How to find cubic residues $\bmod p$ using WolframAlpha?

Question

How to find cubic residues $\bmod p$ using WolframAlpha?

Just type in "quadratic residues modulo p" and you're done, but typing in "cubic residues modulo p" does nothing.

Logically, "x^3 modulo p" should work, but it doesn't. Neither does "x^3 modulo p, x=0..42".

Answer 1

Replace $p$ and $p-1$ in this query by the appropriate values:

Sort[Map[Mod[#^3,p]&, Range[p-1]]]

This will give a multiset (couting how many elements a particular element is a cube of), but it will list all and work within the computation time limit.

Breakdown:

Sort[                             Sort the result of
     Map[                         applying
         Mod[#^3,p]&              the function x -> x^3 mod p
                    ,Range[p-1]]] to all elements of Z_p^*

The Mathematica function DeleteDuplicates can be applied if desired.

For $p=1009$ we get

{1, 1, 1, 3, 3, 3, 8, 8, 8, 9, 9, 9, 13, 13, 13, 14, 14, 14, 19, 19, 19, 
20, 20, 20, 23, 23, 23, 24, 24, 24, 27, 27, 27, 35, 35, 35, 39, 39, 39, 
42, 42, 42, 43, 43, 43, 44, 44, 44, 47, 47, 47, 50, 50, 50, 57, 57, 57, 
59, 59, 59, 60, 60, 60, 61, 61, 61, 62, 62, 62, 64, 64, 64, 68, 68, 68, 
69, 69, 69, 72, 72, 72, 73, 73, 73, 74, 74, 74, 77, 77, 77, 81, 81, 81, 
82, 82, 82, 103, 103, 103, 104, 104, 104, 105, 105, 105, 110, 110, 110, 
112, 112, 112, 113, 113, 113, 116, 116, 116, 117, 117, 117, 119, 119, 119, 
125, 125, 125, 126, 126, 126, 129, 129, 129, 131, 131, 131, 132, 132, 132, 
139, 139, 139, 141, 141, 141, 149, 149, 149, 150, 150, 150, 152, 152, 152, 
155, 155, 155, 157, 157, 157, 160, 160, 160, 169, 169, 169, 170, 170, 170, 
171, 171, 171, 173, 173, 173, 177, 177, 177, 179, 179, 179, 180, 180, 180, 
182, 182, 182, 183, 183, 183, 184, 184, 184, 185, 185, 185, 186, 186, 186, 
191, 191, 191, 192, 192, 192, 196, 196, 196, 203, 203, 203, 204, 204, 204, 
205, 205, 205, 207, 207, 207, 211, 211, 211, 212, 212, 212, 216, 216, 216, 
219, 219, 219, 222, 222, 222, 229, 229, 229, 231, 231, 231, 242, 242, 242, 
243, 243, 243, 246, 246, 246, 247, 247, 247, 254, 254, 254, 260, 260, 260, 
266, 266, 266, 268, 268, 268, 271, 271, 271, 275, 275, 275, 280, 280, 280, 
283, 283, 283, 284, 284, 284, 290, 290, 290, 299, 299, 299, 302, 302, 302, 
309, 309, 309, 312, 312, 312, 315, 315, 315, 316, 316, 316, 322, 322, 322, 
330, 330, 330, 332, 332, 332, 336, 336, 336, 339, 339, 339, 341, 341, 341, 
343, 343, 343, 344, 344, 344, 348, 348, 348, 351, 351, 351, 352, 352, 352, 
353, 353, 353, 356, 356, 356, 357, 357, 357, 359, 359, 359, 361, 361, 361, 
362, 362, 362, 371, 371, 371, 373, 373, 373, 374, 374, 374, 375, 375, 375, 
376, 376, 376, 378, 378, 378, 380, 380, 380, 386, 386, 386, 387, 387, 387, 
388, 388, 388, 393, 393, 393, 394, 394, 394, 396, 396, 396, 397, 397, 397, 
398, 398, 398, 400, 400, 400, 404, 404, 404, 407, 407, 407, 417, 417, 417, 
421, 421, 421, 423, 423, 423, 425, 425, 425, 428, 428, 428, 431, 431, 431, 
433, 433, 433, 436, 436, 436, 437, 437, 437, 447, 447, 447, 450, 450, 450, 
451, 451, 451, 454, 454, 454, 455, 455, 455, 456, 456, 456, 457, 457, 457, 
460, 460, 460, 461, 461, 461, 463, 463, 463, 465, 465, 465, 469, 469, 469, 
471, 471, 471, 472, 472, 472, 478, 478, 478, 479, 479, 479, 480, 480, 480, 
482, 482, 482, 488, 488, 488, 490, 490, 490, 496, 496, 496, 497, 497, 497, 
499, 499, 499, 502, 502, 502, 507, 507, 507, 510, 510, 510, 512, 512, 512, 
513, 513, 513, 519, 519, 519, 521, 521, 521, 527, 527, 527, 529, 529, 529, 
530, 530, 530, 531, 531, 531, 537, 537, 537, 538, 538, 538, 540, 540, 540, 
544, 544, 544, 546, 546, 546, 548, 548, 548, 549, 549, 549, 552, 552, 552, 
553, 553, 553, 554, 554, 554, 555, 555, 555, 558, 558, 558, 559, 559, 559, 
562, 562, 562, 572, 572, 572, 573, 573, 573, 576, 576, 576, 578, 578, 578, 
581, 581, 581, 584, 584, 584, 586, 586, 586, 588, 588, 588, 592, 592, 592, 
602, 602, 602, 605, 605, 605, 609, 609, 609, 611, 611, 611, 612, 612, 612, 
613, 613, 613, 615, 615, 615, 616, 616, 616, 621, 621, 621, 622, 622, 622, 
623, 623, 623, 629, 629, 629, 631, 631, 631, 633, 633, 633, 634, 634, 634, 
635, 635, 635, 636, 636, 636, 638, 638, 638, 647, 647, 647, 648, 648, 648, 
650, 650, 650, 652, 652, 652, 653, 653, 653, 656, 656, 656, 657, 657, 657, 
658, 658, 658, 661, 661, 661, 665, 665, 665, 666, 666, 666, 668, 668, 668, 
670, 670, 670, 673, 673, 673, 677, 677, 677, 679, 679, 679, 687, 687, 687, 
693, 693, 693, 694, 694, 694, 697, 697, 697, 700, 700, 700, 707, 707, 707, 
734, 734, 734, 738, 738, 738, 741, 741, 741, 743, 743, 743, 749, 749, 749, 
755, 755, 755, 762, 762, 762, 763, 763, 763, 766, 766, 766, 767, 767, 767, 
778, 778, 778, 780, 780, 780, 787, 787, 787, 790, 790, 790, 793, 793, 793, 
797, 797, 797, 798, 798, 798, 802, 802, 802, 804, 804, 804, 805, 805, 805,
824, 824, 824, 825, 825, 825, 826, 826, 826, 827, 827, 827, 829, 829, 829, 
830, 830, 830, 832, 832, 832, 836, 836, 836, 838, 838, 838, 839, 839, 839, 
840, 840, 840, 849, 849, 849, 852, 852, 852, 854, 854, 854, 857, 857, 857, 
859, 859, 859, 860, 860, 860, 868, 868, 868, 870, 870, 870, 877, 877, 877, 
878, 878, 878, 880, 880, 880, 883, 883, 883, 884, 884, 884, 890, 890, 890, 
892, 892, 892, 893, 893, 893, 896, 896, 896, 897, 897, 897, 899, 899, 899, 
904, 904, 904, 905, 905, 905, 906, 906, 906, 927, 927, 927, 928, 928, 928, 
932, 932, 932, 935, 935, 935, 936, 936, 936, 937, 937, 937, 940, 940, 940, 
941, 941, 941, 945, 945, 945, 947, 947, 947, 948, 948, 948, 949, 949, 949, 
950, 950, 950, 952, 952, 952, 959, 959, 959, 962, 962, 962, 965, 965, 965, 
966, 966, 966, 967, 967, 967, 970, 970, 970, 974, 974, 974, 982, 982, 982,
996, 996, 996, 1000, 1000, 1000, 1001, 1001, 1001, 1006, 1006, 1006, 1008, 1008, 1008}

Deleting the duplicates gives the list

{1, 3, 8, 9, 13, 14, 19, 20, 23, 24, 27, 35, 39, 42, 43, 
44, 47, 50, 57, 59, 60, 61, 62, 64, 68, 69, 72, 73, 74, 
77, 81, 82, 103, 104, 105, 110, 112, 113, 116, 117, 119, 
125, 126, 129, 131, 132, 139, 141, 149, 150, 152, 155, 
157, 160, 169, 170, 171, 173, 177, 179, 180, 182, 183, 
184, 185, 186, 191, 192, 196, 203, 204, 205, 207, 211, 
212, 216, 219, 222, 229, 231, 242, 243, 246, 247, 254, 
260, 266, 268, 271, 275, 280, 283, 284, 290, 299, 302, 
309, 312, 315, 316, 322, 330, 332, 336, 339, 341, 343, 
344, 348, 351, 352, 353, 356, 357, 359, 361, 362, 371, 
373, 374, 375, 376, 378, 380, 386, 387, 388, 393, 394, 
396, 397, 398, 400, 404, 407, 417, 421, 423, 425, 428, 
431, 433, 436, 437, 447, 450, 451, 454, 455, 456, 457, 
460, 461, 463, 465, 469, 471, 472, 478, 479, 480, 482, 
488, 490, 496, 497, 499, 502, 507, 510, 512, 513, 519, 
521, 527, 529, 530, 531, 537, 538, 540, 544, 546, 548, 
549, 552, 553, 554, 555, 558, 559, 562, 572, 573, 576, 
578, 581, 584, 586, 588, 592, 602, 605, 609, 611, 612, 
613, 615, 616, 621, 622, 623, 629, 631, 633, 634, 635, 
636, 638, 647, 648, 650, 652, 653, 656, 657, 658, 661, 
665, 666, 668, 670, 673, 677, 679, 687, 693, 694, 697, 
700, 707, 734, 738, 741, 743, 749, 755, 762, 763, 766, 
767, 778, 780, 787, 790, 793, 797, 798, 802, 804, 805,
824, 825, 826, 827, 829, 830, 832, 836, 838, 839, 840, 
849, 852, 854, 857, 859, 860, 868, 870, 877, 878, 880, 
883, 884, 890, 892, 893, 896, 897, 899, 904, 905, 906, 
927, 928, 932, 935, 936, 937, 940, 941, 945, 947, 948, 
949, 950, 952, 959, 962, 965, 966, 967, 970, 974, 982, 
996, 1000, 1001, 1006, 1008}