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<h1 class="chapter"> 31. Number Theory </h1>


<table class="menu" border="0" cellspacing="0">
<tr><td align="left" valign="top"><a href="#SEC139">31.1 Functions and Variables for Number Theory</a></td><td>&nbsp;&nbsp;</td><td align="left" valign="top">  
</td></tr>
</table>

<p><a name="Item_003a-Functions-and-Variables-for-Number-Theory"></a>
</p><hr size="6">
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</tr></table>
<h2 class="section"> 31.1 Functions and Variables for Number Theory </h2>

<p><a name="Item_003a-bern"></a>
</p><dl>
<dt><u>Function:</u> <b>bern</b><i> (<var>n</var>)</i>
<a name="IDX1148"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Bernoulli number for integer <var>n</var>.
Bernoulli numbers equal to zero are suppressed if <code>zerobern</code> is <code>false</code>.
</p>
<p>See also <code>burn</code>.
</p>
<pre class="example">(%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
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-bernpoly"></a>
</p><dl>
<dt><u>Function:</u> <b>bernpoly</b><i> (<var>x</var>, <var>n</var>)</i>
<a name="IDX1149"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Bernoulli polynomial in the
variable <var>x</var>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-bfzeta"></a>
</p><dl>
<dt><u>Function:</u> <b>bfzeta</b><i> (<var>s</var>, <var>n</var>)</i>
<a name="IDX1150"></a>
</dt>
<dd><p>Returns the Riemann zeta function for the argument <var>s</var>.
The return value is a big float (bfloat);
<var>n</var> is the number of digits in the return value.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
 &middot;
<a href="maxima_95.html#Category_003a-Numerical-evaluation">Numerical evaluation</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-bfhzeta"></a>
</p><dl>
<dt><u>Function:</u> <b>bfhzeta</b><i> (<var>s</var>, <var>h</var>, <var>n</var>)</i>
<a name="IDX1151"></a>
</dt>
<dd><p>Returns the Hurwitz zeta function for the arguments <var>s</var> and <var>h</var>.
The return value is a big float (bfloat);
<var>n</var> is the number of digits in the return value.
</p>
<p>The Hurwitz zeta function is defined as
</p>
<pre class="example">                        inf
                        ====
                        \        1
         zeta (s,h)  =   &gt;    --------
                        /            s
                        ====  (k + h)
                        k = 0
</pre>
<p><code>load (&quot;bffac&quot;)</code> loads this function.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
 &middot;
<a href="maxima_95.html#Category_003a-Numerical-evaluation">Numerical evaluation</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-binomial"></a>
</p><dl>
<dt><u>Function:</u> <b>binomial</b><i> (<var>x</var>, <var>y</var>)</i>
<a name="IDX1152"></a>
</dt>
<dd><p>The binomial coefficient <code><var>x</var>!/(<var>y</var>! (<var>x</var> - <var>y</var>)!)</code>.
If <var>x</var> and <var>y</var> are integers, then the numerical value of the binomial
coefficient is computed.
If <var>y</var>, or <var>x - y</var>, is an integer,
the binomial coefficient is expressed as a polynomial.
</p>
<p>Examples:
</p>
<pre class="example">(%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)
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-burn"></a>
</p><dl>
<dt><u>Function:</u> <b>burn</b><i> (<var>n</var>)</i>
<a name="IDX1153"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Bernoulli number for integer <var>n</var>.
<code>burn</code> may be more efficient than <code>bern</code> for large, isolated <var>n</var>
(perhaps <var>n</var> greater than 105 or so), as <code>bern</code> computes all the Bernoulli numbers up to index <var>n</var> before returning.
</p>

<p><code>burn</code> exploits the observation that (rational) Bernoulli numbers can be
approximated by (transcendental) zetas with tolerable efficiency.
</p>
<p><code>load (&quot;bffac&quot;)</code> loads this function.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-cf"></a>
</p><dl>
<dt><u>Function:</u> <b>cf</b><i> (<var>expr</var>)</i>
<a name="IDX1154"></a>
</dt>
<dd><p>Converts <var>expr</var> into a continued fraction.
<var>expr</var> 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 <code>cf</code>.
</p>
<p><code>cf</code> evaluates its arguments after binding <code>listarith</code> to <code>false</code>.
<code>cf</code> returns a continued fraction, represented as a list.
</p>
<p>A continued fraction <code>a + 1/(b + 1/(c + ...))</code>
is represented by the list <code>[a, b, c, ...]</code>.
The list elements <code>a</code>, <code>b</code>, <code>c</code>, ... must evaluate to integers.
<var>expr</var> may also contain <code>sqrt (n)</code> where <code>n</code> is an integer.
In this case <code>cf</code> will give as many
terms of the continued fraction as the value of the variable
<code>cflength</code> times the period.
</p>
<p>A continued fraction can be evaluated to a number
by evaluating the arithmetic representation
returned by <code>cfdisrep</code>.
See also <code>cfexpand</code> for another way to evaluate a continued fraction.
</p>
<p>See also <code>cfdisrep</code>, <code>cfexpand</code>, and <code>cflength</code>.
</p>
<p>Examples:
</p>
<ul>
<li>
<var>expr</var> is an expression comprising continued fractions and square roots of integers.

<pre class="example">(%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]
</pre>
</li><li>
<code>cflength</code> controls how many periods of the continued fraction
are computed for algebraic, irrational numbers.

<pre class="example">(%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]
</pre>
</li><li>
A continued fraction can be evaluated by evaluating the arithmetic representation
returned by <code>cfdisrep</code>.

<pre class="example">(%i1) cflength: 3$
(%i2) cfdisrep (cf (sqrt (3)))$
(%i3) ev (%, numer);
(%o3)                   1.731707317073171
</pre>
</li><li>
Maxima does not know about operations on continued fractions outside of <code>cf</code>.

<pre class="example">(%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]
</pre>
</li></ul>

<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Continued-fractions">Continued fractions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-cfdisrep"></a>
</p><dl>
<dt><u>Function:</u> <b>cfdisrep</b><i> (<var>list</var>)</i>
<a name="IDX1155"></a>
</dt>
<dd><p>Constructs and returns an ordinary arithmetic expression
of the form <code>a + 1/(b + 1/(c + ...))</code>
from the list representation of a continued fraction <code>[a, b, c, ...]</code>.
</p>
<pre class="example">(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1)                     [1, 1, 1, 2]
(%i2) cfdisrep (%);
                                  1
(%o2)                     1 + ---------
                                    1
                              1 + -----
                                      1
                                  1 + -
                                      2
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Continued-fractions">Continued fractions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-cfexpand"></a>
</p><dl>
<dt><u>Function:</u> <b>cfexpand</b><i> (<var>x</var>)</i>
<a name="IDX1156"></a>
</dt>
<dd><p>Returns a matrix of the numerators and denominators of the
last (column 1) and next-to-last (column 2) convergents of the continued fraction <var>x</var>.
</p>
<pre class="example">(%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
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Continued-fractions">Continued fractions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-cflength"></a>
</p><dl>
<dt><u>Option variable:</u> <b>cflength</b>
<a name="IDX1157"></a>
</dt>
<dd><p>Default value: 1
</p>
<p><code>cflength</code> controls the number of terms of the continued
fraction the function <code>cf</code> will give, as the value <code>cflength</code> times the
period.  Thus the default is to give one period.
</p>
<pre class="example">(%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]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Continued-fractions">Continued fractions</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-divsum"></a>
</p><dl>
<dt><u>Function:</u> <b>divsum</b><i> (<var>n</var>, <var>k</var>)</i>
<a name="IDX1158"></a>
</dt>
<dt><u>Function:</u> <b>divsum</b><i> (<var>n</var>)</i>
<a name="IDX1159"></a>
</dt>
<dd><p><code>divsum (<var>n</var>, <var>k</var>)</code> returns the sum of the divisors of <var>n</var>
raised to the <var>k</var>'th power.
</p>
<p><code>divsum (<var>n</var>)</code> returns the sum of the divisors of <var>n</var>.
</p>
<pre class="example">(%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
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-euler"></a>
</p><dl>
<dt><u>Function:</u> <b>euler</b><i> (<var>n</var>)</i>
<a name="IDX1160"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Euler number for nonnegative integer <var>n</var>.
</p>
<p>For the Euler-Mascheroni constant, see <code>%gamma</code>.
</p>
<pre class="example">(%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]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-_0025gamma"></a>
</p><dl>
<dt><u>Constant:</u> <b>%gamma</b>
<a name="IDX1161"></a>
</dt>
<dd><p>The Euler-Mascheroni constant, 0.5772156649015329 ....
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Constants">Constants</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-factorial"></a>
</p><dl>
<dt><u>Function:</u> <b>factorial</b><i> (<var>x</var>)</i>
<a name="IDX1162"></a>
</dt>
<dd><p>Represents the factorial function. Maxima treats <code>factorial (<var>x</var>)</code> the same as <code><var>x</var>!</code>.
See <code>!</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-factorial_005fexpand"></a>
</p><dl>
<dt><u>Option variable:</u> <b>factorial_expand</b>
<a name="IDX1163"></a>
</dt>
<dd><p>Default value: false
</p>
<p>The option variable <code>factorial_expand</code> controls the simplification of 
expressions like <code>(n+1)!</code>, where <code>n</code> is an integer. 
See <code>!</code> for an example.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Gamma-and-factorial-functions">Gamma and factorial functions</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-fib"></a>
</p><dl>
<dt><u>Function:</u> <b>fib</b><i> (<var>n</var>)</i>
<a name="IDX1164"></a>
</dt>
<dd><p>Returns the <var>n</var>'th Fibonacci number.
<code>fib(0)</code> equal to 0 and <code>fib(1)</code> equal to 1,
and
<code>fib (-<var>n</var>)</code> equal to <code>(-1)^(<var>n</var> + 1) * fib(<var>n</var>)</code>.
</p>
<p>After calling <code>fib</code>,
<code>prevfib</code> is equal to <code>fib (<var>x</var> - 1)</code>,
the Fibonacci number preceding the last one computed.
</p>
<pre class="example">(%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]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-fibtophi"></a>
</p><dl>
<dt><u>Function:</u> <b>fibtophi</b><i> (<var>expr</var>)</i>
<a name="IDX1165"></a>
</dt>
<dd><p>Expresses Fibonacci numbers in <var>expr</var> in terms of the constant <code>%phi</code>,
which is <code>(1 + sqrt(5))/2</code>, approximately 1.61803399.
</p>
<p>Examples:
</p>
<pre class="example">(%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
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-ifactors"></a>
</p><dl>
<dt><u>Function:</u> <b>ifactors</b><i> (<var>n</var>)</i>
<a name="IDX1166"></a>
</dt>
<dd><p>For a positive integer <var>n</var> returns the factorization of <var>n</var>. If
<code>n=p1^e1..pk^nk</code> is the decomposition of <var>n</var> into prime
factors, ifactors returns <code>[[p1, e1], ... , [pk, ek]]</code>.
</p>
<p>Factorization methods used are trial divisions by primes up to 9973,
Pollard's rho method and elliptic curve method.
</p>
<pre class="example">(%i1) ifactors(51575319651600);
(%o1)     [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
(%i2) apply(&quot;*&quot;, map(lambda([u], u[1]^u[2]), %));
(%o2)                        51575319651600
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-inrt"></a>
</p><dl>
<dt><u>Function:</u> <b>inrt</b><i> (<var>x</var>, <var>n</var>)</i>
<a name="IDX1167"></a>
</dt>
<dd><p>Returns the integer <var>n</var>'th root of the absolute value of <var>x</var>.
</p>
<pre class="example">(%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]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-inv_005fmod"></a>
</p><dl>
<dt><u>Function:</u> <b>inv_mod</b><i> (<var>n</var>, <var>m</var>)</i>
<a name="IDX1168"></a>
</dt>
<dd><p>Computes the inverse of <var>n</var> modulo <var>m</var>. 
<code>inv_mod (n,m)</code> returns <code>false</code>, 
if <var>n</var> is a zero divisor modulo <var>m</var>.
</p>
<pre class="example">(%i1) inv_mod(3, 41);
(%o1)                           14
(%i2) ratsimp(3^-1), modulus=41;
(%o2)                           14
(%i3) inv_mod(3, 42);
(%o3)                          false
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-jacobi"></a>
</p><dl>
<dt><u>Function:</u> <b>jacobi</b><i> (<var>p</var>, <var>q</var>)</i>
<a name="IDX1169"></a>
</dt>
<dd><p>Returns the Jacobi symbol of <var>p</var> and <var>q</var>.
</p>
<pre class="example">(%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]
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-lcm"></a>
</p><dl>
<dt><u>Function:</u> <b>lcm</b><i> (<var>expr_1</var>, ..., <var>expr_n</var>)</i>
<a name="IDX1170"></a>
</dt>
<dd><p>Returns the least common multiple of its arguments.
The arguments may be general expressions as well as integers.
</p>
<p><code>load (&quot;functs&quot;)</code> loads this function.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-minfactorial"></a>
</p><dl>
<dt><u>Function:</u> <b>minfactorial</b><i> (<var>expr</var>)</i>
<a name="IDX1171"></a>
</dt>
<dd><p>Examines <var>expr</var> for occurrences of two factorials
which differ by an integer.
<code>minfactorial</code> then turns one into a polynomial times the other.
</p>

<pre class="example">(%i1) n!/(n+2)!;
                               n!
(%o1)                       --------
                            (n + 2)!
(%i2) minfactorial (%);
                                1
(%o2)                    ---------------
                         (n + 1) (n + 2)
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-next_005fprime"></a>
</p><dl>
<dt><u>Function:</u> <b>next_prime</b><i> (<var>n</var>)</i>
<a name="IDX1172"></a>
</dt>
<dd><p>Returns the smallest prime bigger than <var>n</var>.
</p>
<pre class="example">(%i1) next_prime(27);
(%o1)                       29
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-partfrac"></a>
</p><dl>
<dt><u>Function:</u> <b>partfrac</b><i> (<var>expr</var>, <var>var</var>)</i>
<a name="IDX1173"></a>
</dt>
<dd><p>Expands the expression <var>expr</var> in partial fractions
with respect to the main variable <var>var</var>.  <code>partfrac</code> 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.
</p>
<pre class="example">(%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)
</pre></dd></dl>

<p><a name="Item_003a-power_005fmod"></a>
</p><dl>
<dt><u>Function:</u> <b>power_mod</b><i> (<var>a</var>, <var>n</var>, <var>m</var>)</i>
<a name="IDX1174"></a>
</dt>
<dd><p>Uses a modular algorithm to compute <code>a^n mod m</code> 
where <var>a</var> and <var>n</var> are integers and <var>m</var> is a positive integer. 
If <var>n</var> is negative, <code>inv_mod</code> is used to find the modular inverse.
</p>
<pre class="example">(%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
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-primep"></a>
</p><dl>
<dt><u>Function:</u> <b>primep</b><i> (<var>n</var>)</i>
<a name="IDX1175"></a>
</dt>
<dd><p>Primality test. If <code>primep (<var>n</var>)</code> returns <code>false</code>, <var>n</var> is a
composite number and if it returns <code>true</code>, <var>n</var> is a prime number
with very high probability.
</p>
<p>For <var>n</var> less than 341550071728321 a deterministic version of
Miller-Rabin's test is used. If <code>primep (<var>n</var>)</code> returns
<code>true</code>, then <var>n</var> is a prime number.
</p>
<p>For <var>n</var> bigger than 341550071728321 <code>primep</code> uses
<code>primep_number_of_tests</code> Miller-Rabin's pseudo-primality tests and
one Lucas pseudo-primality test. The probability that <var>n</var> will pass
one Miller-Rabin test is less than 1/4. Using the default value 25 for
<code>primep_number_of_tests</code>, the probability of <var>n</var> beeing
composite is much smaller that 10^-15.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-primep_005fnumber_005fof_005ftests"></a>
</p><dl>
<dt><u>Option variable:</u> <b>primep_number_of_tests</b>
<a name="IDX1176"></a>
</dt>
<dd><p>Default value: 25
</p>
<p>Number of Miller-Rabin's tests used in <code>primep</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Predicate-functions">Predicate functions</a>
 &middot;
<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>


</dd></dl>

<p><a name="Item_003a-prev_005fprime"></a>
</p><dl>
<dt><u>Function:</u> <b>prev_prime</b><i> (<var>n</var>)</i>
<a name="IDX1177"></a>
</dt>
<dd><p>Returns the greatest prime smaller than <var>n</var>.
</p>
<pre class="example">(%i1) prev_prime(27);
(%o1)                       23
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-qunit"></a>
</p><dl>
<dt><u>Function:</u> <b>qunit</b><i> (<var>n</var>)</i>
<a name="IDX1178"></a>
</dt>
<dd><p>Returns the principal unit of the real quadratic number field
<code>sqrt (<var>n</var>)</code> where <var>n</var> is an integer,
i.e., the element whose norm is unity.
This amounts to solving Pell's equation <code>a^2 - <var>n</var> b^2 = 1</code>.
</p>
<pre class="example">(%i1) qunit (17);
(%o1)                     sqrt(17) + 4
(%i2) expand (% * (sqrt(17) - 4));
(%o2)                           1
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-totient"></a>
</p><dl>
<dt><u>Function:</u> <b>totient</b><i> (<var>n</var>)</i>
<a name="IDX1179"></a>
</dt>
<dd><p>Returns the number of integers less than or equal to <var>n</var> which
are relatively prime to <var>n</var>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-zerobern"></a>
</p><dl>
<dt><u>Option variable:</u> <b>zerobern</b>
<a name="IDX1180"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>zerobern</code> is <code>false</code>, <code>bern</code> excludes the Bernoulli numbers
and <code>euler</code> excludes the Euler numbers which are equal to zero. 
See <code>bern</code> and <code>euler</code>.
</p>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-zeta"></a>
</p><dl>
<dt><u>Function:</u> <b>zeta</b><i> (<var>n</var>)</i>
<a name="IDX1181"></a>
</dt>
<dd><p>Returns the Riemann zeta function. If <var>n</var> 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 <code>zeta%pi</code> has to be
<code>true</code> in addition (See <code>zeta%pi</code>). For a floating point or bigfloat 
number the Riemann zeta function is evaluated numerically. Maxima returns a noun
form <code>zeta (<var>n</var>)</code> for all other arguments, including rational 
noninteger, and complex arguments, or for even integers, if <code>zeta%pi</code> has 
the value <code>false</code>.
</p>
<p><code>zeta(1)</code> is undefined, but Maxima knows the limit 
<code>limit(zeta(x), x, 1)</code> from above and below.
</p>
<p>The Riemann zeta function distributes over lists, matrices, and equations.
</p>
<p>See also <code>bfzeta</code> and <code>zeta%pi</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) zeta([-2,-1,0,0.5,2,3,1+%i]);
                                              2
             1     1                       %pi
(%o1)  [0, - --, - -, - 1.460354508809587, ----, zeta(3), zeta(%i + 1)]
             12    2                        6 

(%i2) limit(zeta(x),x,1,plus);
(%o2)                                 inf
(%i3) limit(zeta(x),x,1,minus);
(%o3)                                minf
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>

<p><a name="Item_003a-zeta_0025pi"></a>
</p><dl>
<dt><u>Option variable:</u> <b>zeta%pi</b>
<a name="IDX1182"></a>
</dt>
<dd><p>Default value: <code>true</code>
</p>
<p>When <code>zeta%pi</code> is <code>true</code>, <code>zeta</code> returns an expression 
proportional to <code>%pi^n</code> for even integer <code>n</code>. Otherwise, <code>zeta</code> 
returns a noun form <code>zeta (n)</code> for even integer <code>n</code>.
</p>
<p>Examples:
</p>
<pre class="example">(%i1) zeta%pi: true$
(%i2) zeta (4);
                                 4
                              %pi
(%o2)                         ----
                               90
(%i3) zeta%pi: false$
(%i4) zeta (4);
(%o4)                        zeta(4)
</pre>
<div class=categorybox>


<p>Categories:&nbsp;&nbsp;<a href="maxima_95.html#Category_003a-Number-theory">Number theory</a>
</p>
</div>

</dd></dl>


<p><a name="Item_003a-Symmetries"></a>
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