Number Theory
bern (n) — Function
Returns the n’th Bernoulli number for integer n.
Bernoulli numbers equal to zero are suppressed if zerobern is
false.
See also burn.
maxima
(%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 1 1 5 691 7
(%o4) [1, - -, -, - --, --, - --, --, - ----, -]
2 6 30 42 30 66 2730 6
See also: burn.
bernpoly (x, n) — Function
Returns the n’th Bernoulli polynomial in the variable x.
bfhzeta (s, h, n) — Function
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
$$\zeta \left(s,h\right) = \sum_{k=0}^\infty {1 \over \left(k+h\right)^{s}}$$
inf
====
\ 1
zeta (s,h) = > --------
/ s
==== (k + h)
k = 0
load ("bffac") loads this function.
bfzeta (s, n) — Function
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.
burn (n) — Function
Returns a rational number, which is an approximation of the n’th Bernoulli
number for integer n. burn exploits the observation that
(rational) Bernoulli numbers can be approximated by (transcendental) zetas with
tolerable efficiency:
n - 1 1 - 2 n
(- 1) 2 zeta(2 n) (2 n)!
B(2 n) = ------------------------------------
2 n
%pi
burn may be more efficient than bern for large, isolated n
as bern computes all the Bernoulli numbers up to index n before
returning. burn invokes the approximation for even integers n > 255. For odd integers and n <= 255 the function bern is called.
load ("bffac") loads this function. See also bern.
See also: bern.
cf (expr) — Function
Computes a continued fraction approximation.
expr is an expression comprising continued fractions,
square roots of integers, and literal real numbers
(integers, rational numbers, ordinary floats, and bigfloats).
cf computes exact expansions for rational numbers,
but expansions are truncated at ratepsilon for ordinary floats
and 10^(-fpprec) for bigfloats.
Operands in the expression may be combined with arithmetic operators.
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:
expr is an expression comprising continued fractions and square roots of integers.
(%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.
maxima
(%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]
A continued fraction can be evaluated by evaluating the arithmetic
representation returned by cfdisrep.
maxima
(%i1) cflength: 3$
(%i2) cfdisrep (cf (sqrt (3)))$
(%i3) ev (%, numer);
(%o3) 1.7317073170731707
Maxima does not know about operations on continued fractions outside of
cf.
maxima
(%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]
See also: cflength, cfdisrep, cfexpand.
cfdisrep (list) — Function
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, ...].
maxima
(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1) [1, 1, 1, 2]
(%i2) cfdisrep (%);
1
(%o2) 1 + ---------
1
1 + -----
1
1 + -
2
cfexpand (x) — Function
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.
maxima
(%i1) cf (rat (ev (%pi, numer)));
rat: replaced 3.141592653589793 by 80143857/25510582 = 3.1415926535897927
(%o1) [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14]
(%i2) cfexpand (%);
[ 80143857 5419351 ]
(%o2) [ ]
[ 25510582 1725033 ]
(%i3) %[1,1]/%[2,1], numer;
(%o3) 3.1415926535897927
cflength — Variable
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.
maxima
(%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]
divsum (n, k) — Function
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.
maxima
(%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
euler (n) — Function
Returns the n’th Euler number for nonnegative integer n.
Euler numbers equal to zero are suppressed if zerobern is
false.
For the Euler-Mascheroni constant, see %gamma.
maxima
(%i1) zerobern: true$
(%i2) map (euler, [0, 1, 2, 3, 4, 5, 6]);
(%o2) [1, 0, - 1, 0, 5, 0, - 61]
(%i3) zerobern: false$
(%i4) map (euler, [0, 1, 2, 3, 4, 5, 6]);
(%o4) [1, - 1, 5, - 61, 1385, - 50521, 2702765]
factors_only — Variable
Default value: false
Controls the value returned by ifactors. The default false
causes ifactors to provide information about multiplicities of the
computed prime factors. If factors_only is set to true,
ifactors returns nothing more than a list of prime factors.
Example: See ifactors.
See also: ifactors.
fib (n) — Function
Returns the n’th Fibonacci number.
fib(0) is equal to 0 and fib(1) equal to 1, and
fib (-n) equal to (-1)^(n + 1) * fib(n).
maxima
(%i1) map (fib, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
(%o1) [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]
fibtophi (expr) — Function
Expresses Fibonacci numbers in expr in terms of the constant %phi,
which is (1 + sqrt(5))/2, approximately 1.61803399.
Examples:
maxima
(%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
ifactors (n) — Function
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 and p-1 method and elliptic curves.
If the variable ifactor_verbose is set to true
ifactor produces detailed output about what it is doing including
immediate feedback as soon as a factor has been found.
The value returned by ifactors is controlled by the option variable factors_005fonly.
The default false causes ifactors to provide information about
the multiplicities of the computed prime factors. If factors_only
is set to true, ifactors simply returns the list of
prime factors.
maxima
(%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
(%i3) ifactors(51575319651600), factors_only : true;
(%o3) [2, 3, 5, 1583, 9050207]
See also: factors_only.
igcdex (n, k) — Function
Returns a list [a, b, u] where u is the greatest
common divisor of n and k, and u is equal to
a n + b k. The arguments n and k
must be integers.
igcdex implements the Euclidean algorithm. See also gcdex.
The command load("gcdex") loads the function.
Examples:
maxima
(%i1) load("gcdex")$
(%i2) igcdex(30,18);
(%o2) [- 1, 2, 6]
(%i3) igcdex(1526757668, 7835626735736);
(%o3) [845922341123, - 164826435, 4]
(%i4) igcdex(fib(20), fib(21));
(%o4) [4181, - 2584, 1]
See also: gcdex.
inrt (x, n) — Function
Returns the integer n’th root of the absolute value of x.
maxima
(%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]
inv_mod (n, m) — Function
Computes the inverse of n modulo m.
inv_mod (n,m) returns false,
if n is a zero divisor modulo m.
maxima
(%i1) inv_mod(3, 41);
(%o1) 14
(%i2) ratsimp(3^-1), modulus = 41;
(%o2) 14
(%i3) inv_mod(3, 42);
(%o3) false
isqrt (x) — Function
Returns the “integer square root” of the absolute value of x, which is an integer.
jacobi (p, q) — Function
Returns the Jacobi symbol of p and q.
maxima
(%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]
lcm (expr_1, …, expr_n) — Function
Returns the least common multiple of its arguments. The arguments may be general expressions as well as integers.
load ("functs") loads this function.
lucas (n) — Function
Returns the n’th Lucas number.
lucas(0) is equal to 2 and lucas(1) equal to 1, and
in general, lucas(n) = lucas(n-1) + lucas(n-2). Also
lucas(-n) is equal to (-1)^(-n) * lucas(n).
maxima
(%i1) map (lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
(%o1) [7, - 4, 3, - 1, 2, 1, 3, 4, 7, 11, 18, 29, 47]
mod (x, y) — Function
If x and y are real numbers and y is nonzero, return
x - y * floor(x / y). Further for all real
x, we have mod (x, 0) = x. For a discussion of the
definition mod (x, 0) = x, see Section 3.4, of
“Concrete Mathematics,” by Graham, Knuth, and Patashnik. The function
mod (x, 1) is a sawtooth function with period 1 with
mod (1, 1) = 0 and mod (0, 1) = 0.
To find the principal argument (a number in the interval (-%pi, %pi]) of
a complex number, use the function
x |-> %pi - mod (%pi - x, 2*%pi), where x is an
argument.
When x and y are constant expressions (10 * %pi, for
example), mod uses the same big float evaluation scheme that floor
and ceiling uses. Again, it’s possible, although unlikely, that
mod could return an erroneous value in such cases.
For nonnumerical arguments x or y, mod knows several
simplification rules:
maxima
(%i1) mod (x, 0);
(%o1) x
(%i2) mod (a*x, a*y);
(%o2) a mod(x, y)
(%i3) mod (0, x);
(%o3) 0
next_prime (n) — Function
Returns the smallest prime bigger than n.
maxima
(%i1) next_prime(27);
(%o1) 29
partfrac (expr, var) — Function
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.
partfrac ignores the value true of the option variable
keepfloat.
maxima
(%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)
power_mod (a, n, m) — Function
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.
maxima
(%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
prev_prime (n) — Function
Returns the greatest prime smaller than n.
maxima
(%i1) prev_prime(27);
(%o1) 23
primep (n) — Function
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 3317044064679887385961981 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 3317044064679887385961981 primep uses
primep_number_of_tests Miller-Rabin’s pseudo-primality tests and one
Lucas pseudo-primality test. The probability that a non-prime 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 being
composite is much smaller that 10^-15.
primep_number_of_tests — Variable
Default value: 25
Number of Miller-Rabin’s tests used in primep.
primes (start, end) — Function
Returns the list of all primes from start to end.
maxima
(%i1) primes(3, 7);
(%o1) [3, 5, 7]
qunit (n) — Function
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.
maxima
(%i1) qunit (17);
(%o1) sqrt(17) + 4
(%i2) expand (% * (sqrt(17) - 4));
(%o2) 1
solve_congruences ([r_1, …, r_n], [m_1, …, m_n]) — Function
Solves the system of congruences x = r_1 mod m_1, …, x = r_n mod m_n.
The remainders r_n may be arbitrary integers while the moduli m_n have to be
positive and pairwise coprime integers.
maxima
(%i1) mods : [1000, 1001, 1003, 1007];
(%o1) [1000, 1001, 1003, 1007]
(%i2) lreduce('gcd, mods);
(%o2) 1
(%i3) x : random(apply("*", mods));
(%o3) 685124877004
(%i4) rems : map(lambda([z], mod(x, z)), mods);
(%o4) [4, 568, 54, 624]
(%i5) solve_congruences(rems, mods);
(%o5) 685124877004
(%i6) solve_congruences([1, 2], [3, n]);
(%o6) solve_congruences([1, 2], [3, n])
(%i7) %, n = 4;
(%o7) 10
totient (n) — Function
Returns the number of integers less than or equal to n which are relatively prime to n.
zerobern — Variable
Default value: true
When zerobern is false, bern excludes the Bernoulli numbers
and euler excludes the Euler numbers which are equal to zero.
See bern and euler.
zeta (n) — Function
Returns the Riemann zeta function. If n is a negative integer, 0, or a
positive even integer, the Riemann zeta function simplifies to an exact value.
For a positive even integer the option variable zeta%pi has to be
true in addition (See zeta%pi). For a floating point or bigfloat
number the Riemann zeta function is evaluated numerically. Maxima returns a
noun form zeta (n) for all other arguments, including rational
noninteger, and complex arguments, or for even integers, if zeta%pi has
the value false.
zeta(1) is undefined, but Maxima knows the limit
limit(zeta(x), x, 1) from above and below.
The Riemann zeta function distributes over lists, matrices, and equations.
See also bfzeta and zeta_0025pi.
Examples:
maxima
(%i1) zeta([-2, -1, 0, 0.5, 2, 3,1+%i]);
2
1 1 %pi
(%o1) [0, - --, - -, - 1.4603545088095862, ----, zeta(3),
12 2 6
zeta(%i + 1)]
(%i2) limit(zeta(x),x,1,plus);
(%o2) inf
(%i3) limit(zeta(x),x,1,minus);
(%o3) minf
See also: bfzeta, zeta%pi.
zeta%pi — Variable
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.
Examples:
maxima
(%i1) zeta%pi: true$
(%i2) zeta (4);
4
%pi
(%o2) ----
90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4) zeta(4)
See also: true.
zn_add_table (n) — Function
Shows an addition table of all elements in (Z/nZ).
See also zn_mult_table, zn_005fpower_005ftable.
See also: zn_mult_table, zn_power_table.
zn_carmichael_lambda (n) — Function
Returns 1 if n is 1 and otherwise
the greatest characteristic factor of the totient of n.
For remarks and examples see zn_005fcharacteristic_005ffactors.
See also: zn_characteristic_factors.
zn_characteristic_factors (n) — Function
Returns a list containing the characteristic factors of the totient of n.
Using the characteristic factors a multiplication group modulo n can be expressed as a group direct product of cyclic subgroups.
In case the group itself is cyclic the list only contains the totient
and using zn_primroot a generator can be computed.
If the totient splits into more than one characteristic factors
zn_factor_generators finds generators of the corresponding subgroups.
Each of the r factors in the list divides the right following factors.
For the last factor f_r therefore holds a^f_r = 1 (mod n)
for all a coprime to n.
This factor is also known as Carmichael function or Carmichael lambda.
If n > 2, then totient(n)/2^r is the number of quadratic residues,
and each of these has 2^r square roots.
See also totient, zn_primroot, zn_005ffactor_005fgenerators.
Examples:
The multiplication group modulo 14 is cyclic and its 6 elements
can be generated by a primitive root.
maxima
(%i1) [zn_characteristic_factors(14), phi: totient(14)];
(%o1) [[6], 6]
(%i2) [zn_factor_generators(14), g: zn_primroot(14)];
(%o2) [[3], 3]
(%i3) M14: makelist(power_mod(g,i,14), i,0,phi-1);
(%o3) [1, 3, 9, 13, 11, 5]
The multiplication group modulo 15 is not cyclic and its 8 elements
can be generated by two factor generators.
maxima
(%i1) [[f1,f2]: zn_characteristic_factors(15), totient(15)];
(%o1) [[2, 4], 8]
(%i2) [[g1,g2]: zn_factor_generators(15), zn_primroot(15)];
(%o2) [[11, 7], false]
(%i3) UG1: makelist(power_mod(g1,i,15), i,0,f1-1);
(%o3) [1, 11]
(%i4) UG2: makelist(power_mod(g2,i,15), i,0,f2-1);
(%o4) [1, 7, 4, 13]
(%i5) M15: create_list(mod(i*j,15), i,UG1, j,UG2);
(%o5) [1, 7, 4, 13, 11, 2, 14, 8]
For the last characteristic factor 4 it holds that a^4 = 1 (mod 15)
for all a in M15.
M15 has two characteristic factors and therefore 8/2^2 quadratic residues,
and each of these has 2^2 square roots.
maxima
(%i1) zn_power_table(15);
[ 1 1 1 1 ]
[ ]
[ 2 4 8 1 ]
[ ]
[ 4 1 4 1 ]
[ ]
[ 7 4 13 1 ]
(%o1) [ ]
[ 8 4 2 1 ]
[ ]
[ 11 1 11 1 ]
[ ]
[ 13 4 7 1 ]
[ ]
[ 14 1 14 1 ]
(%i2) map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
(%o2) [[1, 4, 11, 14], [2, 7, 8, 13]]
See also: totient, zn_primroot, zn_factor_generators.
zn_determinant (matrix, p) — Function
Uses the technique of LU-decomposition to compute the determinant of matrix over (Z/pZ). p must be a prime.
However if the determinant is equal to zero the LU-decomposition might fail.
In that case zn_determinant computes the determinant non-modular
and reduces thereafter.
See also zn_005finvert_005fby_005flu.
Examples:
maxima
(%i1) m : matrix([1,3],[2,4]);
[ 1 3 ]
(%o1) [ ]
[ 2 4 ]
(%i2) zn_determinant(m, 5);
(%o2) 3
(%i3) m : matrix([2,4,1],[3,1,4],[4,3,2]);
[ 2 4 1 ]
[ ]
(%o3) [ 3 1 4 ]
[ ]
[ 4 3 2 ]
(%i4) zn_determinant(m, 5);
(%o4) 0
See also: zn_invert_by_lu.
zn_factor_generators (n) — Function
Returns a list containing factor generators corresponding to the characteristic factors of the totient of n.
For remarks and examples see zn_005fcharacteristic_005ffactors.
See also: zn_characteristic_factors.
zn_invert_by_lu (matrix, p) — Function
Uses the technique of LU-decomposition to compute the modular inverse of
matrix over (Z/pZ). p must be a prime and matrix
invertible. zn_invert_by_lu returns false if matrix
is not invertible.
See also zn_005fdeterminant.
Example:
maxima
(%i1) m : matrix([1,3],[2,4]);
[ 1 3 ]
(%o1) [ ]
[ 2 4 ]
(%i2) zn_determinant(m, 5);
(%o2) 3
(%i3) mi : zn_invert_by_lu(m, 5);
[ 3 4 ]
(%o3) [ ]
[ 1 2 ]
(%i4) matrixmap(lambda([a], mod(a, 5)), m . mi);
[ 1 0 ]
(%o4) [ ]
[ 0 1 ]
See also: zn_determinant.
zn_log (a, g, n) — Function
Computes the discrete logarithm. Let (Z/nZ)* be a cyclic group, g a
primitive root modulo n or a generator of a subgroup of (Z/nZ)*
and let a be a member of this group.zn_log (a, g, n) then solves the congruence g^x = a mod n.
Please note that if a is not a power of g modulo n,
zn_log will not terminate.
The applied algorithm needs a prime factorization of zn_order(g) resp. totient(n)
in case g is a primitive root modulo n.
A precomputed list of factors of zn_order(g) might be used as the optional fourth argument.
This list must be of the same form as the list returned by ifactors(zn_order(g))
using the default option factors_only : false.
However, compared to the running time of the logarithm algorithm
providing the list of factors has only a quite small effect.
The algorithm uses a Pohlig-Hellman-reduction and Pollard’s Rho-method for
discrete logarithms. The running time of zn_log primarily depends on the
bitlength of the greatest prime factor of zn_order(g).
See also zn_primroot, zn_order, ifactors, totient.
Examples:
zn_log (a, g, n) solves the congruence g^x = a mod n.
maxima
(%i1) n : 22$
(%i2) g : zn_primroot(n);
(%o2) 7
(%i3) ord_7 : zn_order(7, n);
(%o3) 10
(%i4) powers_7 : makelist(power_mod(g, x, n), x, 0, ord_7 - 1);
(%o4) [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
(%i5) zn_log(9, g, n);
(%o5) 8
(%i6) map(lambda([x], zn_log(x, g, n)), powers_7);
(%o6) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
(%i7) ord_5 : zn_order(5, n);
(%o7) 5
(%i8) powers_5 : makelist(power_mod(5,x,n), x, 0, ord_5 - 1);
(%o8) [1, 5, 3, 15, 9]
(%i9) zn_log(9, 5, n);
(%o9) 4
The optional fourth argument must be of the same form as the list returned by
ifactors(zn_order(g)).
The running time primarily depends on the bitlength of the totient’s greatest prime factor.
maxima
(%i1) (p : 2^127-1, primep(p));
(%o1) true
(%i2) ifs : ifactors(p - 1)$
(%i3) g : zn_primroot(p, ifs);
(%o3) 43
(%i4) a : power_mod(g, 4711, p)$
(%i5) zn_log(a, g, p, ifs);
(%o5) 4711
(%i6) f_max : last(ifs);
(%o6) [77158673929, 1]
(%i7) ord_5 : zn_order(5,p,ifs)$
(%i8) (p - 1)/ord_5;
(%o8) 73
(%i9) ifs_5 : ifactors(ord_5)$
(%i10) a : power_mod(5, 4711, p)$
(%i11) zn_log(a, 5, p, ifs_5);
(%o11) 4711
See also: zn_primroot, zn_order, ifactors, totient.
zn_mult_table (n) — Function
Without the optional argument gcd zn_mult_table(n) shows a
multiplication table of all elements in (Z/nZ)* which are all elements
coprime to n.
The optional second argument gcd allows to select a specific
subset of (Z/nZ). If gcd is an integer, a multiplication table of
all residues x with gcd(x,n) =gcd are returned.
Additionally row and column headings are added for better readability.
If necessary, these can be easily removed by submatrix(1, table, 1).
If gcd is set to all, the table is printed for all non-zero
elements in (Z/nZ).
The second example shows an alternative way to create a multiplication table for subgroups.
See also zn_add_table, zn_005fpower_005ftable.
Examples:
The default table shows all elements in (Z/nZ)* and allows to demonstrate and study basic properties of modular multiplication groups. E.g. the principal diagonal contains all quadratic residues, each row and column contains every element, the tables are symmetric, etc..
If gcd is set to all, the table is printed for all non-zero
elements in (Z/nZ).
maxima
(%i1) zn_mult_table(8);
[ 1 3 5 7 ]
[ ]
[ 3 1 7 5 ]
(%o1) [ ]
[ 5 7 1 3 ]
[ ]
[ 7 5 3 1 ]
(%i2) zn_mult_table(8, all);
[ 1 2 3 4 5 6 7 ]
[ ]
[ 2 4 6 0 2 4 6 ]
[ ]
[ 3 6 1 4 7 2 5 ]
[ ]
(%o2) [ 4 0 4 0 4 0 4 ]
[ ]
[ 5 2 7 4 1 6 3 ]
[ ]
[ 6 4 2 0 6 4 2 ]
[ ]
[ 7 6 5 4 3 2 1 ]
If gcd is an integer, row and column headings are added for better readability.
If the subset chosen by gcd is a group there is another way to create
a multiplication table. An isomorphic mapping from a group with 1 as
identity builds a table which is easy to read. The mapping is accomplished via CRT.
In the second version of T36_4 the identity, here 28, is placed in
the top left corner, just like in table T9.
maxima
(%i1) T36_4: zn_mult_table(36,4);
[ * 4 8 16 20 28 32 ]
[ ]
[ 4 16 32 28 8 4 20 ]
[ ]
[ 8 32 28 20 16 8 4 ]
[ ]
(%o1) [ 16 28 20 4 32 16 8 ]
[ ]
[ 20 8 16 32 4 20 28 ]
[ ]
[ 28 4 8 16 20 28 32 ]
[ ]
[ 32 20 4 8 28 32 16 ]
(%i2) T9: zn_mult_table(36/4);
[ 1 2 4 5 7 8 ]
[ ]
[ 2 4 8 1 5 7 ]
[ ]
[ 4 8 7 2 1 5 ]
(%o2) [ ]
[ 5 1 2 7 8 4 ]
[ ]
[ 7 5 1 8 4 2 ]
[ ]
[ 8 7 5 4 2 1 ]
(%i3) T36_4: matrixmap(lambda([x], solve_congruences([0,x],[4,9])), T9);
[ 28 20 4 32 16 8 ]
[ ]
[ 20 4 8 28 32 16 ]
[ ]
[ 4 8 16 20 28 32 ]
(%o3) [ ]
[ 32 28 20 16 8 4 ]
[ ]
[ 16 32 28 8 4 20 ]
[ ]
[ 8 16 32 4 20 28 ]
See also: zn_add_table, zn_power_table.
zn_nth_root (x, n, m) — Function
Returns a list with all n-th roots of x from the multiplication
subgroup of (Z/mZ) which contains x, or false, if x
is no n-th power modulo m or not contained in any multiplication
subgroup of (Z/mZ).
x is an element of a multiplication subgroup modulo m, if the
greatest common divisor g = gcd(x,m) is coprime to m/g.
zn_nth_root is based on an algorithm by Adleman, Manders and Miller
and on theorems about modulo multiplication groups by Daniel Shanks.
The algorithm needs a prime factorization of the modulus m.
So in case the factorization of m is known, the list of factors
can be passed as the fourth argument. This optional argument
must be of the same form as the list returned by ifactors(m)
using the default option factors_only: false.
Examples:
A power table of the multiplication group modulo 14
followed by a list of lists containing all n-th roots of 1
with n from 1 to 6.
maxima
(%i1) zn_power_table(14);
[ 1 1 1 1 1 1 ]
[ ]
[ 3 9 13 11 5 1 ]
[ ]
[ 5 11 13 9 3 1 ]
(%o1) [ ]
[ 9 11 1 9 11 1 ]
[ ]
[ 11 9 1 11 9 1 ]
[ ]
[ 13 1 13 1 13 1 ]
(%i2) makelist(zn_nth_root(1,n,14), n,1,6);
(%o2) [[1], [1, 13], [1, 9, 11], [1, 13], [1],
[1, 3, 5, 9, 11, 13]]
In the following example x is not coprime to m, but is a member of a multiplication subgroup of (Z/mZ) and any n-th root is a member of the same subgroup.
The residue class 3 is no member of any multiplication subgroup of (Z/63Z)
and is therefore not returned as a third root of 27.
Here zn_power_table shows all residues x in (Z/63Z)
with gcd(x,63) = 9. This subgroup is isomorphic to (Z/7Z)*
and its identity 36 is computed via CRT.
maxima
(%i1) m: 7*9$
(%i2) zn_power_table(m,9);
[ 9 18 36 9 18 36 ]
[ ]
[ 18 9 36 18 9 36 ]
[ ]
[ 27 36 27 36 27 36 ]
(%o2) [ ]
[ 36 36 36 36 36 36 ]
[ ]
[ 45 9 27 18 54 36 ]
[ ]
[ 54 18 27 9 45 36 ]
(%i3) zn_nth_root(27,3,m);
(%o3) [27, 45, 54]
(%i4) id7:1$ id63_9: solve_congruences([id7,0],[7,9]);
(%o5) 36
In the following RSA-like example, where the modulus N is squarefree,
i.e. it splits into
exclusively first power factors, every x from 0 to N-1
is contained in a multiplication subgroup.
The process of decryption needs the e-th root.
e is coprime to totient(N) and therefore the e-th root is unique.
In this case zn_nth_root effectively performs CRT-RSA.
(Please note that flatten removes braces but no solutions.)
maxima
(%i1) [p,q,e]: [5,7,17]$ N: p*q$
(%i3) xs: makelist(x,x,0,N-1)$
(%i4) ys: map(lambda([x],power_mod(x,e,N)),xs)$
(%i5) zs: flatten(map(lambda([y], zn_nth_root(y,e,N)), ys))$
(%i6) is(zs = xs);
(%o6) true
In the following example the factorization of the modulus is known and passed as the fourth argument.
maxima
(%i1) p: 2^107-1$ q: 2^127-1$ N: p*q$
(%i4) ibase: obase: 16$
(%i5) msg: 11223344556677889900aabbccddeeff$
(%i6) enc: power_mod(msg, 10001, N);
(%o6) 1A8DB7892AE588BDC2BE25DD5107A425001FE9C82161ABC673241C8B383
(%i7) zn_nth_root(enc, 10001, N, [[p,1],[q,1]]);
(%o7) [11223344556677889900AABBCCDDEEFF]
zn_order (x, n) — Function
Returns the order of x if it is a unit of the finite group (Z/nZ)*
or returns false. x is a unit modulo n if it is coprime to n.
The applied algorithm needs a prime factorization of totient(n). This factorization
might be time consuming in some cases and it can be useful to factor first
and then to pass the list of factors to zn_log as the third argument.
The list must be of the same form as the list returned by ifactors(totient(n))
using the default option factors_only : false.
See also zn_primroot, ifactors, totient.
Examples:
zn_order computes the order of the unit x in (Z/nZ)*.
maxima
(%i1) n: 22$
(%i2) g: zn_primroot(n);
(%o2) 7
(%i3) units_22: sublist(makelist(i,i,1,21), lambda([x], gcd(x,n)=1));
(%o3) [1, 3, 5, 7, 9, 13, 15, 17, 19, 21]
(%i4) (ord_7: zn_order(7, n)) = totient(n);
(%o4) 10 = 10
(%i5) powers_7: makelist(power_mod(g,i,n), i,0,ord_7 - 1);
(%o5) [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
(%i6) map(lambda([x], zn_order(x, n)), powers_7);
(%o6) [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
(%i7) map(lambda([x], ord_7/gcd(x,ord_7)), makelist(i,i,0,ord_7-1));
(%o7) [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
(%i8) totient(totient(n));
(%o8) 4
The optional third argument must be of the same form as the list returned by
ifactors(totient(n)).
maxima
(%i1) (p : 2^142 + 217, primep(p));
(%o1) true
(%i2) ifs: ifactors( totient(p) )$
(%i3) g: zn_primroot(p, ifs);
(%o3) 3
(%i4) is( (ord_3 : zn_order(g, p, ifs)) = totient(p) );
(%o4) true
(%i5) map(lambda([x], ord_3/zn_order(x,p,ifs)), makelist(i,i,2,15));
(%o5) [22, 1, 44, 10, 5, 2, 22, 2, 8, 2, 1, 1, 20, 1]
See also: zn_primroot, ifactors, totient.
zn_power_table (n) — Function
Without any optional argument zn_power_table(n)
shows a power table of all elements in (Z/nZ)*
which are all residue classes coprime to n.
The exponent loops from 1 to the greatest characteristic factor of
totient(n) (also known as Carmichael function or Carmichael lambda)
and the table ends with a column of ones on the right side.
The optional second argument gcd allows to select powers of a specific
subset of (Z/nZ). If gcd is an integer, powers of all residue
classes x with gcd(x,n) =gcd are returned,
i.e. the default value for gcd is 1.
If gcd is set to all, the table contains powers of all elements
in (Z/nZ).
If the optional third argument max_exp is given, the exponent loops from
1 to max_exp.
See also zn_add_table, zn_005fmult_005ftable.
Examples:
The default which is gcd= 1 allows to demonstrate and study basic
theorems of e.g. Fermat and Euler.
The argument gcd allows to select subsets of (Z/nZ) and to study
multiplication subgroups and isomorphisms.
E.g. the groups G10 and G10_2 are under multiplication both
isomorphic to G5. 1 is the identity in G5.
So are 1 resp. 6 the identities in G10 resp. G10_2.
There are corresponding mappings for primitive roots, n-th roots, etc..
maxima
(%i1) zn_power_table(10);
[ 1 1 1 1 ]
[ ]
[ 3 9 7 1 ]
(%o1) [ ]
[ 7 9 3 1 ]
[ ]
[ 9 1 9 1 ]
(%i2) zn_power_table(10,2);
[ 2 4 8 6 ]
[ ]
[ 4 6 4 6 ]
(%o2) [ ]
[ 6 6 6 6 ]
[ ]
[ 8 4 2 6 ]
(%i3) zn_power_table(10,5);
(%o3) [ 5 5 5 5 ]
(%i4) zn_power_table(10,10);
(%o4) [ 0 0 0 0 ]
(%i5) G5: [1,2,3,4];
(%o5) [1, 2, 3, 4]
(%i6) G10_2: map(lambda([x], solve_congruences([0,x],[2,5])), G5);
(%o6) [6, 2, 8, 4]
(%i7) G10: map(lambda([x], power_mod(3, zn_log(x,2,5), 10)), G5);
(%o7) [1, 3, 7, 9]
If gcd is set to all, the table contains powers of all elements
in (Z/nZ).
The third argument max_exp allows to set the highest exponent. The following table shows a very small example of RSA.
maxima
(%i1) linel:100$ N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$
(%i6) zn_power_table(N, all, e*d);
[ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ]
[ ]
[ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ]
[ ]
[ 2 4 8 6 2 4 8 6 2 4 8 6 2 4 8 6 2 4 8 6 2 ]
[ ]
[ 3 9 7 1 3 9 7 1 3 9 7 1 3 9 7 1 3 9 7 1 3 ]
[ ]
[ 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 6 4 ]
(%o6) [ ]
[ 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 ]
[ ]
[ 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ]
[ ]
[ 7 9 3 1 7 9 3 1 7 9 3 1 7 9 3 1 7 9 3 1 7 ]
[ ]
[ 8 4 2 6 8 4 2 6 8 4 2 6 8 4 2 6 8 4 2 6 8 ]
[ ]
[ 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 1 9 ]
See also: zn_add_table, zn_mult_table.
zn_primroot (n) — Function
If the multiplicative group (Z/nZ)* is cyclic, zn_primroot computes the
smallest primitive root modulo n. (Z/nZ)* is cyclic if n is equal to
2, 4, p^k or 2*p^k, where p is prime and
greater than 2 and k is a natural number. zn_primroot
performs an according pretest if the option variable zn_primroot_pretest
(default: false) is set to true. In any case the computation is limited
by the upper bound zn_005fprimroot_005flimit.
If (Z/nZ)* is not cyclic or if there is no primitive root up to
zn_primroot_limit, zn_primroot returns false.
The applied algorithm needs a prime factorization of totient(n). This factorization
might be time consuming in some cases and it can be useful to factor first
and then to pass the list of factors to zn_log as an additional argument.
The list must be of the same form as the list returned by ifactors(totient(n))
using the default option factors_only : false.
See also zn_primroot_p, zn_order, ifactors, totient.
Examples:
zn_primroot computes the smallest primitive root modulo n or returns
false.
maxima
(%i1) n : 14$
(%i2) g : zn_primroot(n);
(%o2) 3
(%i3) zn_order(g, n) = totient(n);
(%o3) 6 = 6
(%i4) n : 15$
(%i5) zn_primroot(n);
(%o5) false
The optional second argument must be of the same form as the list returned by
ifactors(totient(n)).
maxima
(%i1) (p : 2^142 + 217, primep(p));
(%o1) true
(%i2) ifs : ifactors( totient(p) )$
(%i3) g : zn_primroot(p, ifs);
(%o3) 3
(%i4) [time(%o2), time(%o3)];
(%o4) [[3.2], [0.0]]
(%i5) is(zn_order(g, p, ifs) = p - 1);
(%o5) true
(%i6) n : 2^142 + 216$
(%i7) ifs : ifactors(totient(n))$
(%i8) zn_primroot(n, ifs), zn_primroot_limit : 200, zn_primroot_verbose : true;
`zn_primroot' stopped at zn_primroot_limit = 200
(%o8) false
See also: zn_primroot_pretest, zn_primroot_limit, zn_primroot_p, zn_order, ifactors, totient.
zn_primroot_limit — Variable
Default value: 1000
If zn_primroot cannot find a primitive root, it stops at this upper bound.
If the option variable zn_primroot_verbose (default: false) is
set to true, a message will be printed when zn_primroot_limit is reached.
See also: zn_primroot, zn_primroot_verbose.
zn_primroot_p (x, n) — Function
Checks whether x is a primitive root in the multiplicative group (Z/nZ)*.
The applied algorithm needs a prime factorization of totient(n). This factorization
might be time consuming and in case zn_primroot_p will be consecutively
applied to a list of candidates it can be useful to factor first and then to
pass the list of factors to zn_log as a third argument.
The list must be of the same form as the list returned by ifactors(totient(n))
using the default option factors_only : false.
See also zn_primroot, zn_order, ifactors, totient.
Examples:
zn_primroot_p as a predicate function.
maxima
(%i1) n : 14$
(%i2) units_14 : sublist(makelist(i,i,1,13), lambda([i], gcd(i, n) = 1));
(%o2) [1, 3, 5, 9, 11, 13]
(%i3) zn_primroot_p(13, n);
(%o3) false
(%i4) sublist(units_14, lambda([x], zn_primroot_p(x, n)));
(%o4) [3, 5]
(%i5) map(lambda([x], zn_order(x, n)), units_14);
(%o5) [1, 6, 6, 3, 3, 2]
The optional third argument must be of the same form as the list returned by
ifactors(totient(n)).
maxima
(%i1) (p: 2^142 + 217, primep(p));
(%o1) true
(%i2) ifs: ifactors( totient(p) )$
(%i3) sublist(makelist(i,i,1,50), lambda([x], zn_primroot_p(x,p,ifs)));
(%o3) [3, 12, 13, 15, 21, 24, 26, 27, 29, 33, 38, 42, 48]
(%i4) [time(%o2), time(%o3)];
(%o4) [[3.01], [0.03]]
See also: zn_primroot, zn_order, ifactors, totient.
zn_primroot_pretest — Variable
Default value: false
The multiplicative group (Z/nZ)* is cyclic if n is equal to
2, 4, p^k or 2*p^k, where p is prime and
greater than 2 and k is a natural number.
zn_primroot_pretest controls whether zn_primroot will check
if one of these cases occur before it computes the smallest primitive root.
Only if zn_primroot_pretest is set to true this pretest will be
performed.
See also: zn_primroot.
zn_primroot_verbose — Variable
Default value: false
Controls whether zn_primroot prints a message when reaching
zn_005fprimroot_005flimit.
See also: zn_primroot, zn_primroot_limit.