| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
| 31.1 Functions and Variables for Number Theory |
| [ < ] | [ > ] | [ << ] | [ Up ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
Returns the n'th Bernoulli number for integer n.
Bernoulli numbers equal to zero are suppressed if zerobern is false.
See also burn.
(%i1) zerobern: true$
(%i2) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
1 1 1 1 1
(%o2) [1, - -, -, 0, - --, 0, --, 0, - --]
2 6 30 42 30
(%i3) zerobern: false$
(%i4) map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
1 1 1 5 691 7 3617 43867
(%o4) [1, - -, -, - --, --, - ----, -, - ----, -----]
2 6 30 66 2730 6 510 798
@ref{Category: Number theory}
Returns the n'th Bernoulli polynomial in the variable x.
@ref{Category: Number theory}
Returns the Riemann zeta function for the argument s. The return value is a big float (bfloat); n is the number of digits in the return value.
@ref{Category: Number theory} · @ref{Category: Numerical evaluation}
Returns the Hurwitz zeta function for the arguments s and h. The return value is a big float (bfloat); n is the number of digits in the return value.
The Hurwitz zeta function is defined as
sum ((k+h)^-s, k, 0, inf)
load ("bffac") loads this function.
@ref{Category: Number theory} · @ref{Category: Numerical evaluation}
The binomial coefficient x!/(y! (x - y)!).
If x and y are integers, then the numerical value of the binomial
coefficient is computed.
If y, or x - y, is an integer,
the binomial coefficient is expressed as a polynomial.
Examples:
(%i1) binomial (11, 7);
(%o1) 330
(%i2) 11! / 7! / (11 - 7)!;
(%o2) 330
(%i3) binomial (x, 7);
(x - 6) (x - 5) (x - 4) (x - 3) (x - 2) (x - 1) x
(%o3) -------------------------------------------------
5040
(%i4) binomial (x + 7, x);
(x + 1) (x + 2) (x + 3) (x + 4) (x + 5) (x + 6) (x + 7)
(%o4) -------------------------------------------------------
5040
(%i5) binomial (11, y);
(%o5) binomial(11, y)
@ref{Category: Number theory}
Returns the n'th Bernoulli number for integer n.
burn may be more efficient than bern for large, isolated n
(perhaps n greater than 105 or so), as bern computes all the Bernoulli numbers up to index n before returning.
burn exploits the observation that (rational) Bernoulli numbers can be
approximated by (transcendental) zetas with tolerable efficiency.
load ("bffac") loads this function.
@ref{Category: Number theory}
Converts expr into a continued fraction.
expr is an expression
comprising continued fractions and square roots of integers.
Operands in the expression may be combined with arithmetic operators.
Aside from continued fractions and square roots,
factors in the expression must be integer or rational numbers.
Maxima does not know about operations on continued fractions outside of cf.
cf evaluates its arguments after binding listarith to false.
cf returns a continued fraction, represented as a list.
A continued fraction a + 1/(b + 1/(c + ...))
is represented by the list [a, b, c, ...].
The list elements a, b, c, ... must evaluate to integers.
expr may also contain sqrt (n) where n is an integer.
In this case cf will give as many
terms of the continued fraction as the value of the variable
cflength times the period.
A continued fraction can be evaluated to a number
by evaluating the arithmetic representation
returned by cfdisrep.
See also cfexpand for another way to evaluate a continued fraction.
See also cfdisrep, cfexpand, and cflength.
Examples:
(%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]); (%o1) [59, 17, 2, 1, 1, 1, 27] (%i2) cf ((3/17)*[1, -2, 5]/sqrt(11) + (8/13)); (%o2) [0, 1, 1, 1, 3, 2, 1, 4, 1, 9, 1, 9, 2]
cflength controls how many periods of the continued fraction
are computed for algebraic, irrational numbers.
(%i1) cflength: 1$ (%i2) cf ((1 + sqrt(5))/2); (%o2) [1, 1, 1, 1, 2] (%i3) cflength: 2$ (%i4) cf ((1 + sqrt(5))/2); (%o4) [1, 1, 1, 1, 1, 1, 1, 2] (%i5) cflength: 3$ (%i6) cf ((1 + sqrt(5))/2); (%o6) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
cfdisrep.
(%i1) cflength: 3$ (%i2) cfdisrep (cf (sqrt (3)))$ (%i3) ev (%, numer); (%o3) 1.731707317073171
cf.
(%i1) cf ([1,1,1,1,1,2] * 3); (%o1) [4, 1, 5, 2] (%i2) cf ([1,1,1,1,1,2]) * 3; (%o2) [3, 3, 3, 3, 3, 6]
@ref{Category: Continued fractions}
Constructs and returns an ordinary arithmetic expression
of the form a + 1/(b + 1/(c + ...))
from the list representation of a continued fraction [a, b, c, ...].
(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1) [1, 1, 1, 2]
(%i2) cfdisrep (%);
1
(%o2) 1 + ---------
1
1 + -----
1
1 + -
2
@ref{Category: Continued fractions}
Returns a matrix of the numerators and denominators of the last (column 1) and next-to-last (column 2) convergents of the continued fraction x.
(%i1) cf (rat (ev (%pi, numer)));
`rat' replaced 3.141592653589793 by 103993/33102 =3.141592653011902
(%o1) [3, 7, 15, 1, 292]
(%i2) cfexpand (%);
[ 103993 355 ]
(%o2) [ ]
[ 33102 113 ]
(%i3) %[1,1]/%[2,1], numer;
(%o3) 3.141592653011902
@ref{Category: Continued fractions}
Default value: 1
cflength controls the number of terms of the continued
fraction the function cf will give, as the value cflength times the
period. Thus the default is to give one period.
(%i1) cflength: 1$ (%i2) cf ((1 + sqrt(5))/2); (%o2) [1, 1, 1, 1, 2] (%i3) cflength: 2$ (%i4) cf ((1 + sqrt(5))/2); (%o4) [1, 1, 1, 1, 1, 1, 1, 2] (%i5) cflength: 3$ (%i6) cf ((1 + sqrt(5))/2); (%o6) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
@ref{Category: Continued fractions}
divsum (n, k) returns the sum of the divisors of n
raised to the k'th power.
divsum (n) returns the sum of the divisors of n.
(%i1) divsum (12); (%o1) 28 (%i2) 1 + 2 + 3 + 4 + 6 + 12; (%o2) 28 (%i3) divsum (12, 2); (%o3) 210 (%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2; (%o4) 210
@ref{Category: Number theory}
Returns the n'th Euler number for nonnegative integer n.
For the Euler-Mascheroni constant, see %gamma.
(%i1) map (euler, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]); (%o1) [1, 0, - 1, 0, 5, 0, - 61, 0, 1385, 0, - 50521]
@ref{Category: Number theory}
The Euler-Mascheroni constant, 0.5772156649015329 ....
@ref{Category: Constants}
Represents the factorial function. Maxima treats factorial (x) the same as x!.
See !.
@ref{Category: Gamma and factorial functions}
Returns the n'th Fibonacci number.
fib(0) equal to 0 and fib(1) equal to 1,
and
fib (-n) equal to (-1)^(n + 1) * fib(n).
After calling fib,
prevfib is equal to fib (x - 1),
the Fibonacci number preceding the last one computed.
(%i1) map (fib, [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10]); (%o1) [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55]
@ref{Category: Number theory}
Expresses Fibonacci numbers in expr in terms of the constant %phi,
which is (1 + sqrt(5))/2, approximately 1.61803399.
Examples:
(%i1) fibtophi (fib (n));
n n
%phi - (1 - %phi)
(%o1) -------------------
2 %phi - 1
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2) - fib(n + 1) + fib(n) + fib(n - 1)
(%i3) fibtophi (%);
n + 1 n + 1 n n
%phi - (1 - %phi) %phi - (1 - %phi)
(%o3) - --------------------------- + -------------------
2 %phi - 1 2 %phi - 1
n - 1 n - 1
%phi - (1 - %phi)
+ ---------------------------
2 %phi - 1
(%i4) ratsimp (%);
(%o4) 0
@ref{Category: Number theory}
For a positive integer n returns the factorization of n. If
n=p1^e1..pk^nk is the decomposition of n into prime
factors, ifactors returns [[p1, e1], ... , [pk, ek]].
Factorization methods used are trial divisions by primes up to 9973, Pollard's rho method and elliptic curve method.
(%i1) ifactors(51575319651600);
(%o1) [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
(%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
(%o2) 51575319651600
@ref{Category: Number theory}
Returns the integer n'th root of the absolute value of x.
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$ (%i2) map (lambda ([a], inrt (10^a, 3)), l); (%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]
@ref{Category: Number theory}
Computes the inverse of n modulo m.
inv_mod (n,m) returns false,
if n is a zero divisor modulo m.
(%i1) inv_mod(3, 41); (%o1) 14 (%i2) ratsimp(3^-1), modulus=41; (%o2) 14 (%i3) inv_mod(3, 42); (%o3) false
@ref{Category: Number theory}
Returns the Jacobi symbol of p and q.
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$ (%i2) map (lambda ([a], jacobi (a, 9)), l); (%o2) [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]
@ref{Category: Number theory}
Returns the least common multiple of its arguments. The arguments may be general expressions as well as integers.
load ("functs") loads this function.
@ref{Category: Number theory}
Examines expr for occurrences of two factorials
which differ by an integer.
minfactorial then turns one into a polynomial times the other.
(%i1) n!/(n+2)!;
n!
(%o1) --------
(n + 2)!
(%i2) minfactorial (%);
1
(%o2) ---------------
(n + 1) (n + 2)
@ref{Category: Number theory}
Returns the smallest prime bigger than n.
(%i1) next_prime(27); (%o1) 29
@ref{Category: Number theory}
Expands the expression expr in partial fractions
with respect to the main variable var. partfrac does a complete
partial fraction decomposition. The algorithm employed is based on
the fact that the denominators of the partial fraction expansion (the
factors of the original denominator) are relatively prime. The
numerators can be written as linear combinations of denominators, and
the expansion falls out.
(%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
2 2 1
(%o1) ----- - ----- + --------
x + 2 x + 1 2
(x + 1)
(%i2) ratsimp (%);
x
(%o2) - -------------------
3 2
x + 4 x + 5 x + 2
(%i3) partfrac (%, x);
2 2 1
(%o3) ----- - ----- + --------
x + 2 x + 1 2
(x + 1)
Uses a modular algorithm to compute a^n mod m
where a and n are integers and m is a positive integer.
If n is negative, inv_mod is used to find the modular inverse.
(%i1) power_mod(3, 15, 5); (%o1) 2 (%i2) mod(3^15,5); (%o2) 2 (%i3) power_mod(2, -1, 5); (%o3) 3 (%i4) inv_mod(2,5); (%o4) 3
@ref{Category: Number theory}
Primality test. If primep (n) returns false, n is a
composite number and if it returns true, n is a prime number
with very high probability.
For n less than 10^16 a deterministic version of Miller-Rabin's
test is used. If primep (n) returns true, then n is a
prime number.
For n bigger than 10^16 primep uses
primep_number_of_tests Miller-Rabin's pseudo-primality tests
and one Lucas pseudo-primality test. The probability that n will
pass one Miller-Rabin test is less than 1/4. Using the default value 25 for
primep_number_of_tests, the probability of n beeing
composite is much smaller that 10^-15.
@ref{Category: Predicate functions} · @ref{Category: Number theory}
Default value: 25
Number of Miller-Rabin's tests used in primep.
@ref{Category: Predicate functions} · @ref{Category: Number theory}
Returns the greatest prime smaller than n.
(%i1) prev_prime(27); (%o1) 23
@ref{Category: Number theory}
Returns the principal unit of the real quadratic number field
sqrt (n) where n is an integer,
i.e., the element whose norm is unity.
This amounts to solving Pell's equation a^2 - n b^2 = 1.
(%i1) qunit (17); (%o1) sqrt(17) + 4 (%i2) expand (% * (sqrt(17) - 4)); (%o2) 1
@ref{Category: Number theory}
Returns the number of integers less than or equal to n which are relatively prime to n.
@ref{Category: Number theory}
Default value: true
When zerobern is false,
bern excludes the Bernoulli numbers which are equal to zero.
See bern.
@ref{Category: Number theory}
Returns the Riemann zeta function if x is a negative integer, 0, 1,
or a positive even number,
and returns a noun form zeta (n) for all other arguments,
including rational noninteger, floating point, and complex arguments.
See also bfzeta and zeta%pi.
(%i1) map (zeta, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5]);
2 4
1 1 1 %pi %pi
(%o1) [0, ---, 0, - --, - -, inf, ----, zeta(3), ----, zeta(5)]
120 12 2 6 90
@ref{Category: Number theory}
Default value: true
When zeta%pi is true, zeta returns an expression
proportional to %pi^n for even integer n.
Otherwise, zeta returns a noun form zeta (n)
for even integer n.
(%i1) zeta%pi: true$
(%i2) zeta (4);
4
%pi
(%o2) ----
90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4) zeta(4)
@ref{Category: Number theory}
| [ << ] | [ >> ] | [Top] | [Contents] | [Index] | [ ? ] |
This document was generated by root on July, 13 2009 using texi2html 1.76.