Today I'm 28285 days old and one of the properties of this number is that it's magnanimous, meaning that inserting a "+" between any two digits produces a prime. In this case we have:
- 2+8285 = 8287 is prime
- 28+285 = 313 is prime
- 282+85 = 367 is prime
- 2828+5 = 2833 is prime
I review these sorts of numbers in my post titled Magnanimous Numbers from December of 2020. This got me thinking about subtraction instead of addition or to put it formally and applying the absolute value operator:
| Minuend - Subtrahend | = | Difference |
Take as an example, the number 28041 where we have:
- | 2 - 8041 | = 8039 is prime
- | 28 - 41 | = 13 is prime
- | 280 - 41 | = 239 is prime
- | 2804 - 1 | = 2803 is prime
Here is the full list of 230 such numbers up to 40000 (permalink):
13, 14, 16, 18, 20, 24, 25, 27, 29, 30, 31, 35, 36, 38, 41, 42, 46, 47, 49, 50, 52, 53, 57, 58, 61, 63, 64, 68, 69, 70, 72, 74, 75, 79, 81, 83, 85, 86, 92, 94, 96, 97, 103, 108, 114, 118, 130, 132, 138, 154, 174, 190, 198, 207, 209, 225, 245, 263, 269, 285, 301, 310, 332, 334, 356, 370, 376, 392, 409, 421, 441, 447, 463, 465, 487, 503, 507, 518, 536, 552, 558, 601, 623, 629, 643, 665, 667, 689, 709, 710, 714, 730, 736, 754, 774, 790, 796, 801, 845, 867, 907, 912, 956, 970, 992, 1003, 1114, 1152, 1158, 1332, 1354, 1510, 1614, 1758, 1930, 1992, 1998, 2007, 2009, 2025, 2241, 2421, 2465, 2603, 2609, 2663, 2685, 2729, 2825, 2841, 2865, 3154, 3310, 3370, 3376, 3392, 3436, 3512, 3596, 3634, 3730, 3790, 3956, 3970, 4021, 4047, 4087, 4201, 4245, 4341, 4401, 4467, 4603, 4663, 4665, 4687, 4801, 4867, 5052, 5156, 5514, 5592, 5658, 5952, 5996, 6001, 6023, 6029, 6043, 6065, 6067, 6089, 6163, 6203, 6245, 6269, 6365, 6563, 6607, 6623, 6689, 6863, 7009, 7134, 7270, 7314, 7390, 7396, 7570, 7734, 7936, 8069, 8241, 8265, 8285, 8607, 8625, 8667, 8669, 8847, 8865, 8885, 9092, 9290, 9356, 9736, 9896, 9976, 9992, 10003, 11532, 11598, 11952, 15154, 17574, 20007, 20063, 20841, 22041, 22401, 24029, 24203, 24263, 24623, 26009, 26265, 27029, 28041, 33190, 37396
With multiplication we would need to add 1 or subtract 1. Let's take 28034 as an example of the former where we have:
- 2 x 8034 + 1 = 16069 is prime
- 28 x 34 + 1 = 953 is prime
- 280 x 34 + 1 = 9521 is prime
- 2803 x 4 + 1 = 11213 is prime
The 421 numbers that satisfy in the range up to 40000 are (permalink):
11, 12, 14, 16, 21, 22, 23, 25, 26, 28, 29, 32, 34, 36, 41, 43, 44, 47, 49, 52, 56, 58, 61, 62, 63, 65, 66, 67, 74, 76, 82, 85, 89, 92, 94, 98, 101, 104, 106, 112, 116, 118, 128, 136, 142, 152, 166, 178, 182, 202, 203, 205, 209, 221, 223, 229, 236, 244, 248, 254, 256, 263, 265, 274, 281, 298, 302, 306, 326, 332, 336, 352, 374, 376, 392, 394, 401, 407, 425, 428, 434, 448, 449, 484, 487, 502, 508, 512, 526, 542, 548, 556, 562, 566, 584, 586, 601, 603, 607, 616, 625, 626, 632, 647, 652, 658, 661, 663, 666, 704, 706, 718, 728, 734, 748, 766, 778, 794, 805, 812, 829, 832, 844, 865, 874, 884, 902, 934, 958, 968, 982, 992, 1001, 1004, 1006, 1018, 1052, 1106, 1108, 1156, 1162, 1178, 1192, 1226, 1228, 1256, 1312, 1352, 1448, 1498, 1568, 1616, 1658, 1708, 1726, 1756, 1862, 1886, 1982, 2002, 2003, 2009, 2023, 2026, 2065, 2074, 2086, 2176, 2204, 2216, 2221, 2228, 2243, 2281, 2306, 2336, 2384, 2386, 2443, 2524, 2576, 2645, 2648, 2686, 2716, 2774, 2776, 2849, 2876, 2998, 3002, 3006, 3034, 3074, 3136, 3152, 3224, 3244, 3302, 3404, 3412, 3526, 3542, 3566, 3574, 3592, 3704, 3734, 3932, 4001, 4007, 4034, 4043, 4078, 4102, 4142, 4168, 4234, 4265, 4274, 4304, 4402, 4423, 4447, 4504, 4685, 4724, 4807, 4825, 4918, 4982, 5008, 5036, 5066, 5162, 5198, 5206, 5302, 5344, 5366, 5506, 5518, 5534, 5542, 5558, 5594, 5608, 5612, 5708, 5726, 5786, 5918, 6001, 6003, 6007, 6056, 6076, 6112, 6166, 6178, 6182, 6245, 6263, 6268, 6382, 6412, 6443, 6452, 6467, 6518, 6536, 6601, 6605, 6607, 6632, 6667, 6766, 6802, 6902, 7004, 7006, 7094, 7118, 7126, 7244, 7384, 7424, 7516, 7606, 7706, 7886, 7918, 7948, 8005, 8042, 8056, 8186, 8201, 8249, 8261, 8266, 8324, 8332, 8462, 8474, 8516, 8536, 8542, 8602, 8662, 8794, 8912, 8942, 8984, 9002, 9034, 9112, 9128, 9158, 9254, 9278, 9428, 9494, 9502, 9532, 9652, 9704, 9748, 9932, 9992, 10004, 10018, 10028, 10136, 10312, 10336, 10448, 11102, 11276, 11482, 11552, 11578, 11608, 11662, 12026, 12206, 12656, 12728, 13256, 13312, 14002, 14098, 14158, 14548, 14758, 14798, 14812, 14968, 15208, 15856, 15952, 16142, 17126, 18116, 18536, 18736, 18962, 19012, 19402, 19768, 19832, 20002, 20021, 20254, 20849, 20866, 21778, 22178, 22616, 22778, 22948, 23834, 23876, 24043, 24118, 24334, 24424, 24824, 25228, 25298, 25366, 25556, 25844, 26005, 26516, 26665, 27028, 27356, 27784, 28034, 28516, 28874, 29708, 30274, 30362, 30662, 30704, 32024, 32066, 32234, 33106, 33182, 33784, 34492, 35206, 36676, 38032, 38362, 38512, 38734, 39532
If instead we subtract 1, we get the following 400 numbers in the range up to 40000 (permalink):
13, 14, 16, 18, 22, 23, 24, 26, 27, 29, 31, 32, 34, 36, 38, 41, 42, 43, 45, 46, 48, 54, 56, 61, 62, 63, 64, 65, 67, 68, 69, 72, 76, 81, 83, 84, 86, 89, 92, 96, 98, 103, 106, 108, 114, 124, 138, 154, 162, 168, 174, 184, 198, 203, 204, 207, 209, 212, 222, 236, 242, 264, 269, 296, 301, 302, 306, 308, 324, 334, 336, 338, 384, 386, 398, 402, 405, 406, 421, 426, 427, 441, 456, 463, 468, 492, 496, 504, 512, 522, 536, 548, 572, 588, 596, 601, 603, 604, 607, 608, 618, 629, 638, 642, 643, 664, 667, 684, 702, 706, 714, 726, 762, 784, 786, 792, 801, 803, 806, 809, 825, 834, 845, 846, 863, 876, 902, 908, 912, 948, 956, 962, 972, 986, 992, 996, 1006, 1038, 1062, 1084, 1104, 1152, 1308, 1314, 1368, 1422, 1504, 1524, 1548, 1608, 1614, 1654, 1662, 1692, 1734, 1752, 1824, 1854, 1864, 1884, 1968, 1992, 2003, 2007, 2012, 2036, 2052, 2064, 2112, 2274, 2427, 2465, 2486, 2609, 2612, 2664, 2724, 2736, 2805, 3002, 3008, 3086, 3198, 3234, 3304, 3318, 3368, 3408, 3498, 3504, 3596, 3624, 3634, 3638, 3668, 3786, 3926, 3976, 4005, 4006, 4021, 4026, 4083, 4152, 4158, 4188, 4207, 4221, 4246, 4272, 4306, 4326, 4396, 4407, 4458, 4692, 4801, 4818, 4843, 4906, 4962, 4992, 5004, 5102, 5148, 5168, 5172, 5202, 5214, 5334, 5376, 5418, 5462, 5492, 5538, 5604, 5622, 5756, 5784, 6001, 6004, 6008, 6012, 6043, 6045, 6067, 6098, 6264, 6308, 6334, 6368, 6402, 6465, 6542, 6603, 6665, 6714, 6754, 6774, 6834, 6858, 6912, 6998, 7002, 7152, 7242, 7422, 7452, 7476, 7546, 7554, 7564, 7806, 7876, 7896, 7926, 7996, 8003, 8006, 8063, 8154, 8184, 8198, 8289, 8445, 8483, 8598, 8685, 8784, 8948, 9296, 9542, 9672, 9756, 9806, 9836, 9902, 9972, 9976, 10014, 10024, 10062, 10234, 10422, 10548, 10662, 11088, 11118, 11202, 12004, 12504, 12522, 12634, 13308, 13638, 14022, 14448, 14592, 14958, 15114, 15438, 15504, 15648, 15684, 16008, 16038, 16654, 16962, 17742, 18054, 18468, 19398, 20007, 20012, 20112, 20154, 20205, 20289, 20427, 20724, 20784, 21102, 21212, 21272, 21774, 21792, 21962, 22014, 22122, 22164, 22212, 22242, 22401, 22605, 22821, 22824, 22962, 23486, 24086, 24602, 25122, 25386, 26004, 26412, 26574, 26669, 27036, 27252, 27542, 27756, 27912, 28056, 28284, 28896, 29072, 29292, 29402, 29702, 30086, 30098, 30604, 30848, 31234, 32004, 32514, 33336, 33354, 33368, 33726, 34908, 35706, 36124, 36168, 37134, 37204, 37356, 38036, 38184, 39006, 39048
Let's take the last member, 39048, as an example:
- 3 x 9048 - 1 = 27143 is prime
- 39 x 48 - 1 = 1871 is prime
- 390 x 48 - 1 = 18719 is prime
- 3904 x 8 - 1 = 31231 is prime
No comments:
Post a Comment