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29. Number Theory


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29.1 Functions and Variables for Number Theory

Function: bern (n)

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   1     1   5     691   7
(%o4)           [1, - -, -, - --, --, - --, --, - ----, -]
                      2  6    30  42    30  66    2730  6

Categories:  Number theory

Function: bernpoly (x, n)

Returns the n'th Bernoulli polynomial in the variable x.

Categories:  Number theory

Function: bfzeta (s, n)

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.

Function: bfhzeta (s, h, n)

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

                        inf
                        ====
                        \        1
         zeta (s,h)  =   >    --------
                        /            s
                        ====  (k + h)
                        k = 0

load ("bffac") loads this function.

Function: burn (n)

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.

Categories:  Number theory

Function: chinese ([r_1, …, r_n], [m_1, …, m_n])

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.

(%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) chinese(rems, mods);
(%o5)                         685124877004
(%i6) chinese([1, 2], [3, n]);
(%o6)                    chinese([1, 2], [3, n])
(%i7) %, n = 4;
(%o7)                              10

Categories:  Number theory

Function: cf (expr)

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:

Categories:  Continued fractions

Function: cfdisrep (list)

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

Categories:  Continued fractions

Function: cfexpand (x)

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

Categories:  Continued fractions

Option variable: cflength

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]

Categories:  Continued fractions

Function: divsum (n, k)
Function: divsum (n)

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

Categories:  Number theory

Function: euler (n)

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.

(%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]

Categories:  Number theory

Option variable: factors_only

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 .

Categories:  Number theory

Function: fib (n)

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]

Categories:  Number theory

Function: fibtophi (expr)

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

Categories:  Number theory

Function: ifactors (n)

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.

The value returned by ifactors is controlled by the option variable factors_only . 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.

(%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]

Categories:  Number theory

Function: igcdex (n, k)

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:

(%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]

Categories:  Number theory

Function: inrt (x, n)

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]

Categories:  Number theory

Function: inv_mod (n, m)

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

Categories:  Number theory

Function: isqrt (x)

Returns the "integer square root" of the absolute value of x, which is an integer.

Categories:  Mathematical functions

Function: jacobi (p, q)

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]

Categories:  Number theory

Function: lcm (expr_1, …, expr_n)

Returns the least common multiple of its arguments. The arguments may be general expressions as well as integers.

load ("functs") loads this function.

Categories:  Number theory

Function: mod (x, y)

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:

(%i1) mod (x, 0);
(%o1)                           x
(%i2) mod (a*x, a*y);
(%o2)                      a mod(x, y)
(%i3) mod (0, x);
(%o3)                           0

Categories:  Mathematical functions

Function: next_prime (n)

Returns the smallest prime bigger than n.

(%i1) next_prime(27);
(%o1)                       29

Categories:  Number theory

Function: partfrac (expr, var)

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)

Function: power_mod (a, n, m)

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

Categories:  Number theory

Function: primep (n)

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 341550071728321 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 341550071728321 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 beeing composite is much smaller that 10^-15.

Option variable: primep_number_of_tests

Default value: 25

Number of Miller-Rabin's tests used in primep.

Categories:  Number theory

Function: prev_prime (n)

Returns the greatest prime smaller than n.

(%i1) prev_prime(27);
(%o1)                       23

Categories:  Number theory

Function: qunit (n)

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

Categories:  Number theory

Function: totient (n)

Returns the number of integers less than or equal to n which are relatively prime to n.

Categories:  Number theory

Option variable: zerobern

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.

Categories:  Number theory

Function: zeta (n)

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%pi.

Examples:

(%i1) zeta([-2, -1, 0, 0.5, 2, 3, 1+%i]);
                                             2
            1     1                       %pi
(%o1) [0, - --, - -, - 1.460354508809586, ----, 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

Categories:  Number theory

Option variable: zeta%pi

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:

(%i1) zeta%pi: true$
(%i2) zeta (4);
                                 4
                              %pi
(%o2)                         ----
                               90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4)                        zeta(4)

Categories:  Number theory

Function: zn_log (a, g, n)
Function: zn_log (a, g, n, [[p1, e1], …, [pk, ek]])

Computes the discrete logarithm. Let (Z/nZ)* be a cyclic group, g a primitive root modulo n and let a be a member of this group. zn_log (a, g, n) then solves the congruence g^x = a mod n.

The applied algorithm needs a prime factorization of totient(n). This factorization might be time consuming as well and in some cases it can be useful to factor first and then to pass the list of factors to zn_log as the fourth argument. The list must be of the same form as the list returned by ifactors(totient(n)) using the default option factors_only : false.

The algorithm uses a Pohlig-Hellman-reduction and Pollard's Rho-method for discrete logarithms. The run time of zn_log primarily depends on the bitlength of the totient's greatest prime factor.

See also zn_primroot , zn_order , ifactors , totient .

Examples:

zn_log (a, g, n) solves the congruence g^x = a mod n.

(%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(21, g, n);
(%o5)                               5
(%i6) map(lambda([x], zn_log(x, g, n)), powers_7);
(%o6)                [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

The optional fourth argument must be of the same form as the list returned by ifactors(totient(n)). The run time primarily depends on the bitlength of the totient's greatest prime factor.

(%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, 1234567890, p)$
(%i5) zn_log(a, g, p, ifs);
(%o5)                          1234567890
(%i6) time(%o5);  
(%o6)                            [1.204]
(%i7) f_max : last(ifs);
(%o7)                       [77158673929, 1]
(%i8) slength( printf(false, "~b", f_max[1]) );
(%o8)                              37

Categories:  Number theory

Function: zn_order (x, n)
Function: zn_order (x, n, [[p1, e1], …, [pk, ek]])

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)*.

(%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)).

(%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]

Categories:  Number theory

Function: zn_primroot (n)
Function: zn_primroot (n, [[p1, e1], …, [pk, ek]])

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_primroot_limit .

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.

(%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)).

(%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)                    [[15.556972], [0.004]]
(%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

Categories:  Number theory

Option variable: zn_primroot_limit

Default value: 1000

If zn_primroot cannot find a primitve 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.

Categories:  Number theory

Function: zn_primroot_p (x, n)
Function: zn_primroot_p (x, n, [[p1, e1], …, [pk, ek]])

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.

(%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)).

(%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)                   [[7.748484], [0.036002]]

Option variable: zn_primroot_pretest

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.

Categories:  Number theory

Option variable: zn_primroot_verbose

Default value: false

Controls whether zn_primroot prints a message when reaching zn_primroot_limit .

Categories:  Number theory


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