Zbiory zawierające liczby pierwsze
: 30 mar 2025, o 23:25
autor: ksetlak
Raz na kilka miesięcy lubię w Excelu pobawić się liczbami pierwszymi. Kiedyś znalazłem sposób na przyspieszenie algorytmu Fermata (sprawdzony za pomocą skryptu PHP, działa). Dzisiaj chciałbym tylko pokazać 3 zbiory, które cieszą moje oczy po tygodniu zabawy 
Najpierw definiuję zakres (np. 100, 1100 albo 2100), potem liczę, a na końcu po posortowaniu i ręcznym sprawdzeniu pierwszości mam coś takiego:
W trzecim zbiorze nie chciało mi się sprawdzać 2063 elementów, dlatego sprawdziłem jedynie początkowe i końcowe.
Spodziewam się, że w większych przedziałach będzie więcej "czerwonych" liczb na końcu danego zbioru, więc użyteczność tych zbiorów może być średnia, ale i tak ładnie to wygląda w porównaniu ze zbiorami, które generowałem wcześniej
Dodano po 20 godzinach 4 minutach 17 sekundach:
Napisałem skrypt w PHP robiący test pierwszości dla liczb z wklejonej listy 2063 elementów
Nie jestem najlepszy w programowanie i musiałem kombinować, ale udało się. Trzeba było przekształcić mój zbiór na poniższy, odejmując 1 w Excelu.
Podsumowując, w trzecim zbiorze jest 1217 liczb pierwszych, prawie 59% z 2063.
Sposób na tworzenie tych zbiorów jest fajny, bo wykorzystuje mój ulubiony ciąg liczb pierwszych (\(\displaystyle{ 3,5,7,11,13,17...}\)) oraz wielokrotności liczby 2 (\(\displaystyle{ 2,4,8,16,32,64...}\)). Bardzo dużo jest w środku rozwiązań, które optymalizują skuteczność. Taki zgrabny sposób, bez udziwnień.
Najpierw definiuję zakres (np. 100, 1100 albo 2100), potem liczę, a na końcu po posortowaniu i ręcznym sprawdzeniu pierwszości mam coś takiego:
https://ksetlak.pl/matematyka/20250330%20odkrycie%20wersja%20publikuj.pdfW trzecim zbiorze nie chciało mi się sprawdzać 2063 elementów, dlatego sprawdziłem jedynie początkowe i końcowe.
Spodziewam się, że w większych przedziałach będzie więcej "czerwonych" liczb na końcu danego zbioru, więc użyteczność tych zbiorów może być średnia, ale i tak ładnie to wygląda w porównaniu ze zbiorami, które generowałem wcześniej
Dodano po 20 godzinach 4 minutach 17 sekundach:
Napisałem skrypt w PHP robiący test pierwszości dla liczb z wklejonej listy 2063 elementów
https://ksetlak.pl/matematyka/test-pierwszosci.phpNie jestem najlepszy w programowanie i musiałem kombinować, ale udało się. Trzeba było przekształcić mój zbiór na poniższy, odejmując 1 w Excelu.
Ukryta treść:
14980,14976,14968,14956,14952,14940,14932,14928,14920,14908,14896,14892,14880,14872,14868,14856,14848,14836,14820,14812,14808,14800,14796,14788,14760,14752,14740,14736,14716,14712,14700,14688,14680,14668,14656,14652,14632,14628,14620,14616,14592,14568,14560,14556,14548,14536,14532,14512,14500,14488,14476,14472,14460,14452,14448,14428,14400,14392,14388,14380,14352,14340,14320,14316,14308,14296,14280,14268,14256,14248,14236,14220,14212,14196,14176,14172,14152,14148,14136,14128,14116,14100,14092,14088,14080,14056,14040,14032,14016,14008,13996,13972,13968,13932,13920,13900,13888,13876,13872,13860,13840,13828,13816,13812,13800,13788,13780,13776,13752,13732,13728,13720,13708,13696,13692,13668,13660,13648,13632,13612,13596,13576,13560,13548,13536,13528,13512,13500,13492,13476,13468,13440,13420,13396,13392,13372,13368,13360,13356,13336,13308,13300,13296,13288,13260,13252,13248,13240,13200,13192,13176,13156,13140,13132,13120,13108,13080,13072,13060,13048,13032,13020,12996,12988,12976,12972,12952,12948,12936,12916,12912,12892,12888,12876,12840,12832,12828,12820,12808,12796,12780,12768,12756,12712,12708,12688,12672,12652,12640,12628,12612,12588,12580,12568,12556,12552,12540,12516,12496,12460,12456,12448,12436,12432,12420,12412,12408,12400,12376,12372,12360,12292,12280,12268,12252,12216,12196,12168,12160,12156,12148,12136,12120,12112,12108,12096,12072,12052,12048,12040,12036,12028,12016,11992,11968,11952,11928,11916,11908,11896,11880,11872,11856,11848,11820,11812,11800,11796,11788,11776,11772,11760,11740,11700,11688,11680,11676,11628,11620,11616,11596,11572,11568,11548,11532,11520,11508,11496,11460,11448,11440,11436,11412,11392,11368,11352,11320,11292,11280,11272,11268,11260,11256,11236,11212,11200,11188,11152,11148,11116,11112,11068,11056,11040,11028,11020,11016,11008,10992,10980,10960,10956,10936,10908,10888,10876,10852,10848,10840,10836,10816,10792,10788,10756,10752,10740,10732,10728,10708,10692,10656,10608,10600,10588,10572,10540,10536,10528,10512,10480,10476,10468,10452,10440,10428,10420,10396,10392,10348,10336,10332,10320,10312,10300,10272,10260,10252,10248,10236,10228,10216,10192,10180,10168,10140,10132,10120,10096,10080,10068,10060,10008,9972,9952,9936,9928,9916,9912,9900,9856,9832,9828,9796,9780,9772,9748,9720,9700,9696,9688,9672,9660,9648,9640,9636,9628,9616,9600,9592,9588,9576,9556,9552,9532,9516,9508,9480,9460,9436,9432,9420,9412,9408,9376,9348,9340,9328,9312,9300,9292,9256,9240,9220,9208,9196,9180,9160,9136,9108,9100,9088,9076,9072,9016,9012,8992,8976,8968,8940,8932,8916,8892,8880,8856,8848,8836,8796,8776,8772,8740,8688,8680,8676,8652,8640,8632,8608,8596,8580,8572,8556,8512,8508,8500,8496,8472,8416,8412,8400,8388,8376,8368,8356,8352,8328,8320,8296,8292,8272,8268,8236,8232,8200,8188,8148,8116,8100,8092,8088,8080,8076,8032,8016,7992,7980,7960,7956,7948,7936,7932,7920,7900,7896,7872,7852,7848,7828,7800,7788,7780,7768,7752,7740,7728,7696,7672,7668,7660,7632,7620,7612,7596,7576,7560,7548,7536,7492,7480,7476,7456,7428,7420,7408,7372,7368,7348,7320,7312,7308,7288,7260,7252,7236,7200,7192,7180,7176,7168,7156,7152,7128,7120,7080,7036,7012,7008,7000,6988,6976,6972,6960,6916,6912,6900,6868,6856,6832,6828,6816,6792,6760,6756,6748,6736,6708,6700,6696,6660,6612,6592,6580,6576,6556,6552,6532,6528,6508,6496,6492,6472,6468,6448,6436,6420,6360,6340,6336,6316,6300,6288,6268,6232,6220,6196,6172,6168,6132,6120,6112,6100,6088,6052,6048,6028,6000,5988,5980,5976,5968,5920,5916,5892,5880,5848,5832,5820,5808,5772,5740,5736,5728,5716,5712,5700,5688,5668,5656,5632,5608,5572,5568,5556,5548,5512,5496,5448,5440,5436,5428,5416,5412,5388,5380,5352,5292,5272,5260,5236,5208,5196,5176,5152,5148,5140,5128,5112,5100,5080,5076,5068,5040,5028,5016,5008,4996,4992,4972,4968,4936,4912,4896,4860,4852,4840,4816,4812,4800,4788,4776,4720,4708,4680,4672,4656,4636,4620,4596,4560,4552,4548,4540,4492,4468,4456,4452,4420,4408,4396,4368,4348,4320,4312,4252,4240,4228,4216,4200,4188,4180,4176,4152,4140,4132,4128,4116,4072,4060,4056,4048,4012,4008,4000,3976,3972,3960,3952,3900,3888,3876,3868,3852,3832,3768,3736,3732,3720,3712,3708,3700,3696,3676,3648,3636,3628,3612,3592,3580,3540,3532,3528,3516,3456,3448,3432,3412,3400,3388,3372,3360,3336,3328,3316,3312,3300,3292,3280,3276,3256,3252,3220,3208,3168,3160,3136,3108,3052,3040,3000,2992,2956,2940,2928,2920,2908,2880,2868,2856,2836,2832,2812,2788,2772,2752,2748,2728,2712,2688,2668,2620,2616,2608,2592,2580,2572,2556,2520,2500,2488,2476,2472,2460,2440,2416,2412,2392,2380,2368,2356,2340,2328,2308,2296,2292,2280,2268,2256,2236,2220,2172,2160,2116,2112,2076,2068,2052,2032,1972,1956,1920,1912,1908,1888,1876,1872,1860,1852,1816,1800,1776,1740,1708,1696,1692,1668,1656,1636,1632,1612,1608,1596,1576,1548,1540,1492,1488,1456,1452,1432,1428,1408,1368,1360,1356,1348,1332,1300,1288,1276,1272,1248,1216,1200,1192,1180,1120,1116,1096,1092,1080,1072,1060,1048,1032,1020,1012,988,976,960,936,928,900,876,856,808,796,772,760,756,712,696,660,640,628,600,592,588,576,556,540,528,520,508,448,420,400,396,372,352,348,316,312,276,228,192,180,148,112,100,96,60,52,40,36,16,14946,14938,14922,14898,14890,14866,14850,14842,14826,14802,14778,14770,14746,14730,14722,14698,14682,14658,14626,14610,14602,14586,14578,14562,14506,14490,14466,14458,14418,14410,14386,14362,14346,14322,14298,14290,14250,14242,14226,14218,14170,14122,14106,14098,14082,14058,14050,14010,13986,13962,13938,13930,13906,13890,13882,13842,13786,13770,13762,13746,13690,13666,13626,13618,13602,13578,13546,13522,13498,13482,13458,13426,13410,13378,13338,13330,13290,13282,13258,13242,13218,13186,13170,13162,13146,13098,13066,13050,13018,13002,12978,12930,12922,12850,12826,12786,12762,12738,12730,12706,12666,12642,12618,12610,12586,12562,12546,12538,12490,12450,12442,12426,12402,12378,12370,12322,12306,12282,12250,12210,12178,12138,12106,12082,12058,12042,12010,11986,11970,11938,11922,11866,11826,11778,11770,11730,11722,11706,11698,11658,11602,11586,11578,11562,11506,11490,11482,11466,11386,11370,11338,11298,11266,11250,11226,11202,11146,11130,11106,11082,11050,11026,10978,10962,10938,10930,10890,10882,10858,10818,10810,10770,10762,10738,10666,10650,10642,10626,10602,10578,10546,10522,10498,10410,10402,10362,10330,10290,10266,10242,10210,10162,10146,10122,10098,10090,10066,10018,9978,9906,9898,9882,9858,9850,9826,9810,9802,9786,9738,9730,9706,9570,9546,9522,9490,9418,9378,9322,9306,9298,9282,9258,9226,9210,9202,9178,9130,9090,9082,9066,9058,9042,9018,8970,8922,8890,8842,8818,8802,8778,8746,8730,8698,8682,8626,8610,8586,8578,8562,8538,8530,8506,8466,8386,8362,8346,8338,8242,8226,8218,8178,8130,8122,8082,8050,8026,8002,7978,7906,7882,7866,7858,7810,7770,7722,7690,7626,7570,7546,7530,7522,7506,7498,7458,7410,7386,7362,7290,7282,7218,7210,7122,7098,7066,7042,7026,7018,7002,6970,6946,6906,6898,6858,6802,6762,6738,6690,6682,6666,6658,6618,6570,6562,6498,6490,6466,6450,6442,6402,6370,6298,6202,6186,6162,6130,6066,6058,6042,6010,5946,5938,5922,5890,5866,5842,5826,5778,5770,5682,5658,5650,5626,5610,5586,5530,5506,5490,5482,5458,5442,5418,5370,5346,5322,5266,5250,5226,5178,5146,5122,5106,5002,4930,4890,4858,4842,4818,4810,4786,4698,4650,4642,4578,4546,4530,4482,4426,4386,4378,4362,4330,4306,4282,4266,4258,4242,4218,4186,4170,4162,4138,4098,4090,4050,4018,4002,3946,3906,3858,3850,3826,3810,3802,3738,3682,3666,3642,3610,3586,3570,3498,3466,3426,3402,3378,3346,3306,3258,3202,3186,3162,3138,3130,3018,3010,2970,2938,2922,2866,2850,2818,2770,2746,2698,2682,2658,2578,2538,2530,2466,2362,2346,2338,2290,2266,2250,2202,2178,2146,2130,2098,2010,2002,1986,1978,1930,1818,1810,1786,1762,1738,1722,1698,1690,1642,1626,1578,1570,1530,1522,1458,1450,1386,1362,1282,1218,1186,1170,1162,1146,1138,1050,1018,970,946,906,898,882,858,850,826,786,778,730,690,682,642,586,562,546,522,490,466,442,378,330,322,306,250,226,210,178,138,106,82,58,14878,14862,14830,14782,14766,14718,14686,14670,14638,14590,14542,14526,14478,14446,14430,14382,14350,14302,14238,14206,14190,14158,14142,14110,13998,13966,13918,13902,13822,13806,13758,13710,13678,13630,13582,13566,13486,13470,13438,13422,13326,13230,13198,13182,13150,13102,13086,13006,12958,12910,12862,12846,12798,12766,12750,12670,12558,12526,12510,12478,12366,12318,12238,12222,12190,12142,12078,12030,11982,11950,11902,11838,11806,11742,11662,11646,11566,11550,11502,11470,11422,11358,11326,11310,11278,11182,11118,11086,11022,10990,10942,10846,10830,10686,10638,10558,10510,10462,10446,10398,10318,10270,10222,10206,10158,10110,10078,10062,9966,9886,9870,9838,9790,9742,9726,9630,9598,9550,9486,9406,9342,9262,9198,9150,9102,9070,9006,8958,8926,8862,8830,8718,8638,8542,8526,8446,8430,8398,8382,8302,8190,8158,8142,8110,7998,7966,7950,7918,7758,7726,7662,7582,7518,7486,7438,7390,7278,7246,7198,7150,7086,7038,6942,6910,6862,6846,6766,6750,6702,6622,6606,6526,6510,6462,6318,6286,6270,6238,6190,6142,6078,6030,5982,5806,5790,5710,5646,5566,5518,5470,5406,5310,5278,5230,5182,5166,5118,5022,4942,4798,4782,4750,4702,4686,4638,4606,4590,4558,4462,4446,4398,4126,4078,4030,3966,3822,3742,3630,3598,3582,3550,3502,3438,3406,3390,3342,3246,3166,3150,3118,3102,3070,3022,2926,2830,2766,2670,2622,2590,2542,2478,2446,2382,2350,2238,2206,2158,2142,2110,2062,2046,1998,1918,1758,1710,1678,1662,1470,1438,1422,1342,1246,1230,1150,1086,1038,990,942,798,750,718,702,606,526,430,366,238,126,78,46,30,14742,14710,14646,14550,14518,14422,14358,14326,14262,14166,14070,14038,13942,13878,13846,13750,13686,13590,13462,13398,13366,13302,13270,13206,12982,12918,12822,12790,12630,12598,12502,12406,12342,12246,12150,12118,11958,11926,11862,11830,11638,11446,11382,11350,11286,11190,11158,10998,10902,10806,10710,10678,10582,10518,10486,10326,10102,10038,10006,9942,9718,9622,9462,9430,9366,9270,9142,9046,8950,8886,8790,8662,8598,8470,8310,8278,8118,8086,7990,7926,7830,7702,7638,7606,7542,7350,7222,7158,7030,6966,6870,6678,6646,6358,6262,6102,6006,5910,5878,5782,5622,5526,5430,5398,5302,5206,5142,5110,4918,4758,4726,4662,4566,4470,4438,4246,4182,4086,3958,3798,3670,3510,3382,3286,3190,3126,2998,2902,2838,2710,2646,2422,2262,2070,2038,1878,1846,1782,1750,1590,1366,1302,1270,1206,982,918,886,822,502,438,310,150,22,14470,14406,14278,14086,14022,13830,13702,13638,13510,13318,13126,13062,12870,12742,12678,12486,12358,12166,11910,11782,11718,11590,11526,11398,10950,10822,10630,10566,10246,10182,9990,9798,9670,9478,9286,9222,8902,8838,8710,8646,8262,7878,7750,7686,7558,7366,7302,6982,6790,6598,6406,6342,6150,6022,5958,5638,5190,5062,4998,4870,4422,4230,3910,3846,3718,3526,3270,3078,2886,2758,2566,2310,2182,1926,1606,1542,1222,1158,966,838,646,390,262,198,70,13926,13798,13542,13158,13030,12646,12390,12262,12006,11622,11238,11110,10726,10470,10342,9958,9702,9318,8806,8550,8422,8166,8038,7782,6886,6630,6246,6118,5478,5350,4966,4582,4326,3942,3558,3430,2790,2662,2406,2278,1510,742,486,358,102,12838,12582,12070,11302,11046,10278,9766,9510,8998,8230,7462,7206,6438,5926,5670,4902,4390,3622,2598,2086,1830,1318,1062,550,10662,10150,9126,7590,7078,5542,4518,4006,2982,1446,6310,5286,3238,166
Sposób na tworzenie tych zbiorów jest fajny, bo wykorzystuje mój ulubiony ciąg liczb pierwszych (\(\displaystyle{ 3,5,7,11,13,17...}\)) oraz wielokrotności liczby 2 (\(\displaystyle{ 2,4,8,16,32,64...}\)). Bardzo dużo jest w środku rozwiązań, które optymalizują skuteczność. Taki zgrabny sposób, bez udziwnień.