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Primi finti

Teoria dei numeri 

Il termine “primi finti” è una mia (arbitraria) traduzione dell’inglese “prime pretender”: forse non linguisticamente impeccabile, ma rende l’idea.

I primi finti sono i numeri composti n che soddisfano l’equazione bnb mod n. E’ una condizione meno restrittiva di quella del “piccolo” teorema di Fermat; in particolare non è richiesto che b e n siano primi tra loro. Comprendono quindi gli pseudoprimi di Fermat, i divisori della base e molti altri numeri; in particolare per ogni base maggiore di 1 vi sono infiniti primi finti che non sono pseudoprimi di Fermat, né divisori della base.

 

Ogni numero composto n è primo finto nelle basi che sono multipli di n, nelle basi della forma kn + 1 e, se è dispari, in quelle della forma kn – 1.

 

La tabella seguente mostra i primi finti fino a 20 nelle basi fino a 1000.

Primi finti

Basi

4

4, 5, 8, 9, 12, 13, 16, 17, 20, 21, 24, 25, 28, 29, 32, 33, 36, 37, 40, 41, 44, 45, 48, 49, 52, 53, 56, 57, 60, 61, 64, 65, 68, 69, 72, 73, 76, 77, 80, 81, 84, 85, 88, 89, 92, 93, 96, 97, 100, 101, 104, 105, 108, 109, 112, 113, 116, 117, 120, 121, 124, 125, 128, 129, 132, 133, 136, 137, 140, 141, 144, 145, 148, 149, 152, 153, 156, 157, 160, 161, 164, 165, 168, 169, 172, 173, 176, 177, 180, 181, 184, 185, 188, 189, 192, 193, 196, 197, 200, 201, 204, 205, 208, 209, 212, 213, 216, 217, 220, 221, 224, 225, 228, 229, 232, 233, 236, 237, 240, 241, 244, 245, 248, 249, 252, 253, 256, 257, 260, 261, 264, 265, 268, 269, 272, 273, 276, 277, 280, 281, 284, 285, 288, 289, 292, 293, 296, 297, 300, 301, 304, 305, 308, 309, 312, 313, 316, 317, 320, 321, 324, 325, 328, 329, 332, 333, 336, 337, 340, 341, 344, 345, 348, 349, 352, 353, 356, 357, 360, 361, 364, 365, 368, 369, 372, 373, 376, 377, 380, 381, 384, 385, 388, 389, 392, 393, 396, 397, 400, 401, 404, 405, 408, 409, 412, 413, 416, 417, 420, 421, 424, 425, 428, 429, 432, 433, 436, 437, 440, 441, 444, 445, 448, 449, 452, 453, 456, 457, 460, 461, 464, 465, 468, 469, 472, 473, 476, 477, 480, 481, 484, 485, 488, 489, 492, 493, 496, 497, 500, 501, 504, 505, 508, 509, 512, 513, 516, 517, 520, 521, 524, 525, 528, 529, 532, 533, 536, 537, 540, 541, 544, 545, 548, 549, 552, 553, 556, 557, 560, 561, 564, 565, 568, 569, 572, 573, 576, 577, 580, 581, 584, 585, 588, 589, 592, 593, 596, 597, 600, 601, 604, 605, 608, 609, 612, 613, 616, 617, 620, 621, 624, 625, 628, 629, 632, 633, 636, 637, 640, 641, 644, 645, 648, 649, 652, 653, 656, 657, 660, 661, 664, 665, 668, 669, 672, 673, 676, 677, 680, 681, 684, 685, 688, 689, 692, 693, 696, 697, 700, 701, 704, 705, 708, 709, 712, 713, 716, 717, 720, 721, 724, 725, 728, 729, 732, 733, 736, 737, 740, 741, 744, 745, 748, 749, 752, 753, 756, 757, 760, 761, 764, 765, 768, 769, 772, 773, 776, 777, 780, 781, 784, 785, 788, 789, 792, 793, 796, 797, 800, 801, 804, 805, 808, 809, 812, 813, 816, 817, 820, 821, 824, 825, 828, 829, 832, 833, 836, 837, 840, 841, 844, 845, 848, 849, 852, 853, 856, 857, 860, 861, 864, 865, 868, 869, 872, 873, 876, 877, 880, 881, 884, 885, 888, 889, 892, 893, 896, 897, 900, 901, 904, 905, 908, 909, 912, 913, 916, 917, 920, 921, 924, 925, 928, 929, 932, 933, 936, 937, 940, 941, 944, 945, 948, 949, 952, 953, 956, 957, 960, 961, 964, 965, 968, 969, 972, 973, 976, 977, 980, 981, 984, 985, 988, 989, 992, 993, 996, 997, 1000

6

3, 4, 6, 7, 9, 10, 12, 13, 15, 16, 18, 19, 21, 22, 24, 25, 27, 28, 30, 31, 33, 34, 36, 37, 39, 40, 42, 43, 45, 46, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 102, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120, 121, 123, 124, 126, 127, 129, 130, 132, 133, 135, 136, 138, 139, 141, 142, 144, 145, 147, 148, 150, 151, 153, 154, 156, 157, 159, 160, 162, 163, 165, 166, 168, 169, 171, 172, 174, 175, 177, 178, 180, 181, 183, 184, 186, 187, 189, 190, 192, 193, 195, 196, 198, 199, 201, 202, 204, 205, 207, 208, 210, 211, 213, 214, 216, 217, 219, 220, 222, 223, 225, 226, 228, 229, 231, 232, 234, 235, 237, 238, 240, 241, 243, 244, 246, 247, 249, 250, 252, 253, 255, 256, 258, 259, 261, 262, 264, 265, 267, 268, 270, 271, 273, 274, 276, 277, 279, 280, 282, 283, 285, 286, 288, 289, 291, 292, 294, 295, 297, 298, 300, 301, 303, 304, 306, 307, 309, 310, 312, 313, 315, 316, 318, 319, 321, 322, 324, 325, 327, 328, 330, 331, 333, 334, 336, 337, 339, 340, 342, 343, 345, 346, 348, 349, 351, 352, 354, 355, 357, 358, 360, 361, 363, 364, 366, 367, 369, 370, 372, 373, 375, 376, 378, 379, 381, 382, 384, 385, 387, 388, 390, 391, 393, 394, 396, 397, 399, 400, 402, 403, 405, 406, 408, 409, 411, 412, 414, 415, 417, 418, 420, 421, 423, 424, 426, 427, 429, 430, 432, 433, 435, 436, 438, 439, 441, 442, 444, 445, 447, 448, 450, 451, 453, 454, 456, 457, 459, 460, 462, 463, 465, 466, 468, 469, 471, 472, 474, 475, 477, 478, 480, 481, 483, 484, 486, 487, 489, 490, 492, 493, 495, 496, 498, 499, 501, 502, 504, 505, 507, 508, 510, 511, 513, 514, 516, 517, 519, 520, 522, 523, 525, 526, 528, 529, 531, 532, 534, 535, 537, 538, 540, 541, 543, 544, 546, 547, 549, 550, 552, 553, 555, 556, 558, 559, 561, 562, 564, 565, 567, 568, 570, 571, 573, 574, 576, 577, 579, 580, 582, 583, 585, 586, 588, 589, 591, 592, 594, 595, 597, 598, 600, 601, 603, 604, 606, 607, 609, 610, 612, 613, 615, 616, 618, 619, 621, 622, 624, 625, 627, 628, 630, 631, 633, 634, 636, 637, 639, 640, 642, 643, 645, 646, 648, 649, 651, 652, 654, 655, 657, 658, 660, 661, 663, 664, 666, 667, 669, 670, 672, 673, 675, 676, 678, 679, 681, 682, 684, 685, 687, 688, 690, 691, 693, 694, 696, 697, 699, 700, 702, 703, 705, 706, 708, 709, 711, 712, 714, 715, 717, 718, 720, 721, 723, 724, 726, 727, 729, 730, 732, 733, 735, 736, 738, 739, 741, 742, 744, 745, 747, 748, 750, 751, 753, 754, 756, 757, 759, 760, 762, 763, 765, 766, 768, 769, 771, 772, 774, 775, 777, 778, 780, 781, 783, 784, 786, 787, 789, 790, 792, 793, 795, 796, 798, 799, 801, 802, 804, 805, 807, 808, 810, 811, 813, 814, 816, 817, 819, 820, 822, 823, 825, 826, 828, 829, 831, 832, 834, 835, 837, 838, 840, 841, 843, 844, 846, 847, 849, 850, 852, 853, 855, 856, 858, 859, 861, 862, 864, 865, 867, 868, 870, 871, 873, 874, 876, 877, 879, 880, 882, 883, 885, 886, 888, 889, 891, 892, 894, 895, 897, 898, 900, 901, 903, 904, 906, 907, 909, 910, 912, 913, 915, 916, 918, 919, 921, 922, 924, 925, 927, 928, 930, 931, 933, 934, 936, 937, 939, 940, 942, 943, 945, 946, 948, 949, 951, 952, 954, 955, 957, 958, 960, 961, 963, 964, 966, 967, 969, 970, 972, 973, 975, 976, 978, 979, 981, 982, 984, 985, 987, 988, 990, 991, 993, 994, 996, 997, 999, 1000

8

8, 9, 16, 17, 24, 25, 32, 33, 40, 41, 48, 49, 56, 57, 64, 65, 72, 73, 80, 81, 88, 89, 96, 97, 104, 105, 112, 113, 120, 121, 128, 129, 136, 137, 144, 145, 152, 153, 160, 161, 168, 169, 176, 177, 184, 185, 192, 193, 200, 201, 208, 209, 216, 217, 224, 225, 232, 233, 240, 241, 248, 249, 256, 257, 264, 265, 272, 273, 280, 281, 288, 289, 296, 297, 304, 305, 312, 313, 320, 321, 328, 329, 336, 337, 344, 345, 352, 353, 360, 361, 368, 369, 376, 377, 384, 385, 392, 393, 400, 401, 408, 409, 416, 417, 424, 425, 432, 433, 440, 441, 448, 449, 456, 457, 464, 465, 472, 473, 480, 481, 488, 489, 496, 497, 504, 505, 512, 513, 520, 521, 528, 529, 536, 537, 544, 545, 552, 553, 560, 561, 568, 569, 576, 577, 584, 585, 592, 593, 600, 601, 608, 609, 616, 617, 624, 625, 632, 633, 640, 641, 648, 649, 656, 657, 664, 665, 672, 673, 680, 681, 688, 689, 696, 697, 704, 705, 712, 713, 720, 721, 728, 729, 736, 737, 744, 745, 752, 753, 760, 761, 768, 769, 776, 777, 784, 785, 792, 793, 800, 801, 808, 809, 816, 817, 824, 825, 832, 833, 840, 841, 848, 849, 856, 857, 864, 865, 872, 873, 880, 881, 888, 889, 896, 897, 904, 905, 912, 913, 920, 921, 928, 929, 936, 937, 944, 945, 952, 953, 960, 961, 968, 969, 976, 977, 984, 985, 992, 993, 1000

9

8, 9, 10, 17, 18, 19, 26, 27, 28, 35, 36, 37, 44, 45, 46, 53, 54, 55, 62, 63, 64, 71, 72, 73, 80, 81, 82, 89, 90, 91, 98, 99, 100, 107, 108, 109, 116, 117, 118, 125, 126, 127, 134, 135, 136, 143, 144, 145, 152, 153, 154, 161, 162, 163, 170, 171, 172, 179, 180, 181, 188, 189, 190, 197, 198, 199, 206, 207, 208, 215, 216, 217, 224, 225, 226, 233, 234, 235, 242, 243, 244, 251, 252, 253, 260, 261, 262, 269, 270, 271, 278, 279, 280, 287, 288, 289, 296, 297, 298, 305, 306, 307, 314, 315, 316, 323, 324, 325, 332, 333, 334, 341, 342, 343, 350, 351, 352, 359, 360, 361, 368, 369, 370, 377, 378, 379, 386, 387, 388, 395, 396, 397, 404, 405, 406, 413, 414, 415, 422, 423, 424, 431, 432, 433, 440, 441, 442, 449, 450, 451, 458, 459, 460, 467, 468, 469, 476, 477, 478, 485, 486, 487, 494, 495, 496, 503, 504, 505, 512, 513, 514, 521, 522, 523, 530, 531, 532, 539, 540, 541, 548, 549, 550, 557, 558, 559, 566, 567, 568, 575, 576, 577, 584, 585, 586, 593, 594, 595, 602, 603, 604, 611, 612, 613, 620, 621, 622, 629, 630, 631, 638, 639, 640, 647, 648, 649, 656, 657, 658, 665, 666, 667, 674, 675, 676, 683, 684, 685, 692, 693, 694, 701, 702, 703, 710, 711, 712, 719, 720, 721, 728, 729, 730, 737, 738, 739, 746, 747, 748, 755, 756, 757, 764, 765, 766, 773, 774, 775, 782, 783, 784, 791, 792, 793, 800, 801, 802, 809, 810, 811, 818, 819, 820, 827, 828, 829, 836, 837, 838, 845, 846, 847, 854, 855, 856, 863, 864, 865, 872, 873, 874, 881, 882, 883, 890, 891, 892, 899, 900, 901, 908, 909, 910, 917, 918, 919, 926, 927, 928, 935, 936, 937, 944, 945, 946, 953, 954, 955, 962, 963, 964, 971, 972, 973, 980, 981, 982, 989, 990, 991, 998, 999, 1000

10

5, 6, 10, 11, 15, 16, 20, 21, 25, 26, 30, 31, 35, 36, 40, 41, 45, 46, 50, 51, 55, 56, 60, 61, 65, 66, 70, 71, 75, 76, 80, 81, 85, 86, 90, 91, 95, 96, 100, 101, 105, 106, 110, 111, 115, 116, 120, 121, 125, 126, 130, 131, 135, 136, 140, 141, 145, 146, 150, 151, 155, 156, 160, 161, 165, 166, 170, 171, 175, 176, 180, 181, 185, 186, 190, 191, 195, 196, 200, 201, 205, 206, 210, 211, 215, 216, 220, 221, 225, 226, 230, 231, 235, 236, 240, 241, 245, 246, 250, 251, 255, 256, 260, 261, 265, 266, 270, 271, 275, 276, 280, 281, 285, 286, 290, 291, 295, 296, 300, 301, 305, 306, 310, 311, 315, 316, 320, 321, 325, 326, 330, 331, 335, 336, 340, 341, 345, 346, 350, 351, 355, 356, 360, 361, 365, 366, 370, 371, 375, 376, 380, 381, 385, 386, 390, 391, 395, 396, 400, 401, 405, 406, 410, 411, 415, 416, 420, 421, 425, 426, 430, 431, 435, 436, 440, 441, 445, 446, 450, 451, 455, 456, 460, 461, 465, 466, 470, 471, 475, 476, 480, 481, 485, 486, 490, 491, 495, 496, 500, 501, 505, 506, 510, 511, 515, 516, 520, 521, 525, 526, 530, 531, 535, 536, 540, 541, 545, 546, 550, 551, 555, 556, 560, 561, 565, 566, 570, 571, 575, 576, 580, 581, 585, 586, 590, 591, 595, 596, 600, 601, 605, 606, 610, 611, 615, 616, 620, 621, 625, 626, 630, 631, 635, 636, 640, 641, 645, 646, 650, 651, 655, 656, 660, 661, 665, 666, 670, 671, 675, 676, 680, 681, 685, 686, 690, 691, 695, 696, 700, 701, 705, 706, 710, 711, 715, 716, 720, 721, 725, 726, 730, 731, 735, 736, 740, 741, 745, 746, 750, 751, 755, 756, 760, 761, 765, 766, 770, 771, 775, 776, 780, 781, 785, 786, 790, 791, 795, 796, 800, 801, 805, 806, 810, 811, 815, 816, 820, 821, 825, 826, 830, 831, 835, 836, 840, 841, 845, 846, 850, 851, 855, 856, 860, 861, 865, 866, 870, 871, 875, 876, 880, 881, 885, 886, 890, 891, 895, 896, 900, 901, 905, 906, 910, 911, 915, 916, 920, 921, 925, 926, 930, 931, 935, 936, 940, 941, 945, 946, 950, 951, 955, 956, 960, 961, 965, 966, 970, 971, 975, 976, 980, 981, 985, 986, 990, 991, 995, 996, 1000

12

4, 9, 12, 13, 16, 21, 24, 25, 28, 33, 36, 37, 40, 45, 48, 49, 52, 57, 60, 61, 64, 69, 72, 73, 76, 81, 84, 85, 88, 93, 96, 97, 100, 105, 108, 109, 112, 117, 120, 121, 124, 129, 132, 133, 136, 141, 144, 145, 148, 153, 156, 157, 160, 165, 168, 169, 172, 177, 180, 181, 184, 189, 192, 193, 196, 201, 204, 205, 208, 213, 216, 217, 220, 225, 228, 229, 232, 237, 240, 241, 244, 249, 252, 253, 256, 261, 264, 265, 268, 273, 276, 277, 280, 285, 288, 289, 292, 297, 300, 301, 304, 309, 312, 313, 316, 321, 324, 325, 328, 333, 336, 337, 340, 345, 348, 349, 352, 357, 360, 361, 364, 369, 372, 373, 376, 381, 384, 385, 388, 393, 396, 397, 400, 405, 408, 409, 412, 417, 420, 421, 424, 429, 432, 433, 436, 441, 444, 445, 448, 453, 456, 457, 460, 465, 468, 469, 472, 477, 480, 481, 484, 489, 492, 493, 496, 501, 504, 505, 508, 513, 516, 517, 520, 525, 528, 529, 532, 537, 540, 541, 544, 549, 552, 553, 556, 561, 564, 565, 568, 573, 576, 577, 580, 585, 588, 589, 592, 597, 600, 601, 604, 609, 612, 613, 616, 621, 624, 625, 628, 633, 636, 637, 640, 645, 648, 649, 652, 657, 660, 661, 664, 669, 672, 673, 676, 681, 684, 685, 688, 693, 696, 697, 700, 705, 708, 709, 712, 717, 720, 721, 724, 729, 732, 733, 736, 741, 744, 745, 748, 753, 756, 757, 760, 765, 768, 769, 772, 777, 780, 781, 784, 789, 792, 793, 796, 801, 804, 805, 808, 813, 816, 817, 820, 825, 828, 829, 832, 837, 840, 841, 844, 849, 852, 853, 856, 861, 864, 865, 868, 873, 876, 877, 880, 885, 888, 889, 892, 897, 900, 901, 904, 909, 912, 913, 916, 921, 924, 925, 928, 933, 936, 937, 940, 945, 948, 949, 952, 957, 960, 961, 964, 969, 972, 973, 976, 981, 984, 985, 988, 993, 996, 997, 1000

14

7, 8, 14, 15, 21, 22, 28, 29, 35, 36, 42, 43, 49, 50, 56, 57, 63, 64, 70, 71, 77, 78, 84, 85, 91, 92, 98, 99, 105, 106, 112, 113, 119, 120, 126, 127, 133, 134, 140, 141, 147, 148, 154, 155, 161, 162, 168, 169, 175, 176, 182, 183, 189, 190, 196, 197, 203, 204, 210, 211, 217, 218, 224, 225, 231, 232, 238, 239, 245, 246, 252, 253, 259, 260, 266, 267, 273, 274, 280, 281, 287, 288, 294, 295, 301, 302, 308, 309, 315, 316, 322, 323, 329, 330, 336, 337, 343, 344, 350, 351, 357, 358, 364, 365, 371, 372, 378, 379, 385, 386, 392, 393, 399, 400, 406, 407, 413, 414, 420, 421, 427, 428, 434, 435, 441, 442, 448, 449, 455, 456, 462, 463, 469, 470, 476, 477, 483, 484, 490, 491, 497, 498, 504, 505, 511, 512, 518, 519, 525, 526, 532, 533, 539, 540, 546, 547, 553, 554, 560, 561, 567, 568, 574, 575, 581, 582, 588, 589, 595, 596, 602, 603, 609, 610, 616, 617, 623, 624, 630, 631, 637, 638, 644, 645, 651, 652, 658, 659, 665, 666, 672, 673, 679, 680, 686, 687, 693, 694, 700, 701, 707, 708, 714, 715, 721, 722, 728, 729, 735, 736, 742, 743, 749, 750, 756, 757, 763, 764, 770, 771, 777, 778, 784, 785, 791, 792, 798, 799, 805, 806, 812, 813, 819, 820, 826, 827, 833, 834, 840, 841, 847, 848, 854, 855, 861, 862, 868, 869, 875, 876, 882, 883, 889, 890, 896, 897, 903, 904, 910, 911, 917, 918, 924, 925, 931, 932, 938, 939, 945, 946, 952, 953, 959, 960, 966, 967, 973, 974, 980, 981, 987, 988, 994, 995

15

4, 5, 6, 9, 10, 11, 14, 15, 16, 19, 20, 21, 24, 25, 26, 29, 30, 31, 34, 35, 36, 39, 40, 41, 44, 45, 46, 49, 50, 51, 54, 55, 56, 59, 60, 61, 64, 65, 66, 69, 70, 71, 74, 75, 76, 79, 80, 81, 84, 85, 86, 89, 90, 91, 94, 95, 96, 99, 100, 101, 104, 105, 106, 109, 110, 111, 114, 115, 116, 119, 120, 121, 124, 125, 126, 129, 130, 131, 134, 135, 136, 139, 140, 141, 144, 145, 146, 149, 150, 151, 154, 155, 156, 159, 160, 161, 164, 165, 166, 169, 170, 171, 174, 175, 176, 179, 180, 181, 184, 185, 186, 189, 190, 191, 194, 195, 196, 199, 200, 201, 204, 205, 206, 209, 210, 211, 214, 215, 216, 219, 220, 221, 224, 225, 226, 229, 230, 231, 234, 235, 236, 239, 240, 241, 244, 245, 246, 249, 250, 251, 254, 255, 256, 259, 260, 261, 264, 265, 266, 269, 270, 271, 274, 275, 276, 279, 280, 281, 284, 285, 286, 289, 290, 291, 294, 295, 296, 299, 300, 301, 304, 305, 306, 309, 310, 311, 314, 315, 316, 319, 320, 321, 324, 325, 326, 329, 330, 331, 334, 335, 336, 339, 340, 341, 344, 345, 346, 349, 350, 351, 354, 355, 356, 359, 360, 361, 364, 365, 366, 369, 370, 371, 374, 375, 376, 379, 380, 381, 384, 385, 386, 389, 390, 391, 394, 395, 396, 399, 400, 401, 404, 405, 406, 409, 410, 411, 414, 415, 416, 419, 420, 421, 424, 425, 426, 429, 430, 431, 434, 435, 436, 439, 440, 441, 444, 445, 446, 449, 450, 451, 454, 455, 456, 459, 460, 461, 464, 465, 466, 469, 470, 471, 474, 475, 476, 479, 480, 481, 484, 485, 486, 489, 490, 491, 494, 495, 496, 499, 500, 501, 504, 505, 506, 509, 510, 511, 514, 515, 516, 519, 520, 521, 524, 525, 526, 529, 530, 531, 534, 535, 536, 539, 540, 541, 544, 545, 546, 549, 550, 551, 554, 555, 556, 559, 560, 561, 564, 565, 566, 569, 570, 571, 574, 575, 576, 579, 580, 581, 584, 585, 586, 589, 590, 591, 594, 595, 596, 599, 600, 601, 604, 605, 606, 609, 610, 611, 614, 615, 616, 619, 620, 621, 624, 625, 626, 629, 630, 631, 634, 635, 636, 639, 640, 641, 644, 645, 646, 649, 650, 651, 654, 655, 656, 659, 660, 661, 664, 665, 666, 669, 670, 671, 674, 675, 676, 679, 680, 681, 684, 685, 686, 689, 690, 691, 694, 695, 696, 699, 700, 701, 704, 705, 706, 709, 710, 711, 714, 715, 716, 719, 720, 721, 724, 725, 726, 729, 730, 731, 734, 735, 736, 739, 740, 741, 744, 745, 746, 749, 750, 751, 754, 755, 756, 759, 760, 761, 764, 765, 766, 769, 770, 771, 774, 775, 776, 779, 780, 781, 784, 785, 786, 789, 790, 791, 794, 795, 796, 799, 800, 801, 804, 805, 806, 809, 810, 811, 814, 815, 816, 819, 820, 821, 824, 825, 826, 829, 830, 831, 834, 835, 836, 839, 840, 841, 844, 845, 846, 849, 850, 851, 854, 855, 856, 859, 860, 861, 864, 865, 866, 869, 870, 871, 874, 875, 876, 879, 880, 881, 884, 885, 886, 889, 890, 891, 894, 895, 896, 899, 900, 901, 904, 905, 906, 909, 910, 911, 914, 915, 916, 919, 920, 921, 924, 925, 926, 929, 930, 931, 934, 935, 936, 939, 940, 941, 944, 945, 946, 949, 950, 951, 954, 955, 956, 959, 960, 961, 964, 965, 966, 969, 970, 971, 974, 975, 976, 979, 980, 981, 984, 985, 986, 989, 990, 991, 994, 995, 996, 999, 1000

16

16, 17, 32, 33, 48, 49, 64, 65, 80, 81, 96, 97, 112, 113, 128, 129, 144, 145, 160, 161, 176, 177, 192, 193, 208, 209, 224, 225, 240, 241, 256, 257, 272, 273, 288, 289, 304, 305, 320, 321, 336, 337, 352, 353, 368, 369, 384, 385, 400, 401, 416, 417, 432, 433, 448, 449, 464, 465, 480, 481, 496, 497, 512, 513, 528, 529, 544, 545, 560, 561, 576, 577, 592, 593, 608, 609, 624, 625, 640, 641, 656, 657, 672, 673, 688, 689, 704, 705, 720, 721, 736, 737, 752, 753, 768, 769, 784, 785, 800, 801, 816, 817, 832, 833, 848, 849, 864, 865, 880, 881, 896, 897, 912, 913, 928, 929, 944, 945, 960, 961, 976, 977, 992, 993

18

9, 10, 18, 19, 27, 28, 36, 37, 45, 46, 54, 55, 63, 64, 72, 73, 81, 82, 90, 91, 99, 100, 108, 109, 117, 118, 126, 127, 135, 136, 144, 145, 153, 154, 162, 163, 171, 172, 180, 181, 189, 190, 198, 199, 207, 208, 216, 217, 225, 226, 234, 235, 243, 244, 252, 253, 261, 262, 270, 271, 279, 280, 288, 289, 297, 298, 306, 307, 315, 316, 324, 325, 333, 334, 342, 343, 351, 352, 360, 361, 369, 370, 378, 379, 387, 388, 396, 397, 405, 406, 414, 415, 423, 424, 432, 433, 441, 442, 450, 451, 459, 460, 468, 469, 477, 478, 486, 487, 495, 496, 504, 505, 513, 514, 522, 523, 531, 532, 540, 541, 549, 550, 558, 559, 567, 568, 576, 577, 585, 586, 594, 595, 603, 604, 612, 613, 621, 622, 630, 631, 639, 640, 648, 649, 657, 658, 666, 667, 675, 676, 684, 685, 693, 694, 702, 703, 711, 712, 720, 721, 729, 730, 738, 739, 747, 748, 756, 757, 765, 766, 774, 775, 783, 784, 792, 793, 801, 802, 810, 811, 819, 820, 828, 829, 837, 838, 846, 847, 855, 856, 864, 865, 873, 874, 882, 883, 891, 892, 900, 901, 909, 910, 918, 919, 927, 928, 936, 937, 945, 946, 954, 955, 963, 964, 972, 973, 981, 982, 990, 991, 999, 1000

20

5, 16, 20, 21, 25, 36, 40, 41, 45, 56, 60, 61, 65, 76, 80, 81, 85, 96, 100, 101, 105, 116, 120, 121, 125, 136, 140, 141, 145, 156, 160, 161, 165, 176, 180, 181, 185, 196, 200, 201, 205, 216, 220, 221, 225, 236, 240, 241, 245, 256, 260, 261, 265, 276, 280, 281, 285, 296, 300, 301, 305, 316, 320, 321, 325, 336, 340, 341, 345, 356, 360, 361, 365, 376, 380, 381, 385, 396, 400, 401, 405, 416, 420, 421, 425, 436, 440, 441, 445, 456, 460, 461, 465, 476, 480, 481, 485, 496, 500, 501, 505, 516, 520, 521, 525, 536, 540, 541, 545, 556, 560, 561, 565, 576, 580, 581, 585, 596, 600, 601, 605, 616, 620, 621, 625, 636, 640, 641, 645, 656, 660, 661, 665, 676, 680, 681, 685, 696, 700, 701, 705, 716, 720, 721, 725, 736, 740, 741, 745, 756, 760, 761, 765, 776, 780, 781, 785, 796, 800, 801, 805, 816, 820, 821, 825, 836, 840, 841, 845, 856, 860, 861, 865, 876, 880, 881, 885, 896, 900, 901, 905, 916, 920, 921, 925, 936, 940, 941, 945, 956, 960, 961, 965, 976, 980, 981, 985, 996, 1000

 

 

E’ interessante esaminare il minimo primo finto nelle varie basi: dato che il minimo numero di Carmichael, 561, è un primo finto in ogni base, il minimo primo finto non può essere maggiore di 561.

Nel 1958 Schinzel dimostrò che:

  • il minimo primo finto in base b è diverso da 4 o 6 se e solo se b ≡ 2 o 11 mod 12;

  • 561 è il minimo primo finto in infinite basi.

  • al crescere della base b la sequenza dei minimi primi finti è periodica con periodo che divide 561!.

J.H. Conway, R.K. Guy, W.A. Schneeberger e N.J.A. Sloane dimostrarono nel 1997 che:

  • il minimo primo finto in base b è 4 se e solo se b ≡ 0 o 1 mod 4;

  • il minimo primo finto in base b è 6 se e solo se b ≡ 0 o 1 mod 3;

  • 8 è primo finto in base b se e solo se b ≡ 0 o 1 mod 4, quindi se e solo se 4 è primo finto in base b;

  • 9 è primo finto in base b se e solo se b ≡ 0, 1 o 8 mod 9;

  • il minimo primo finto in base b è diverso da 4, 6 o 9 se e solo se b ≡ 2, 11, 14 o 23 mod 36;

  • solo 132 numeri hanno l’onore di essere il minimo primo finto in qualche base;

  • la sequenza dei minimi primi finti è periodica, con periodo 23#277#, un numero di 122 cifre.

 

I numeri che sono i minimi primi finti, ciascuno in infinite basi, sono: 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 38, 39, 46, 49, 51, 57, 58, 62, 65, 69, 74, 82, 85, 86, 87, 91, 93, 94, 106, 111, 118, 121, 122, 123, 129, 133, 134, 141, 142, 145, 146, 158, 159, 166, 169, 177, 178, 183, 185, 194, 201, 202, 205, 206, 213, 214, 217, 218, 219, 226, 237, 249, 254, 259, 262, 265, 267, 274, 278, 289, 291, 298, 301, 302, 303, 305, 309, 314, 321, 326, 327, 334, 339, 341, 346, 358, 361, 362, 365, 381, 382, 386, 393, 394, 398, 411, 417, 422, 427, 445, 446, 447, 451, 453, 454, 458, 466, 469, 471, 478, 481, 482, 485, 489, 501, 502, 505, 511, 514, 519, 526, 529, 537, 538, 542, 543, 545, 553, 554, 561.

Sono tutti semiprimi, tranne 561.

 

La tabella mostra il minimo primo finto per le basi sino a 100 (l’inclusione delle basi 0 e 1 è consueta, ma quantomeno discutibile).

Base

Minimo primo finto

0

0

1

4

2

341

3

6

4

4

5

4

6

6

7

6

8

4

9

4

10

6

11

10

12

4

13

4

14

14

15

6

16

4

17

4

18

6

19

6

20

4

21

4

22

6

23

22

24

4

25

4

26

9

27

6

28

4

29

4

30

6

31

6

32

4

33

4

34

6

35

9

36

4

37

4

38

38

39

6

40

4

41

4

42

6

43

6

44

4

45

4

46

6

47

46

48

4

49

4

50

10

51

6

52

4

53

4

54

6

55

6

56

4

57

4

58

6

59

15

60

4

61

4

62

9

63

6

64

4

65

4

66

6

67

6

68

4

69

4

70

6

71

9

72

4

73

4

74

15

75

6

76

4

77

4

78

6

79

6

80

4

81

4

82

6

83

21

84

4

85

4

86

10

87

6

88

4

89

4

90

6

91

6

92

4

93

4

94

6

95

10

96

4

97

4

98

9

99

6

100

4

 

Bibliografia

  • Conway, John Horton;  Guy, Richard K.;  Schneeberger, W.A.;  Sloane, Neil J.A.;  "The Primary Pretenders" in Acta Arithmetica, n. 78, 1997.
  • Wells, David;  Prime Numbers, John Wiley & Sons, 2005 -

    Una miniera di informazioni sui numeri primi.

Contattami

Potete contattarmi al seguente indirizzo bitman[at]bitman.name per suggerimenti o segnalazioni d'errori relativi a questo articolo.