Don't take life too seriously: it's just a temporary situation

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 composti 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.

Sono anche detti “pseudoprimi deboli”.

 

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

 

I primi finti pari in base 2 non sono molto comuni: il minimo è 161038, scoperto da Lehmer nel 1950, tuttavia sono infiniti, come dimostrato da N.G.W.H. Beeger nel 1951; devono essere della forma 4n + 2 e avere almeno due fattori primi dispari.

I primi finti pari in base 2 minori di 109 sono: 161038, 215326, 2568226, 3020626, 7866046, 9115426, 49699666, 143742226, 161292286, 196116194, 209665666, 213388066, 293974066, 336408382, 377994926, 410857426, 665387746, 667363522, 672655726, 760569694.

Qui trovate i primi finti pari in base 2 minori di 2 • 1015 (Max Alekseyev, R.G.E. Pinch, N.J.A. Sloane, Robert G. Wilson V, The Online Encyclopedia of Integer Sequences http://oeis.org).

 

Possono esistere primi finti pari in base 2 multipli di quadrati di primi p, ma solo se p è un primo di Wieferich e il periodo di  in base 2 ha lunghezza dispari; l’unico primo noto con queste proprietà è 3511. Il minimo primo finto pari in base 2 multiplo di quadrati è: 190213279479817426 = 2 • 7 • 79 • 1951 • 35112 • 7151 (Max Alekseyev, 2017).

Qui trovate i primi finti pari in base 2 multipli di quadrati noti (Max Alekseyev, The Online Encyclopedia of Integer Sequences http://oeis.org).

 

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.