{{{id=3| (x^(12)-1).factor() /// (x^4 - x^2 + 1)*(x^2 + x + 1)*(x^2 - x + 1)*(x^2 + 1)*(x + 1)*(x - 1) }}} {{{id=12| range? ///

Type: <type 'builtin_function_or_method'>

Definition: range( [noargspec] )

Docstring:


range(stop) -> list of integers
range(start, stop[, step]) -> list of integers

Return a list containing an arithmetic progression of integers.
range(i, j) returns [i, i+1, i+2, ..., j-1]; start (!) defaults to 0.
When step is given, it specifies the increment (or decrement).
For example, range(4) returns [0, 1, 2, 3].  The end point is omitted!
These are exactly the valid indices for a list of 4 elements.
}}} {{{id=13| range(1,13) /// [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] }}} {{{id=5| for n in range(1,13): n,(x^n -1).factor() /// (1, x - 1) (2, (x + 1)*(x - 1)) (3, (x^2 + x + 1)*(x - 1)) (4, (x^2 + 1)*(x + 1)*(x - 1)) (5, (x^4 + x^3 + x^2 + x + 1)*(x - 1)) (6, (x^2 + x + 1)*(x^2 - x + 1)*(x + 1)*(x - 1)) (7, (x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)) (8, (x^4 + 1)*(x^2 + 1)*(x + 1)*(x - 1)) (9, (x^6 + x^3 + 1)*(x^2 + x + 1)*(x - 1)) (10, (x^4 + x^3 + x^2 + x + 1)*(x^4 - x^3 + x^2 - x + 1)*(x + 1)*(x - 1)) (11, (x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)) (12, (x^4 - x^2 + 1)*(x^2 + x + 1)*(x^2 - x + 1)*(x^2 + 1)*(x + 1)*(x - 1)) }}} {{{id=14| cyclotomic_polynomial(11) /// x^10 + x^9 + x^8 + x^7 + x^6 + x^5 + x^4 + x^3 + x^2 + x + 1 }}} {{{id=6| from sage.rings.polynomial.cyclotomic import cyclotomic_coeffs /// }}} {{{id=7| cyclotomic_coeffs(12) /// {0: 1, 2: -1, 4: 1} }}} {{{id=8| cyclotomic_coeffs(12, sparse=false) /// [1, 0, -1, 0, 1] }}} {{{id=10| for n in range(10,20): n,cyclotomic_coeffs(n,sparse=false) /// (10, [1, -1, 1, -1, 1]) (11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) (12, [1, 0, -1, 0, 1]) (13, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) (14, [1, -1, 1, -1, 1, -1, 1]) (15, [1, -1, 0, 1, -1, 1, 0, -1, 1]) (16, [1, 0, 0, 0, 0, 0, 0, 0, 1]) (17, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) (18, [1, 0, 0, -1, 0, 0, 1]) (19, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]) }}}

Max and min of coefficients of cyclotomic polynomials

{{{id=17| for n in range(1,100): list= cyclotomic_coeffs(n, sparse=false) n,min(list),max(list) /// (1, -1, 1) (2, 1, 1) (3, 1, 1) (4, 0, 1) (5, 1, 1) (6, -1, 1) (7, 1, 1) (8, 0, 1) (9, 0, 1) (10, -1, 1) (11, 1, 1) (12, -1, 1) (13, 1, 1) (14, -1, 1) (15, -1, 1) (16, 0, 1) (17, 1, 1) (18, -1, 1) (19, 1, 1) (20, -1, 1) (21, -1, 1) (22, -1, 1) (23, 1, 1) (24, -1, 1) (25, 0, 1) (26, -1, 1) (27, 0, 1) (28, -1, 1) (29, 1, 1) (30, -1, 1) (31, 1, 1) (32, 0, 1) (33, -1, 1) (34, -1, 1) (35, -1, 1) (36, -1, 1) (37, 1, 1) (38, -1, 1) (39, -1, 1) (40, -1, 1) (41, 1, 1) (42, -1, 1) (43, 1, 1) (44, -1, 1) (45, -1, 1) (46, -1, 1) (47, 1, 1) (48, -1, 1) (49, 0, 1) (50, -1, 1) (51, -1, 1) (52, -1, 1) (53, 1, 1) (54, -1, 1) (55, -1, 1) (56, -1, 1) (57, -1, 1) (58, -1, 1) (59, 1, 1) (60, -1, 1) (61, 1, 1) (62, -1, 1) (63, -1, 1) (64, 0, 1) (65, -1, 1) (66, -1, 1) (67, 1, 1) (68, -1, 1) (69, -1, 1) (70, -1, 1) (71, 1, 1) (72, -1, 1) (73, 1, 1) (74, -1, 1) (75, -1, 1) (76, -1, 1) (77, -1, 1) (78, -1, 1) (79, 1, 1) (80, -1, 1) (81, 0, 1) (82, -1, 1) (83, 1, 1) (84, -1, 1) (85, -1, 1) (86, -1, 1) (87, -1, 1) (88, -1, 1) (89, 1, 1) (90, -1, 1) (91, -1, 1) (92, -1, 1) (93, -1, 1) (94, -1, 1) (95, -1, 1) (96, -1, 1) (97, 1, 1) (98, -1, 1) (99, -1, 1) }}} {{{id=23| /// }}}