Maxima – 170 – Teoria dei numeri – 2

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Continuo da qui, copio dal Reference Manual, PDF scaricabile da qui, sono a p.515.

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

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.

(%i3) cf (rat (ev (%pi, numer)));

rat: replaced 3.141592653589793 by 80143857/25510582 = 3.141592653589792
(%o3)             [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14]
(%i4) cfexpand (%);
                             [ 80143857  5419351 ]
(%o4)                        [                   ]
                             [ 25510582  1725033 ]
(%i5) %[1,1]/%[2,1], numer;
(%o5)                          3.141592653589792

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.

(%i6) cflength: 1$

(%i7) cf ((1 + sqrt(5))/2);
(%o7)                           [1, 1, 1, 1, 2]
(%i8) cflength: 2$

(%i9) cf ((1 + sqrt(5))/2);
(%o9)                      [1, 1, 1, 1, 1, 1, 1, 2]
(%i10) cflength: 3$

(%i11) cf ((1 + sqrt(5))/2);
(%o11)                 [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2]

divsum (n, k)
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.

(%i12) divsum (12);
(%o12)                                28
(%i13) 1 + 2 + 3 + 4 + 6 + 12;
(%o13)                                28
(%i14) divsum (12, 2);
(%o14)                                210
(%i15) 1^2 + 2^2 + 3^2 + 4^2 + 6^2 + 12^2;
(%o15)                                210

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.

(%i16) zerobern: true$

(%i17) map (euler, [0, 1, 2, 3, 4, 5, 6]);
(%o17)                    [1, 0, - 1, 0, 5, 0, - 61]
(%i18) zerobern: false$

(%i19) map (euler, [0, 1, 2, 3, 4, 5, 6]);
(%o19)             [1, - 1, 5, - 61, 1385, - 50521, 2702765]

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.

fib (n)
Returns the n’th Fibonacci number. fib(0) is equal to 0 and fib(1) equal to 1, and fib (-n) equal to (-1)^(n + 1) * fib(n).

After calling fib, prevfib is equal to fib(n - 1), the Fibonacci number preceding the last one computed.

(%i20) map (fib, [-4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8]);
(%o20)           [- 3, 2, - 1, 1, 0, 1, 1, 2, 3, 5, 8, 13, 21]

fibtophi (expr)
Expresses Fibonacci numbers in expr in terms of the constant %phi, which is (1 + sqrt(5))/2, approximately 1.61803399.

(%i21) fibtophi (fib (n));
                                  n             n
                              %phi  - (1 - %phi)
(%o21)                        -------------------
                                  2 %phi - 1
(%i22) fib (n-1) + fib (n) - fib (n+1);
(%o22)               (- fib(n + 1)) + fib(n) + fib(n - 1)
(%i23) fibtophi (%);
              n + 1             n + 1        n             n
          %phi      - (1 - %phi)         %phi  - (1 - %phi)
(%o23) (- ---------------------------) + -------------------
                  2 %phi - 1                 2 %phi - 1
                                                        n - 1             n - 1
                                                    %phi      - (1 - %phi)
                                                  + ---------------------------
                                                            2 %phi - 1
(%i24) ratsimp (%);
(%o24)                                 0

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.

If the variable ifactor_verbose is set to true ifactor produces detailed output about what it is doing including immediate feedback as soon as a factor has been found.

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

(%i25) ifactors(51575319651600);
(%o25)         [[2, 4], [3, 2], [5, 2], [1583, 1], [9050207, 1]]
(%i26) apply("*", map(lambda([u], u[1]^u[2]), %));
(%o26)                          51575319651600
(%i27) ifactors(51575319651600), factors_only : true;
(%o27)                     [2, 3, 5, 1583, 9050207]

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