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[Maxima-commits] [git] Maxima CAS branch rtoy-number-update-examples created. branch-5_48-base-160-g37a342961
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From: rtoy <rt...@us...> - 2025-09-03 01:27:02
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This is an automated email from the git hooks/post-receive script. It was
generated because a ref change was pushed to the repository containing
the project "Maxima CAS".
The branch, rtoy-number-update-examples has been created
at 37a3429614bd68b5faaa99f7b9b24828501de5d2 (commit)
- Log -----------------------------------------------------------------
commit 37a3429614bd68b5faaa99f7b9b24828501de5d2
Author: Raymond Toy <toy...@gm...>
Date: Tue Sep 2 18:26:34 2025 -0700
Add comment blocks for examples and regenerate them
Also marked some examples with "maxima" to get syntax highlighting.
These particular ones don't get code blocks since they're not really
code examples to be executed.
diff --git a/doc/info/Number.texi b/doc/info/Number.texi
index 9d4fb32d9..ad2fa4f88 100644
--- a/doc/info/Number.texi
+++ b/doc/info/Number.texi
@@ -227,6 +227,7 @@ Examples:
@var{expr} is an expression comprising continued fractions and square roots of
integers.
+@c The second example below no longer works.
@example
(%i1) cf ([5, 3, 1]*[11, 9, 7] + [3, 7]/[4, 3, 2]);
(%o1) [59, 17, 2, 1, 1, 1, 27]
@@ -932,15 +933,23 @@ If @var{n} is negative, @code{inv_mod} is used to find the modular inverse.
@c power_mod(2, -1, 5);
@c inv_mod(2,5);
@c ===end===
-@example
+@example maxima
+@group
(%i1) power_mod(3, 15, 5);
-(%o1) 2
+(%o1) 2
+@end group
+@group
(%i2) mod(3^15,5);
-(%o2) 2
+(%o2) 2
+@end group
+@group
(%i3) power_mod(2, -1, 5);
-(%o3) 3
+(%o3) 3
+@end group
+@group
(%i4) inv_mod(2,5);
-(%o4) 3
+(%o4) 3
+@end group
@end example
@opencatbox{Categories:}
@@ -994,9 +1003,11 @@ Returns the list of all primes from @var{start} to @var{end}.
@c ===beg===
@c primes(3, 7);
@c ===end===
-@example
+@example maxima
+@group
(%i1) primes(3, 7);
-(%o1) [3, 5, 7]
+(%o1) [3, 5, 7]
+@end group
@end example
@opencatbox{Categories:}
@@ -1013,9 +1024,11 @@ Returns the greatest prime smaller than @var{n}.
@c ===beg===
@c prev_prime(27);
@c ===end===
-@example
+@example maxima
+@group
(%i1) prev_prime(27);
-(%o1) 23
+(%o1) 23
+@end group
@end example
@opencatbox{Categories:}
@@ -1036,11 +1049,15 @@ This amounts to solving Pell's equation @code{a^2 - @var{n} b^2 = 1}.
@c qunit (17);
@c expand (% * (sqrt(17) - 4));
@c ===end===
-@example
+@example maxima
+@group
(%i1) qunit (17);
(%o1) sqrt(17) + 4
+@end group
+@group
(%i2) expand (% * (sqrt(17) - 4));
(%o2) 1
+@end group
@end example
@opencatbox{Categories:}
@@ -1234,17 +1251,27 @@ can be generated by two factor generators.
@c UG2: makelist(power_mod(g2,i,15), i,0,f2-1);
@c M15: create_list(mod(i*j,15), i,UG1, j,UG2);
@c ===end===
-@example
+@example maxima
+@group
(%i1) [[f1,f2]: zn_characteristic_factors(15), totient(15)];
-(%o1) [[2, 4], 8]
+(%o1) [[2, 4], 8]
+@end group
+@group
(%i2) [[g1,g2]: zn_factor_generators(15), zn_primroot(15)];
-(%o2) [[11, 7], false]
+(%o2) [[11, 7], false]
+@end group
+@group
(%i3) UG1: makelist(power_mod(g1,i,15), i,0,f1-1);
-(%o3) [1, 11]
+(%o3) [1, 11]
+@end group
+@group
(%i4) UG2: makelist(power_mod(g2,i,15), i,0,f2-1);
-(%o4) [1, 7, 4, 13]
+(%o4) [1, 7, 4, 13]
+@end group
+@group
(%i5) M15: create_list(mod(i*j,15), i,UG1, j,UG2);
-(%o5) [1, 7, 4, 13, 11, 2, 14, 8]
+(%o5) [1, 7, 4, 13, 11, 2, 14, 8]
+@end group
@end example
For the last characteristic factor @code{4} it holds that @code{a^4 = 1 (mod 15)}
@@ -1322,21 +1349,29 @@ Examples:
@c m : matrix([2,4,1],[3,1,4],[4,3,2]);
@c zn_determinant(m, 5);
@c ===end===
-@example
+@example maxima
+@group
(%i1) m : matrix([1,3],[2,4]);
- [ 1 3 ]
-(%o1) [ ]
- [ 2 4 ]
+ [ 1 3 ]
+(%o1) [ ]
+ [ 2 4 ]
+@end group
+@group
(%i2) zn_determinant(m, 5);
-(%o2) 3
+(%o2) 3
+@end group
+@group
(%i3) m : matrix([2,4,1],[3,1,4],[4,3,2]);
- [ 2 4 1 ]
- [ ]
-(%o3) [ 3 1 4 ]
- [ ]
- [ 4 3 2 ]
+ [ 2 4 1 ]
+ [ ]
+(%o3) [ 3 1 4 ]
+ [ ]
+ [ 4 3 2 ]
+@end group
+@group
(%i4) zn_determinant(m, 5);
-(%o4) 0
+(%o4) 0
+@end group
@end example
@opencatbox{Categories:}
@@ -1377,21 +1412,29 @@ Example:
@c mi : zn_invert_by_lu(m, 5);
@c matrixmap(lambda([a], mod(a, 5)), m . mi);
@c ===end===
-@example
+@example maxima
+@group
(%i1) m : matrix([1,3],[2,4]);
- [ 1 3 ]
-(%o1) [ ]
- [ 2 4 ]
+ [ 1 3 ]
+(%o1) [ ]
+ [ 2 4 ]
+@end group
+@group
(%i2) zn_determinant(m, 5);
-(%o2) 3
+(%o2) 3
+@end group
+@group
(%i3) mi : zn_invert_by_lu(m, 5);
- [ 3 4 ]
-(%o3) [ ]
- [ 1 2 ]
+ [ 3 4 ]
+(%o3) [ ]
+ [ 1 2 ]
+@end group
+@group
(%i4) matrixmap(lambda([a], mod(a, 5)), m . mi);
- [ 1 0 ]
-(%o4) [ ]
- [ 0 1 ]
+ [ 1 0 ]
+(%o4) [ ]
+ [ 0 1 ]
+@end group
@end example
@opencatbox{Categories:}
@@ -1441,24 +1484,40 @@ Examples:
@c powers_5 : makelist(power_mod(5,x,n), x, 0, ord_5 - 1);
@c zn_log(9, 5, n);
@c ===end===
-@example
+@example maxima
(%i1) n : 22$
+@group
(%i2) g : zn_primroot(n);
-(%o2) 7
+(%o2) 7
+@end group
+@group
(%i3) ord_7 : zn_order(7, n);
-(%o3) 10
+(%o3) 10
+@end group
+@group
(%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]
+(%o4) [1, 7, 5, 13, 3, 21, 15, 17, 9, 19]
+@end group
+@group
(%i5) zn_log(9, g, n);
-(%o5) 8
+(%o5) 8
+@end group
+@group
(%i6) map(lambda([x], zn_log(x, g, n)), powers_7);
-(%o6) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
+(%o6) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
+@end group
+@group
(%i7) ord_5 : zn_order(5, n);
-(%o7) 5
+(%o7) 5
+@end group
+@group
(%i8) powers_5 : makelist(power_mod(5,x,n), x, 0, ord_5 - 1);
-(%o8) [1, 5, 3, 15, 9]
+(%o8) [1, 5, 3, 15, 9]
+@end group
+@group
(%i9) zn_log(9, 5, n);
-(%o9) 4
+(%o9) 4
+@end group
@end example
The optional fourth argument must be of the same form as the list returned by
@@ -1553,29 +1612,33 @@ elements in (Z/@var{n}Z).
@c zn_mult_table(8);
@c zn_mult_table(8, all);
@c ===end===
-@example
+@example maxima
+@group
(%i1) zn_mult_table(8);
- [ 1 3 5 7 ]
- [ ]
- [ 3 1 7 5 ]
-(%o1) [ ]
- [ 5 7 1 3 ]
- [ ]
- [ 7 5 3 1 ]
+ [ 1 3 5 7 ]
+ [ ]
+ [ 3 1 7 5 ]
+(%o1) [ ]
+ [ 5 7 1 3 ]
+ [ ]
+ [ 7 5 3 1 ]
+@end group
+@group
(%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 ]
+ [ 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 ]
+@end group
@end example
If @var{gcd} is an integer, row and column headings are added for better readability.
@@ -1592,45 +1655,51 @@ the top left corner, just like in table @code{T9}.
@c T9: zn_mult_table(36/4);
@c T36_4: matrixmap(lambda([x], solve_congruences([0,x],[4,9])), T9);
@c ===end===
-@example
+@example maxima
+@group
(%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 ]
+ [ * 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 ]
+@end group
+@group
(%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 ]
+ [ 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 ]
+@end group
+@group
(%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 ]
+ [ 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 ]
+@end group
@end example
@opencatbox{Categories:}
@@ -1672,21 +1741,26 @@ with @var{n} from @code{1} to @code{6}.
@c zn_power_table(14);
@c makelist(zn_nth_root(1,n,14), n,1,6);
@c ===end===
-@example
+@example maxima
+@group
(%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 ]
+ [ 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 ]
+@end group
+@group
(%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]]
+(%o2) [[1], [1, 13], [1, 9, 11], [1, 13], [1],
+ [1, 3, 5, 9, 11, 13]]
+@end group
@end example
In the following example @var{x} is not coprime to @var{m},
@@ -1749,17 +1823,15 @@ In this case @code{zn_nth_root} effectively performs CRT-RSA.
@c zs: flatten(map(lambda([y], zn_nth_root(y,e,N)), ys))$
@c is(zs = xs);
@c ===end===
-@example
+@example 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))$
-
+@group
(%i6) is(zs = xs);
-(%o6) true
+(%o6) true
+@end group
@end example
In the following example the factorization of the modulus is known
@@ -1822,22 +1894,36 @@ Examples:
@c map(lambda([x], ord_7/gcd(x,ord_7)), makelist(i,i,0,ord_7-1));
@c totient(totient(n));
@c ===end===
-@example
+@example maxima
(%i1) n: 22$
+@group
(%i2) g: zn_primroot(n);
-(%o2) 7
+(%o2) 7
+@end group
+@group
(%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]
+@end group
+@group
(%i4) (ord_7: zn_order(7, n)) = totient(n);
(%o4) 10 = 10
+@end group
+@group
(%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]
+@end group
+@group
(%i6) map(lambda([x], zn_order(x, n)), powers_7);
(%o6) [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
+@end group
+@group
(%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]
+(%o7) [1, 10, 5, 10, 5, 2, 5, 10, 5, 10]
+@end group
+@group
(%i8) totient(totient(n));
-(%o8) 4
+(%o8) 4
+@end group
@end example
The optional third argument must be of the same form as the list returned by
@@ -1850,16 +1936,24 @@ The optional third argument must be of the same form as the list returned by
@c is( (ord_3 : zn_order(g, p, ifs)) = totient(p) );
@c map(lambda([x], ord_3/zn_order(x,p,ifs)), makelist(i,i,2,15));
@c ===end===
-@example
+@example maxima
+@group
(%i1) (p : 2^142 + 217, primep(p));
(%o1) true
+@end group
(%i2) ifs: ifactors( totient(p) )$
+@group
(%i3) g: zn_primroot(p, ifs);
(%o3) 3
+@end group
+@group
(%i4) is( (ord_3 : zn_order(g, p, ifs)) = totient(p) );
(%o4) true
+@end group
+@group
(%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]
+@end group
@end example
@opencatbox{Categories:}
@@ -2028,36 +2122,71 @@ Examples:
@code{zn_primroot} computes the smallest primitive root modulo @var{n} or returns
@code{false}.
-@example
+@c ===beg===
+@c n : 14$
+@c g : zn_primroot(n);
+@c zn_order(g, n) = totient(n);
+@c n : 15$
+@c zn_primroot(n);
+@c ===end===
+@example maxima
(%i1) n : 14$
+@group
(%i2) g : zn_primroot(n);
-(%o2) 3
+(%o2) 3
+@end group
+@group
(%i3) zn_order(g, n) = totient(n);
-(%o3) 6 = 6
+(%o3) 6 = 6
+@end group
(%i4) n : 15$
+@group
(%i5) zn_primroot(n);
-(%o5) false
+(%o5) false
+@end group
@end example
The optional second argument must be of the same form as the list returned by
@code{ifactors(totient(n))}.
-@example
+@c NOTE: This is a bit slow. I think the main cause is running
+@c ifactors.
+@c
+@c ===beg===
+@c (p : 2^142 + 217, primep(p));
+@c ifs : ifactors( totient(p) )$
+@c g : zn_primroot(p, ifs);
+@c [time(%o2), time(%o3)];
+@c is(zn_order(g, p, ifs) = p - 1);
+@c n : 2^142 + 216$
+@c ifs : ifactors(totient(n))$
+@c zn_primroot(n, ifs), zn_primroot_limit : 200, zn_primroot_verbose : true;
+@c ===end===
+@example maxima
+@group
(%i1) (p : 2^142 + 217, primep(p));
-(%o1) true
+(%o1) true
+@end group
(%i2) ifs : ifactors( totient(p) )$
+@group
(%i3) g : zn_primroot(p, ifs);
-(%o3) 3
+(%o3) 3
+@end group
+@group
(%i4) [time(%o2), time(%o3)];
-(%o4) [[15.556972], [0.004]]
+(%o4) [[3.2], [0.0]]
+@end group
+@group
(%i5) is(zn_order(g, p, ifs) = p - 1);
-(%o5) true
+(%o5) true
+@end group
(%i6) n : 2^142 + 216$
(%i7) ifs : ifactors(totient(n))$
-(%i8) zn_primroot(n, ifs),
- zn_primroot_limit : 200, zn_primroot_verbose : true;
+@group
+(%i8) zn_primroot(n, ifs), zn_primroot_limit : 200, zn_primroot_verbose : true;
`zn_primroot' stopped at zn_primroot_limit = 200
-(%o8) false
+(%o8) false
+@end group
@end example
@opencatbox{Categories:}
@@ -2100,29 +2229,56 @@ Examples:
@code{zn_primroot_p} as a predicate function.
-@example
+@c ===beg===
+@c n : 14$
+@c units_14 : sublist(makelist(i,i,1,13), lambda([i], gcd(i, n) = 1));
+@c zn_primroot_p(13, n);
+@c sublist(units_14, lambda([x], zn_primroot_p(x, n)));
+@c map(lambda([x], zn_order(x, n)), units_14);
+@c ===end===
+@example maxima
(%i1) n : 14$
+@group
(%i2) units_14 : sublist(makelist(i,i,1,13), lambda([i], gcd(i, n) = 1));
-(%o2) [1, 3, 5, 9, 11, 13]
+(%o2) [1, 3, 5, 9, 11, 13]
+@end group
+@group
(%i3) zn_primroot_p(13, n);
-(%o3) false
+(%o3) false
+@end group
+@group
(%i4) sublist(units_14, lambda([x], zn_primroot_p(x, n)));
-(%o4) [3, 5]
+(%o4) [3, 5]
+@end group
+@group
(%i5) map(lambda([x], zn_order(x, n)), units_14);
-(%o5) [1, 6, 6, 3, 3, 2]
+(%o5) [1, 6, 6, 3, 3, 2]
+@end group
@end example
The optional third argument must be of the same form as the list returned by
@code{ifactors(totient(n))}.
-@example
+@c ===beg===
+@c (p: 2^142 + 217, primep(p));
+@c ifs: ifactors( totient(p) )$
+@c sublist(makelist(i,i,1,50), lambda([x], zn_primroot_p(x,p,ifs)));
+@c [time(%o2), time(%o3)];
+@c ===end===
+@example maxima
+@group
(%i1) (p: 2^142 + 217, primep(p));
(%o1) true
+@end group
(%i2) ifs: ifactors( totient(p) )$
+@group
(%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]
+@end group
+@group
(%i4) [time(%o2), time(%o3)];
-(%o4) [[7.748484], [0.036002]]
+(%o4) [[3.01], [0.03]]
+@end group
@end example
@opencatbox{Categories:}
commit b9389997f2428d79151e6e8e0aacc39d9e01905d
Author: Raymond Toy <toy...@gm...>
Date: Tue Sep 2 15:32:48 2025 -0700
Add comment blocks for examples and regenerate them
Also marked some examples with "maxima" to get syntax highlighting.
These particular ones don't get code blocks since they're not really
code examples to be executed.
There are still examples that need comment blocks so they can be
regenerated.
diff --git a/doc/info/Number.texi b/doc/info/Number.texi
index 58cd02b0c..9d4fb32d9 100644
--- a/doc/info/Number.texi
+++ b/doc/info/Number.texi
@@ -926,6 +926,12 @@ Uses a modular algorithm to compute @code{a^n mod m}
where @var{a} and @var{n} are integers and @var{m} is a positive integer.
If @var{n} is negative, @code{inv_mod} is used to find the modular inverse.
+@c ===beg===
+@c power_mod(3, 15, 5);
+@c mod(3^15,5);
+@c power_mod(2, -1, 5);
+@c inv_mod(2,5);
+@c ===end===
@example
(%i1) power_mod(3, 15, 5);
(%o1) 2
@@ -985,6 +991,9 @@ Number of Miller-Rabin's tests used in @code{primep}.
Returns the list of all primes from @var{start} to @var{end}.
+@c ===beg===
+@c primes(3, 7);
+@c ===end===
@example
(%i1) primes(3, 7);
(%o1) [3, 5, 7]
@@ -1001,6 +1010,9 @@ Returns the list of all primes from @var{start} to @var{end}.
Returns the greatest prime smaller than @var{n}.
+@c ===beg===
+@c prev_prime(27);
+@c ===end===
@example
(%i1) prev_prime(27);
(%o1) 23
@@ -1020,6 +1032,10 @@ Returns the principal unit of the real quadratic number field
i.e., the element whose norm is unity.
This amounts to solving Pell's equation @code{a^2 - @var{n} b^2 = 1}.
+@c ===beg===
+@c qunit (17);
+@c expand (% * (sqrt(17) - 4));
+@c ===end===
@example
(%i1) qunit (17);
(%o1) sqrt(17) + 4
@@ -1188,18 +1204,36 @@ Examples:
The multiplication group modulo @code{14} is cyclic and its @code{6} elements
can be generated by a primitive root.
-@example
+@c ===beg===
+@c [zn_characteristic_factors(14), phi: totient(14)];
+@c [zn_factor_generators(14), g: zn_primroot(14)];
+@c M14: makelist(power_mod(g,i,14), i,0,phi-1);
+@c ===end===
+@example maxima
+@group
(%i1) [zn_characteristic_factors(14), phi: totient(14)];
-(%o1) [[6], 6]
+(%o1) [[6], 6]
+@end group
+@group
(%i2) [zn_factor_generators(14), g: zn_primroot(14)];
-(%o2) [[3], 3]
+(%o2) [[3], 3]
+@end group
+@group
(%i3) M14: makelist(power_mod(g,i,14), i,0,phi-1);
-(%o3) [1, 3, 9, 13, 11, 5]
+(%o3) [1, 3, 9, 13, 11, 5]
+@end group
@end example
The multiplication group modulo @code{15} is not cyclic and its @code{8} elements
can be generated by two factor generators.
+@c ===beg===
+@c [[f1,f2]: zn_characteristic_factors(15), totient(15)];
+@c [[g1,g2]: zn_factor_generators(15), zn_primroot(15)];
+@c UG1: makelist(power_mod(g1,i,15), i,0,f1-1);
+@c UG2: makelist(power_mod(g2,i,15), i,0,f2-1);
+@c M15: create_list(mod(i*j,15), i,UG1, j,UG2);
+@c ===end===
@example
(%i1) [[f1,f2]: zn_characteristic_factors(15), totient(15)];
(%o1) [[2, 4], 8]
@@ -1219,25 +1253,33 @@ for all @code{a} in @code{M15}.
@code{M15} has two characteristic factors and therefore @code{8/2^2} quadratic residues,
and each of these has @code{2^2} square roots.
-@example
-(%i6) zn_power_table(15);
- [ 1 1 1 1 ]
- [ ]
- [ 2 4 8 1 ]
- [ ]
- [ 4 1 4 1 ]
- [ ]
- [ 7 4 13 1 ]
-(%o6) [ ]
- [ 8 4 2 1 ]
- [ ]
- [ 11 1 11 1 ]
- [ ]
- [ 13 4 7 1 ]
- [ ]
- [ 14 1 14 1 ]
-(%i7) map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
-(%o7) [[1, 4, 11, 14], [2, 7, 8, 13]]
+@c ===beg===
+@c zn_power_table(15);
+@c map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
+@c ===end===
+@example maxima
+@group
+(%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 ]
+@end group
+@group
+(%i2) map(lambda([i], zn_nth_root(i,2,15)), [1,4]);
+(%o2) [[1, 4, 11, 14], [2, 7, 8, 13]]
+@end group
@end example
@opencatbox{Categories:}
@@ -1274,6 +1316,12 @@ See also @mrefdot{zn_invert_by_lu}
Examples:
+@c ===beg===
+@c m : matrix([1,3],[2,4]);
+@c zn_determinant(m, 5);
+@c m : matrix([2,4,1],[3,1,4],[4,3,2]);
+@c zn_determinant(m, 5);
+@c ===end===
@example
(%i1) m : matrix([1,3],[2,4]);
[ 1 3 ]
@@ -1323,6 +1371,12 @@ See also @mrefdot{zn_determinant}
Example:
+@c ===beg===
+@c m : matrix([1,3],[2,4]);
+@c zn_determinant(m, 5);
+@c mi : zn_invert_by_lu(m, 5);
+@c matrixmap(lambda([a], mod(a, 5)), m . mi);
+@c ===end===
@example
(%i1) m : matrix([1,3],[2,4]);
[ 1 3 ]
@@ -1376,6 +1430,17 @@ Examples:
@code{zn_log (a, g, n)} solves the congruence @code{g^x = a mod n}.
+@c ===beg===
+@c n : 22$
+@c g : zn_primroot(n);
+@c ord_7 : zn_order(7, n);
+@c powers_7 : makelist(power_mod(g, x, n), x, 0, ord_7 - 1);
+@c zn_log(9, g, n);
+@c map(lambda([x], zn_log(x, g, n)), powers_7);
+@c ord_5 : zn_order(5, n);
+@c powers_5 : makelist(power_mod(5,x,n), x, 0, ord_5 - 1);
+@c zn_log(9, 5, n);
+@c ===end===
@example
(%i1) n : 22$
(%i2) g : zn_primroot(n);
@@ -1400,24 +1465,49 @@ The optional fourth argument must be of the same form as the list returned by
@code{ifactors(zn_order(g))}.
The running time primarily depends on the bitlength of the totient's greatest prime factor.
-@example
+@c ===beg===
+@c (p : 2^127-1, primep(p));
+@c ifs : ifactors(p - 1)$
+@c g : zn_primroot(p, ifs);
+@c a : power_mod(g, 4711, p)$
+@c zn_log(a, g, p, ifs);
+@c f_max : last(ifs);
+@c ord_5 : zn_order(5,p,ifs)$
+@c (p - 1)/ord_5;
+@c ifs_5 : ifactors(ord_5)$
+@c a : power_mod(5, 4711, p)$
+@c zn_log(a, 5, p, ifs_5);
+@c ===end===
+@example maxima
+@group
(%i1) (p : 2^127-1, primep(p));
-(%o1) true
+(%o1) true
+@end group
(%i2) ifs : ifactors(p - 1)$
+@group
(%i3) g : zn_primroot(p, ifs);
-(%o3) 43
+(%o3) 43
+@end group
(%i4) a : power_mod(g, 4711, p)$
+@group
(%i5) zn_log(a, g, p, ifs);
-(%o5) 4711
-(%i6) f_max : last(ifs);
-(%o6) [77158673929, 1]
+(%o5) 4711
+@end group
+@group
+(%i6) f_max : last(ifs);
+(%o6) [77158673929, 1]
+@end group
(%i7) ord_5 : zn_order(5,p,ifs)$
+@group
(%i8) (p - 1)/ord_5;
-(%o8) 73
+(%o8) 73
+@end group
(%i9) ifs_5 : ifactors(ord_5)$
(%i10) a : power_mod(5, 4711, p)$
+@group
(%i11) zn_log(a, 5, p, ifs_5);
-(%o11) 4711
+(%o11) 4711
+@end group
@end example
@opencatbox{Categories:}
@@ -1459,6 +1549,10 @@ each row and column contains every element, the tables are symmetric, etc..
If @var{gcd} is set to @code{all}, the table is printed for all non-zero
elements in (Z/@var{n}Z).
+@c ===beg===
+@c zn_mult_table(8);
+@c zn_mult_table(8, all);
+@c ===end===
@example
(%i1) zn_mult_table(8);
[ 1 3 5 7 ]
@@ -1493,6 +1587,11 @@ identity builds a table which is easy to read. The mapping is accomplished via C
In the second version of @code{T36_4} the identity, here @code{28}, is placed in
the top left corner, just like in table @code{T9}.
+@c ===beg===
+@c T36_4: zn_mult_table(36,4);
+@c T9: zn_mult_table(36/4);
+@c T36_4: matrixmap(lambda([x], solve_congruences([0,x],[4,9])), T9);
+@c ===end===
@example
(%i1) T36_4: zn_mult_table(36,4);
[ * 4 8 16 20 28 32 ]
@@ -1569,6 +1668,10 @@ A power table of the multiplication group modulo @code{14}
followed by a list of lists containing all @var{n}-th roots of @code{1}
with @var{n} from @code{1} to @code{6}.
+@c ===beg===
+@c zn_power_table(14);
+@c makelist(zn_nth_root(1,n,14), n,1,6);
+@c ===end===
@example
(%i1) zn_power_table(14);
[ 1 1 1 1 1 1 ]
@@ -1597,25 +1700,36 @@ Here @code{zn_power_table} shows all residues @code{x} in (Z/63Z)
with @code{gcd(x,63) = 9}. This subgroup is isomorphic to (Z/7Z)*
and its identity @code{36} is computed via CRT.
-@example
+@c ===beg===
+@c m: 7*9$
+@c zn_power_table(m,9);
+@c zn_nth_root(27,3,m);
+@c id7:1$ id63_9: solve_congruences([id7,0],[7,9]);
+@c ===end===
+@example maxima
(%i1) m: 7*9$
-
+@group
(%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 ]
+ [ 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 ]
+@end group
+@group
(%i3) zn_nth_root(27,3,m);
-(%o3) [27, 45, 54]
+(%o3) [27, 45, 54]
+@end group
+@group
(%i4) id7:1$ id63_9: solve_congruences([id7,0],[7,9]);
-(%o5) 36
+(%o5) 36
+@end group
@end example
In the following RSA-like example, where the modulus @code{N} is squarefree,
@@ -1628,6 +1742,13 @@ The process of decryption needs the @code{e}-th root.
In this case @code{zn_nth_root} effectively performs CRT-RSA.
(Please note that @code{flatten} removes braces but no solutions.)
+@c ===beg===
+@c [p,q,e]: [5,7,17]$ N: p*q$
+@c xs: makelist(x,x,0,N-1)$
+@c ys: map(lambda([x],power_mod(x,e,N)),xs)$
+@c zs: flatten(map(lambda([y], zn_nth_root(y,e,N)), ys))$
+@c is(zs = xs);
+@c ===end===
@example
(%i1) [p,q,e]: [5,7,17]$ N: p*q$
@@ -1644,17 +1765,25 @@ In this case @code{zn_nth_root} effectively performs CRT-RSA.
In the following example the factorization of the modulus is known
and passed as the fourth argument.
-@example
+@c ===beg===
+@c p: 2^107-1$ q: 2^127-1$ N: p*q$
+@c ibase: obase: 16$
+@c msg: 11223344556677889900aabbccddeeff$
+@c enc: power_mod(msg, 10001, N);
+@c zn_nth_root(enc, 10001, N, [[p,1],[q,1]]);
+@c ===end===
+@example maxima
(%i1) p: 2^107-1$ q: 2^127-1$ N: p*q$
-
(%i4) ibase: obase: 16$
-
(%i5) msg: 11223344556677889900aabbccddeeff$
-
+@group
(%i6) enc: power_mod(msg, 10001, N);
-(%o6) 1a8db7892ae588bdc2be25dd5107a425001fe9c82161abc673241c8b383
+(%o6) 1A8DB7892AE588BDC2BE25DD5107A425001FE9C82161ABC673241C8B383
+@end group
+@group
(%i7) zn_nth_root(enc, 10001, N, [[p,1],[q,1]]);
-(%o7) [11223344556677889900aabbccddeeff]
+(%o7) [11223344556677889900AABBCCDDEEFF]
+@end group
@end example
@opencatbox{Categories:}
@@ -1683,6 +1812,16 @@ Examples:
@code{zn_order} computes the order of the unit @var{x} in (Z/@var{n}Z)*.
+@c ===beg===
+@c n: 22$
+@c g: zn_primroot(n);
+@c units_22: sublist(makelist(i,i,1,21), lambda([x], gcd(x,n)=1));
+@c (ord_7: zn_order(7, n)) = totient(n);
+@c powers_7: makelist(power_mod(g,i,n), i,0,ord_7 - 1);
+@c map(lambda([x], zn_order(x, n)), powers_7);
+@c map(lambda([x], ord_7/gcd(x,ord_7)), makelist(i,i,0,ord_7-1));
+@c totient(totient(n));
+@c ===end===
@example
(%i1) n: 22$
(%i2) g: zn_primroot(n);
@@ -1704,6 +1843,13 @@ Examples:
The optional third argument must be of the same form as the list returned by
@code{ifactors(totient(n))}.
+@c ===beg===
+@c (p : 2^142 + 217, primep(p));
+@c ifs: ifactors( totient(p) )$
+@c g: zn_primroot(p, ifs);
+@c is( (ord_3 : zn_order(g, p, ifs)) = totient(p) );
+@c map(lambda([x], ord_3/zn_order(x,p,ifs)), makelist(i,i,2,15));
+@c ===end===
@example
(%i1) (p : 2^142 + 217, primep(p));
(%o1) true
@@ -1759,33 +1905,56 @@ isomorphic to @code{G5}. @code{1} is the identity in @code{G5}.
So are @code{1} resp. @code{6} the identities in @code{G10} resp. @code{G10_2}.
There are corresponding mappings for primitive roots, n-th roots, etc..
-@example
+@c ===beg===
+@c zn_power_table(10);
+@c zn_power_table(10,2);
+@c zn_power_table(10,5);
+@c zn_power_table(10,10);
+@c G5: [1,2,3,4];
+@c G10_2: map(lambda([x], solve_congruences([0,x],[2,5])), G5);
+@c G10: map(lambda([x], power_mod(3, zn_log(x,2,5), 10)), G5);
+@c ===end===
+@example maxima
+@group
(%i1) zn_power_table(10);
- [ 1 1 1 1 ]
- [ ]
- [ 3 9 7 1 ]
-(%o1) [ ]
- [ 7 9 3 1 ]
- [ ]
- [ 9 1 9 1 ]
+ [ 1 1 1 1 ]
+ [ ]
+ [ 3 9 7 1 ]
+(%o1) [ ]
+ [ 7 9 3 1 ]
+ [ ]
+ [ 9 1 9 1 ]
+@end group
+@group
(%i2) zn_power_table(10,2);
- [ 2 4 8 6 ]
- [ ]
- [ 4 6 4 6 ]
-(%o2) [ ]
- [ 6 6 6 6 ]
- [ ]
- [ 8 4 2 6 ]
+ [ 2 4 8 6 ]
+ [ ]
+ [ 4 6 4 6 ]
+(%o2) [ ]
+ [ 6 6 6 6 ]
+ [ ]
+ [ 8 4 2 6 ]
+@end group
+@group
(%i3) zn_power_table(10,5);
-(%o3) [ 5 5 5 5 ]
+(%o3) [ 5 5 5 5 ]
+@end group
+@group
(%i4) zn_power_table(10,10);
-(%o4) [ 0 0 0 0 ]
+(%o4) [ 0 0 0 0 ]
+@end group
+@group
(%i5) G5: [1,2,3,4];
-(%o6) [1, 2, 3, 4]
+(%o5) [1, 2, 3, 4]
+@end group
+@group
(%i6) G10_2: map(lambda([x], solve_congruences([0,x],[2,5])), G5);
-(%o6) [6, 2, 8, 4]
+(%o6) [6, 2, 8, 4]
+@end group
+@group
(%i7) G10: map(lambda([x], power_mod(3, zn_log(x,2,5), 10)), G5);
-(%o7) [1, 3, 7, 9]
+(%o7) [1, 3, 7, 9]
+@end group
@end example
If @var{gcd} is set to @code{all}, the table contains powers of all elements
@@ -1794,29 +1963,34 @@ in (Z/@var{n}Z).
The third argument @var{max_exp} allows to set the highest exponent.
The following table shows a very small example of RSA.
-@example
-(%i1) N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$
-
-(%i5) 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 ]
-(%o5) [ ]
- [ 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 ]
+@c ===beg===
+@c linel:100$ N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$
+@c zn_power_table(N, all, e*d);
+@c ===end===
+@example maxima
+(%i1) linel:100$ N:2*5$ phi:totient(N)$ e:7$ d:inv_mod(e,phi)$
+@group
+(%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 ]
+@end group
@end example
@opencatbox{Categories:}
commit c28dd2bb70e067033a4b6d78b15fcda0c6a271f8
Author: Raymond Toy <toy...@gm...>
Date: Mon Sep 1 19:05:11 2025 -0700
Add comment blocks for examples and regenerate them
Also marked some examples with "maxima" to get syntax highlighting.
These particular ones don't get code blocks since they're not really
code examples to be executed.
There are still examples that need comment blocks so they can be
regenerated.
diff --git a/doc/info/Number.texi b/doc/info/Number.texi
index a89e11cf5..58cd02b0c 100644
--- a/doc/info/Number.texi
+++ b/doc/info/Number.texi
@@ -19,17 +19,27 @@ Bernoulli numbers equal to zero are suppressed if @code{zerobern} is
See also @mrefdot{burn}
-@example
+@c ===beg===
+@c zerobern: true$
+@c map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
+@c zerobern: false$
+@c map (bern, [0, 1, 2, 3, 4, 5, 6, 7, 8]);
+@c ===end===
+@example maxima
(%i1) zerobern: true$
+@group
(%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
+ 1 1 1 1 1
+(%o2) [1, - -, -, 0, - --, 0, --, 0, - --]
+ 2 6 30 42 30
+@end group
(%i3) zerobern: false$
+@group
(%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
+ 1 1 1 1 1 5 691 7
+(%o4) [1, - -, -, - --, --, - --, --, - ----, -]
+ 2 6 30 42 30 66 2730 6
+@end group
@end example
@opencatbox{Categories:}
@@ -133,21 +143,44 @@ Solves the system of congruences @code{x = r_1 mod m_1}, @dots{}, @code{x = r_n
The remainders @var{r_n} may be arbitrary integers while the moduli @var{m_n} have to be
positive and pairwise coprime integers.
-@example
+@c ===beg===
+@c mods : [1000, 1001, 1003, 1007];
+@c lreduce('gcd, mods);
+@c x : random(apply("*", mods));
+@c rems : map(lambda([z], mod(x, z)), mods);
+@c solve_congruences(rems, mods);
+@c solve_congruences([1, 2], [3, n]);
+@c %, n = 4;
+@c ===end===
+@example maxima
+@group
(%i1) mods : [1000, 1001, 1003, 1007];
-(%o1) [1000, 1001, 1003, 1007]
+(%o1) [1000, 1001, 1003, 1007]
+@end group
+@group
(%i2) lreduce('gcd, mods);
-(%o2) 1
+(%o2) 1
+@end group
+@group
(%i3) x : random(apply("*", mods));
-(%o3) 685124877004
+(%o3) 685124877004
+@end group
+@group
(%i4) rems : map(lambda([z], mod(x, z)), mods);
-(%o4) [4, 568, 54, 624]
+(%o4) [4, 568, 54, 624]
+@end group
+@group
(%i5) solve_congruences(rems, mods);
-(%o5) 685124877004
+(%o5) 685124877004
+@end group
+@group
(%i6) solve_congruences([1, 2], [3, n]);
-(%o6) solve_congruences([1, 2], [3, n])
+(%o6) solve_congruences([1, 2], [3, n])
+@end group
+@group
(%i7) %, n = 4;
-(%o7) 10
+(%o7) 10
+@end group
@end example
@opencatbox{Categories:}
@@ -205,38 +238,67 @@ integers.
@code{cflength} controls how many periods of the continued fraction
are computed for algebraic, irrational numbers.
-@example
+@c ===beg===
+@c cflength: 1$
+@c cf ((1 + sqrt(5))/2);
+@c cflength: 2$
+@c cf ((1 + sqrt(5))/2);
+@c cflength: 3$
+@c cf ((1 + sqrt(5))/2);
+@c ===end===
+@example maxima
(%i1) cflength: 1$
+@group
(%i2) cf ((1 + sqrt(5))/2);
(%o2) [1, 1, 1, 1, 2]
+@end group
(%i3) cflength: 2$
+@group
(%i4) cf ((1 + sqrt(5))/2);
(%o4) [1, 1, 1, 1, 1, 1, 1, 2]
+@end group
(%i5) cflength: 3$
+@group
(%i6) cf ((1 + sqrt(5))/2);
(%o6) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
+@end group
@end example
@item
A continued fraction can be evaluated by evaluating the arithmetic
representation returned by @mrefdot{cfdisrep}
-@example
+@c ===beg===
+@c cflength: 3$
+@c cfdisrep (cf (sqrt (3)))$
+@c ev (%, numer);
+@c ===end===
+@example maxima
(%i1) cflength: 3$
(%i2) cfdisrep (cf (sqrt (3)))$
+@group
(%i3) ev (%, numer);
-(%o3) 1.731707317073171
+(%o3) 1.7317073170731707
+@end group
@end example
@item
Maxima does not know about operations on continued fractions outside of
@code{cf}.
-@example
+@c ===beg===
+@c cf ([1,1,1,1,1,2] * 3);
+@c cf ([1,1,1,1,1,2]) * 3;
+@c ===end===
+@example maxima
+@group
(%i1) cf ([1,1,1,1,1,2] * 3);
(%o1) [4, 1, 5, 2]
+@end group
+@group
(%i2) cf ([1,1,1,1,1,2]) * 3;
(%o2) [3, 3, 3, 3, 3, 6]
+@end group
@end example
@end itemize
@@ -257,9 +319,16 @@ Constructs and returns an ordinary arithmetic expression
of the form @code{a + 1/(b + 1/(c + ...))}
from the list representation of a continued fraction @code{[a, b, c, ...]}.
-@example
+@c ===beg===
+@c cf ([1, 2, -3] + [1, -2, 1]);
+@c cfdisrep (%);
+@c ===end===
+@example maxima
+@group
(%i1) cf ([1, 2, -3] + [1, -2, 1]);
(%o1) [1, 1, 1, 2]
+@end group
+@group
(%i2) cfdisrep (%);
1
(%o2) 1 + ---------
@@ -268,6 +337,7 @@ from the list representation of a continued fraction @code{[a, b, c, ...]}.
1
1 + -
2
+@end group
@end example
@opencatbox{Categories:}
@@ -282,17 +352,27 @@ from the list representation of a continued fraction @code{[a, b, c, ...]}.
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}.
-@example
+@c ===beg===
+@c cf (rat (ev (%pi, numer)));
+@c cfexpand (%);
+@c %[1,1]/%[2,1], numer;
+@c ===end===
+@example maxima
+@group
(%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 ]
+rat: replaced 3.141592653589793 by 80143857/25510582 = 3.1415926535897927
+(%o1) [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14]
+@end group
+@group
+(%i2) cfexpand (%);
+ [ 80143857 5419351 ]
+(%o2) [ ]
+ [ 25510582 1725033 ]
+@end group
+@group
(%i3) %[1,1]/%[2,1], numer;
-(%o3) 3.141592653011902
+(%o3) 3.1415926535897927
+@end group
@end example
@opencatbox{Categories:}
@@ -309,16 +389,30 @@ Default value: 1
function @code{cf} will give, as the value @code{cflength} times the period.
Thus the default is to give one period.
-@example
+@c ===beg===
+@c cflength: 1$
+@c cf ((1 + sqrt(5))/2);
+@c cflength: 2$
+@c cf ((1 + sqrt(5))/2);
+@c cflength: 3$
+@c cf ((1 + sqrt(5))/2);
+@c ===end===
+@example maxima
(%i1) cflength: 1$
+@group
(%i2) cf ((1 + sqrt(5))/2);
(%o2) [1, 1, 1, 1, 2]
+@end group
(%i3) cflength: 2$
+@group
(%i4) cf ((1 + sqrt(5))/2);
(%o4) [1, 1, 1, 1, 1, 1, 1, 2]
+@end group
(%i5) cflength: 3$
+@group
(%i6) cf ((1 + sqrt(5))/2);
(%o6) [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]
+@end group
@end example
@opencatbox{Categories:}
@@ -337,15 +431,29 @@ raised to the @var{k}'th power.
@code{divsum (@var{n})} returns the sum of the divisors of @var{n}.
-@example
+@c ===beg===
+@c divsum (12);
+@c 1 + 2 + 3 + 4 + 6 + 12;
+@c divsum (12, 2);
+@c 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
+@c ===end===
+@example maxima
+@group
(%i1) divsum (12);
(%o1) 28
+@end group
+@group
(%i2) 1 + 2 + 3 + 4 + 6 + 12;
(%o2) 28
+@end group
+@group
(%i3) divsum (12, 2);
(%o3) 210
+@end group
+@group
(%i4) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
(%o4) 210
+@end group
@end example
@opencatbox{Categories:}
@@ -363,13 +471,23 @@ Euler numbers equal to zero are suppressed if @code{zerobern} is
For the Euler-Mascheroni constant, see @code{%gamma}.
-@example
+@c ===beg===
+@c zerobern: true$
+@c map (euler, [0, 1, 2, 3, 4, 5, 6]);
+@c zerobern: false$
+@c map (euler, [0, 1, 2, 3, 4, 5, 6]);
+@c ===end===
+@example maxima
(%i1) zerobern: true$
+@group
(%i2) map (euler, [0, 1, 2, 3, 4, 5, 6]);
-(%o2) [1, 0, - 1, 0, 5, 0, - 61]
+(%o2) [1, 0, - 1, 0, 5, 0, - 61]
+@end group
(%i3) zerobern: false$
+@group
(%i4) map (euler, [0, 1, 2, 3, 4, 5, 6]);
-(%o4) [1, - 1, 5, - 61, 1385, - 50521, 2702765]
+(%o4) [1, - 1, 5, - 61, 1385, - 50521, 2702765]
+@end group
@end example
@opencatbox{Categories:}
@@ -402,9 +520,14 @@ Returns the @var{n}'th Fibonacci number.
@code{fib(0)} is equal to 0 and @code{fib(1)} equal to 1, and
@code{fib (-@var{n})} equal to @code{(-1)^(@var{n} + 1) * fib(@var{n})}.
-@example
+@c ===beg===
+@c map (fib, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
+@c ===end===
+@example maxima
+@group
(%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]
+(%o1) [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]
+@end group
@end example
@opencatbox{Categories:}
@@ -482,13 +605,24 @@ the multiplicities of the computed prime factors. If @code{factors_only}
is set to @code{true}, @code{ifactors} simply returns the list of
prime factors.
-@example
+@c ===beg===
+@c ifactors(51575319651600);
+@c apply("*", map(lambda([u], u[1]^u[2]), %));
+@c ifactors(51575319651600), factors_only : true;
+@c ===end===
+@example maxima
+@group
(%i1) ifactors(51575319651600);
-(%o1) [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
+(%o1) [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
+@end group
+@group
(%i2) apply("*", map(lambda([u], u[1]^u[2]), %));
-(%o2) 51575319651600
+(%o2) 51575319651600
+@end group
+@group
(%i3) ifactors(51575319651600), factors_only : true;
-(%o3) [2, 3, 5, 1583, 9050207]
+(%o3) [2, 3, 5, 1583, 9050207]
+@end group
@end example
@opencatbox{Categories:}
@@ -511,15 +645,26 @@ The command @code{load("gcdex")} loads the function.
Examples:
-@example
+@c ===beg===
+@c load("gcdex")$
+@c igcdex(30,18);
+@c igcdex(1526757668, 7835626735736);
+@c igcdex(fib(20), fib(21));
+@c ===end===
+@example maxima
(%i1) load("gcdex")$
-
+@group
(%i2) igcdex(30,18);
(%o2) [- 1, 2, 6]
+@end group
+@group
(%i3) igcdex(1526757668, 7835626735736);
(%o3) [845922341123, - 164826435, 4]
+@end group
+@group
(%i4) igcdex(fib(20), fib(21));
(%o4) [4181, - 2584, 1]
+@end group
@end example
@opencatbox{Categories:}
@@ -533,10 +678,16 @@ Examples:
Returns the integer @var{n}'th root of the absolute value of @var{x}.
-@example
+@c ===beg===
+@c l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
+@c map (lambda ([a], inrt (10^a, 3)), l);
+@c ===end===
+@example maxima
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
+@group
(%i2) map (lambda ([a], inrt (10^a, 3)), l);
(%o2) [2, 4, 10, 21, 46, 100, 215, 464, 1000, 2154, 4641, 10000]
+@end group
@end example
@opencatbox{Categories:}
@@ -552,13 +703,24 @@ Computes the inverse of @var{n} modulo @var{m}.
@code{inv_mod (n,m)} returns @code{false},
if @var{n} is a zero divisor modulo @var{m}.
-@example
+@c ===beg===
+@c inv_mod(3, 41);
+@c ratsimp(3^-1), modulus = 41;
+@c inv_mod(3, 42);
+@c ===end===
+@example maxima
+@group
(%i1) inv_mod(3, 41);
-(%o1) 14
+(%o1) 14
+@end group
+@group
(%i2) ratsimp(3^-1), modulus = 41;
-(%o2) 14
+(%o2) 14
+@end group
+@group
(%i3) inv_mod(3, 42);
-(%o3) false
+(%o3) false
+@end group
@end example
@opencatbox{Categories:}
@@ -584,10 +746,16 @@ integer.
Returns the Jacobi symbol of @var{p} and @var{q}.
-@example
+@c ===beg===
+@c l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
+@c map (lambda ([a], jacobi (a, 9)), l);
+@c ===end===
+@example maxima
(%i1) l: [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]$
+@group
(%i2) map (lambda ([a], jacobi (a, 9)), l);
(%o2) [1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0]
+@end group
@end example
@opencatbox{Categories:}
@@ -618,9 +786,14 @@ Returns the @var{n}'th Lucas number.
in general, @code{lucas(@var{n}) = lucas(@var{n}-1) + lucas(@var{n}-2)}. Also
@code{lucas(-@var{n})} is equal to @code{(-1)^(-@var{n}) * lucas(@var{n})}.
-@example
+@c ===beg===
+@c map (lucas, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
+@c ===end===
+@example maxima
+@group
(%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]
+@end group
@end example
@@ -685,9 +858,14 @@ simplification rules:
Returns the smallest prime bigger than @var{n}.
-@example
+@c ===beg===
+@c next_prime(27);
+@c ===end===
+@example maxima
+@group
(%i1) next_prime(27);
-(%o1) 29
+(%o1) 29
+@end group
@end example
@opencatbox{Categories:}
@@ -710,22 +888,33 @@ the expansion falls out.
@code{partfrac} ignores the value @code{true} of the option variable
@code{keepfloat}.
-@example
+@c ===beg===
+@c 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
+@c ratsimp (%);
+@c partfrac (%, x);
+@c ===end===
+@example maxima
+@group
(%i1) 1/(1+x)^2 - 2/(1+x) + 2/(2+x);
2 2 1
(%o1) ----- - ----- + --------
x + 2 x + 1 2
(x + 1)
+@end group
+@group
(%i2) ratsimp (%);
x
(%o2) - -------------------
3 2
x + 4 x + 5 x + 2
+@end group
+@group
(%i3) partfrac (%, x);
2 2 1
(%o3) ----- - ----- + --------
x + 2 x + 1 2
(x + 1)
+@end group
@end example
@end deffn
commit 776e8a93df620e484eba4e1b26334dad80f9a046
Author: Raymond Toy <toy...@gm...>
Date: Mon Sep 1 18:37:23 2025 -0700
Regenerate existing examples
diff --git a/doc/info/Number.texi b/doc/info/Number.texi
index 2628dac3a..a89e11cf5 100644
--- a/doc/info/Number.texi
+++ b/doc/info/Number.texi
@@ -427,14 +427,19 @@ Examples:
@c fibtophi (%);
@c ratsimp (%);
@c ===end===
-@example
+@example maxima
+@group
(%i1) fibtophi (fib (n));
n n
%phi - (1 - %phi)
(%o1) -------------------
2 %phi - 1
+@end group
+@group
(%i2) fib (n-1) + fib (n) - fib (n+1);
(%o2) - fib(n + 1) + fib(n) + fib(n - 1)
+@end group
+@group
(%i3) fibtophi (%);
n + 1 n + 1 n n
%phi - (1 - %phi) %phi - (1 - %phi)
@@ -444,8 +449,11 @@ Examples:
%phi - (1 - %phi)
+ ---------------------------
2 %phi - 1
+@end group
+@group
(%i4) ratsimp (%);
(%o4) 0
+@end group
@end example
@opencatbox{Categories:}
@@ -651,13 +659,19 @@ simplification rules:
@c mod (a*x, a*y);
@c mod (0, x);
@c ===end===
-@example
+@example maxima
+@group
(%i1) mod (x, 0);
(%o1) x
+@end group
+@group
(%i2) mod (a*x, a*y);
(%o2) a mod(x, y)
+@end group
+@group
(%i3) mod (0, x);
(%o3) 0
+@end group
@end example
@opencatbox{Categories:}
@@ -881,17 +895,23 @@ Examples:
@c limit(zeta(x),x,1,plus);
@c limit(zeta(x),x,1,minus);
@c ===end===
-@example
-(%i1) zeta([-2, -1, 0, 0.5, 2, 3, 1+%i]);
- 2
- 1 1 %pi
-(%o1) [0, - --, - -, - 1.460354508809586, ----, zeta(3),
- 12 2 6
+@example maxima
+@group
+(%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)]
+@end group
+@group
(%i2) limit(zeta(x),x,1,plus);
(%o2) inf
+@end group
+@group
(%i3) limit(zeta(x),x,1,minus);
(%o3) minf
+@end group
@end example
@opencatbox{Categories:}
@@ -916,16 +936,20 @@ Examples:
@c zeta%pi: false$
@c zeta (4);
@c ===end===
-@example
+@example maxima
(%i1) zeta%pi: true$
+@group
(%i2) zeta (4);
4
%pi
(%o2) ----
90
+@end group
(%i3) zeta%pi: false$
+@group
(%i4) zeta (4);
(%o4) zeta(4)
+@end group
@end example
@opencatbox{Categories:}
-----------------------------------------------------------------------
hooks/post-receive
--
Maxima CAS
|
