 44ZAAD0 AAD m AAI 45ZAAD0 AAD m AAI D<5bBAA  ABH [ AEG u ABH H50FBBB B(A0A8D@ 8D0A(B BBBK L54FBBA A(D0 (D ABBD L (D ABBK H 64nBBB B(A0A8D@ 8D0A(B BBBK Hl6XnBBB B(A0A8D@ 8D0A(B BBBK D6|BAA  ABD K ABB Z ABC L7$ QBBB B(D0A8G 8A0A(B BBBG @P74AAD   AAF [ CAI N AAH @7BAA D0  AABG ]  AABG @7LBEB A(A0GP 0D(A BBBD D8BDA  ABI ] ABH M ABH Dd8pBDA  ABI ] ABH M ABH @8BEB A(A0GP 0D(A BBBD H8BEE E(D0C8DPG 8D0A(B BBBH H<9(BEE 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A(A0% (A BBBD HBBB B(A0A8NP 8D0A(B BBBA 00C `@ , 0 o0Bx ~ a oo`oo\ogmpy2 2.1.5 - General Multiple-precision arithmetic for Python gmpy2 supports several multiple-precision libraries. Integer and rational arithmetic is provided by the GMP library. Real floating- point arithmetic is provided by the MPFR library. Complex floating- point arithmetic is provided by the MPC library. The integer type 'mpz' has comparable functionality to Python's builtin integers, but is faster for operations on large numbers. A wide variety of additional functions are provided: - bit manipulations - GCD, Extended GCD, LCM - Fibonacci and Lucas sequences - primality testing - powers and integer Nth roots The rational type 'mpq' is equivalent to Python's fractions module, but is faster. The real type 'mpfr' and complex type 'mpc' provide multiple- precision real and complex numbers with user-definable precision, rounding, and exponent range. All the advanced functions from the MPFR and MPC libraries are available. The GMPY2 source code is licensed under LGPL 3 or later. 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