4^a+2^a+1的最大素因子

太阳 · 发表于 2023-5-8 19:25

本帖最后由太阳于 2023-5-8 19:39 编辑

已知：偶数\(a＞18\)，\(\left( 4^a+2^a+1\right)\)的最大质因数是\(c\)
求证：\(\sqrt{4^a+2^a+1}＞c\)
\(\left( 4^a+2^a+1\right)\)分解质因数，验证3000范围内没有找到反例，有些数没有分解质因数，无法验证
如果此命题是正确，也找不到大素数，找大素数困难重重

太阳 · 发表于 2023-5-8 22:20

\(\left( 4^a+2^a+1\right)\)分解所有的质因数，验证3000范围内没有找到反例，基本上确定命题是正确
没有完全分解的复合因子有的大于平方根，无法验证

太阳 · 发表于 2023-5-8 22:27

4^a+2^a+1，检验和验证：a＝81，153，405，369，639，没有找到反例
549，1251，1503，1719，2169，3897。4041，4113，5553，6147

yangchuanju · 发表于 2023-5-9 05:18

太阳发表于 2023-5-8 22:20
\(\left( 4^a+2^a+1\right)\)分解所有的质因数，验证3000范围内没有找到反例，基本上确定命题是正确
没有 ...

没有完全分解的复合因子都小于平方根，其中的最大素因子更小于平方根啦，怎么无法验证呢？

分解情况复合因子 a 4^a+2^a+1对数平方根对数复合因子对数对数差
CF 7519358646...29<300> 1002 603.2641113 301.6320557 299.8761808 1.755874856
CF 1692963625...53<248> 1006 605.6723513 302.8361756 247.2286476 55.60752801
CF 1532924013...81<174> 1010 608.0805912 304.0402956 173.1855206 130.854775
CF 5584289022...89<246> 1014 610.4888312 305.2444156 245.7469679 59.49744772
CF 2913792036...41<304> 1016 611.6929512 305.8464756 303.4644586 2.382017043
CF 5804743853...67<256> 1022 615.3053111 307.6526556 255.7637831 51.88887251
CF 3354994836...21<237> 1030 620.1217911 310.0608955 236.5256919 73.53520368
CF 1249353004...99<274> 1034 622.530031 311.2650155 273.0966852 38.16833035
CF 5167263916...01<311> 1038 624.938271 312.4691355 310.7132606 1.755874856
CF 2242548162...01<189> 1040 626.142391 313.0711955 188.3507418 124.7204537
CF 4068271388...09<286> 1042 627.346511 313.6732555 285.6094099 28.06384557
CF 1439267095...97<285> 1046 629.7547509 314.8773755 284.1581414 30.71923407
CF 4392034930...73<269> 1048 630.9588709 315.4794355 268.6426658 46.83676967
CF 8588614549...87<276> 1054 634.5712309 317.2856154 275.9339231 41.35169232
CF 3253564237...59<299> 1058 636.9794708 318.4897354 298.5123594 19.97737603
CF 1357472144...11<282> 1066 641.7959508 320.8979754 281.1327309 39.76524445
CF 2826277647...11<251> 1070 644.2041907 322.1020954 250.4512148 71.65088054
CF 2316616775...01<285> 1074 646.6124307 323.3062153 284.3648542 38.94136115
CF 6938350262...11<232> 1078 649.0206707 324.5103353 231.8412562 92.66907911
CF 4248332685...01<249> 1080 650.2247906 325.1123953 248.6282185 76.4841768
CF 1768197700...27<314> 1086 653.8371506 326.9185753 313.2475308 13.67104447
CF 1955735176...17<270> 1088 655.0412706 327.5206353 269.29131 58.22932524
CF 1003593096...81<240> 1090 656.2453905 328.1226953 239.0015577 89.12113761
CF 7074471505...19<329> 1094 658.6536305 329.3268153 328.849694 0.477121255
CF 6797888520...27<308> 1098 661.0618705 330.5309352 307.832374 22.6985612
CF 1371533932...53<293> 1102 663.4701104 331.7350552 292.1372066 39.59784867
CF 3929327504...61<205> 1104 664.6742304 332.3371152 204.5943182 127.742797
CF 1234824571...27<261> 1106 665.8783504 332.9391752 260.0916053 72.84756994
CF 3456865135...11<244> 1110 668.2865904 334.1432952 243.5386824 90.60461275
CF 4756354226...11<330> 1114 670.6948303 335.3474152 329.6772742 5.670140979
CF 3577381366...63<300> 1118 673.1030703 336.5515352 299.5535652 36.99796991
CF 1111574879...41<290> 1122 675.5113103 337.7556551 289.0459387 48.70971641
CF 3038462375...19<339> 1126 677.9195502 338.9597751 338.4826539 0.477121255
CF 7299729113...97<327> 1128 679.1236702 339.5618351 326.8633067 12.69852836
CF 7585076326...91<253> 1130 680.3277902 340.1638951 252.87996 87.28393514
CF 1665364180...57<269> 1134 682.7360302 341.3680151 268.2215092 73.14650586
CF 1429842743...01<338> 1136 683.9401501 341.9700751 337.1552883 4.814786799
CF 1225119825...79<329> 1138 685.1442701 342.5721351 328.0881786 14.4839565
CF 1155905463...09<337> 1142 687.5525101 343.776255 336.0629223 7.713332732
CF 4594017653...53<274> 1144 688.7566301 344.378315 273.6621927 70.71612238
CF 6077857239...71<332> 1146 689.9607501 344.980375 331.7837505 13.19662454
CF 6117232749...61<347> 1152 693.57311 346.786555 346.786555 2.01794E-11
CF 2331544026...01<316> 1154 694.77723 347.388615 315.3676436 32.02097138
CF 2903840927...67<339> 1158 697.18547 348.592735 338.4629728 10.12976216
CF 6083451748...61<220> 1160 698.3895899 349.194795 219.7841501 129.4106449
CF 2838884609...51<297> 1162 699.5937099 349.796855 296.4531477 53.34370722
CF 6890256104...01<307> 1166 702.0019499 351.0009749 306.8382354 44.16273958
CF 7215504603...09<330> 1168 703.2060699 351.6030349 329.8582667 21.74476823
CF 1858391835...23<348> 1174 706.8184298 353.4092149 347.2691373 6.140077621
CF 2378660515...01<295> 1178 709.2266698 354.6133349 294.3763325 60.23700243
CF 9157067908...79<321> 1182 711.6349097 355.8174549 320.9617564 34.85569844
CF 3793943167...91<308> 1186 714.0431497 357.0215749 307.5790908 49.44248404
CF 5439443701...01<189> 1190 716.4513897 358.2256948 188.7355545 169.4901404
CF 2538003263...21<307> 1192 717.6555097 358.8277548 306.4044922 52.42326266
CF 1693059190...61<339> 1194 718.8596296 359.4298148 338.2286721 21.20114268
CF 3535306562...67<324> 1198 721.2678696 360.6339348 323.5484271 37.08550773
CF 8625182057...83<344> 1202 723.6761096 361.8380548 343.9357683 17.90228652
CF 1056313119...51<351> 1206 726.0843495 363.0421748 350.0237927 13.0183821
CF 5366273046...91<223> 1210 728.4925895 364.2462948 222.7296728 141.516622
CF 1074331654...17<360> 1214 730.9008295 365.4504147 359.0311384 6.419276364
CF 1657554913...09<320> 1216 732.1049495 366.0524747 319.2194679 46.8330068
CF 2868443070...47<286> 1218 733.3090694 366.6545347 285.4576462 81.19688848
CF 1630157204...11<333> 1222 735.7173094 367.8586547 332.2122295 35.64642521
CF 6117232749...61<347> 1224 736.9214294 368.4607147 346.786555 21.67415969
CF 4745055959...17<360> 1226 738.1255494 369.0627747 359.6762413 9.386533346
CF 1862861008...21<279> 1230 740.5337893 370.2668947 278.2701805 91.99671421
CF 1114739082...89<346> 1234 742.9420293 371.4710146 345.0471732 26.42384142
CF 1454283469...99<350> 1238 745.3502693 372.6751346 349.1626491 23.51248556
CF 9174308267...01<289> 1240 746.5543892 373.2771946 288.9625733 84.31462129
CF 1861344789...59<336> 1242 747.7585092 373.8792546 335.2698268 38.60942779
CF 1227031054...29<307> 1246 750.1667492 375.0833746 306.0888556 68.99451904
CF 2401672728...21<300> 1248 751.3708692 375.6854346 299.3805138 76.30492076
CF 4026423028...51<252> 1250 752.5749892 376.2874946 251.6049194 124.6825752
CF 1849442743...07<309> 1254 754.9832291 377.4916146 308.2670409 69.22457367
CF 1847881433...33<366> 1256 756.1873491 378.0936746 365.2666741 12.82700045
CF 1070507563...51<348> 1258 757.3914691 378.6957345 347.0295897 31.66614481
CF 2646874032...19<380> 1262 759.7997091 379.8998545 379.4227333 0.477121255
CF 1740410322...09<380> 1266 762.207949 381.1039745 379.2406517 1.86332286
CF 3360684944...91<297> 1270 764.616189 382.3080945 296.5264278 85.78166669
CF 4102940057...33<286> 1274 767.024429 383.5122145 285.6130952 97.8991193
CF 2852148991...09<360> 1278 769.4326689 384.7163345 359.4551722 25.26116225
CF 1806557097...19<365> 1282 771.8409089 385.9204544 364.2568517 21.66360275
CF 8619934500...69<377> 1286 774.2491488 387.1245744 376.935504 10.18907046
CF 1408152609...77<259> 1288 775.4532688 387.7266344 258.1486497 129.5779847
CF 1317314976...11<285> 1290 776.6573888 388.3286944 284.1196896 104.2090048
CF 2995430880...63<364> 1294 779.0656288 389.5328144 363.4764593 26.05635509
CF 1356184407...13<324> 1298 781.4738687 390.7369344 323.1323187 67.60461563
CF 2872293586...71<315> 1302 783.8821087 391.9410544 314.4582288 77.48282553
CF 1394231438...61<372> 1304 785.0862287 392.5431143 371.1443349 21.39877947
CF 2794500419...37<378> 1306 786.2903487 393.1451743 377.4463042 15.69887016
CF 3409299273...51<304> 1310 788.6985886 394.3492943 303.5326651 90.81662919
CF 2081618201...01<386> 1312 789.9027086 394.9513543 385.3184011 9.632953235
CF 1751065426...11<366> 1314 791.1068286 395.5534143 365.2433024 30.31011193
CF 1907273744...19<397> 1318 793.5150686 396.7575343 396.280413 0.477121255
CF 3296810823...53<394> 1322 795.9233085 397.9616543 393.518094 4.443560241
CF 1349965213...17<323> 1326 798.3315485 399.1657743 322.1303226 77.03545167
CF 1606435419...81<384> 1328 799.5356685 399.7678342 383.2058633 16.56197097
CF 2663595705...61<220> 1330 800.7397885 400.3698942 219.4254683 180.9444259
CF 8084494418...51<366> 1334 803.1480284 401.5740142 365.9076529 35.66636135
CF 5117579285...21<328> 1336 804.3521484 402.1760742 327.7090646 74.46700963
CF 1437663128...29<394> 1338 805.5562684 402.7781342 393.1576571 9.620477064
CF 1887726310...57<357> 1342 807.9645084 403.9822542 356.275939 47.70631515
CF 9162088200...97<385> 1346 810.3727483 405.1863742 384.9619945 20.2243797
CF 1385395978...01<300> 1350 812.7809883 406.3904941 299.1415739 107.2489202
CF 1248151190...57<307> 1352 813.9851083 406.9925541 306.0962672 100.8962869
CF 1284492755...13<384> 1354 815.1892283 407.5946141 383.1087317 24.48588247
CF 1062152833...91<348> 1358 817.5974682 408.7987341 347.026187 61.7725471
CF 1533490284...61<295> 1360 818.8015882 409.4007941 294.185681 115.2151131
CF 4017861369...93<400> 1362 820.0057082 410.0028541 399.6039949 10.39885915
CF 2300784998...39<411> 1366 822.4139482 411.2069741 410.361876 0.84509804
CF 3981486834...73<373> 1368 823.6180681 411.8090341 372.6000453 39.20898878
CF 2689610136...21<289> 1370 824.8221881 412.4110941 288.4296893 123.9814047
CF 1332979502...97<368> 1374 827.2304281 413.615214 367.1248235 46.49039057
CF 1922934778...69<333> 1376 828.4345481 414.217274 332.2839646 81.93330948
CF 4363626969...17<335> 1378 829.638668 414.819334 334.6398476 80.17948641
CF 1507842456...79<416> 1382 832.046908 416.023454 415.178356 0.84509804
CF 9566181290...93<399> 1384 833.251028 416.625514 398.9807386 17.64477539
CF 9792914458...67<313> 1386 834.455148 417.227574 312.990912 104.236662
CF 7667371329...51<305> 1390 836.8633879 418.431694 304.8846465 113.5470475
CF 1619387109...93<365> 1392 838.0675079 419.033754 364.2093507 54.82440329
CF 4535477541...13<356> 1394 839.2716279 419.635814 355.656623 63.97919093
CF 3214967902...83<415> 1398 841.6798679 420.8399339 414.5071766 6.332757297
CF 6060855634...01<223> 1400 842.8839879 421.4419939 222.7825339 198.65946
CF 1581087411...51<422> 1402 844.0881078 422.0440539 421.1989559 0.84509804
CF 1607622768...77<364> 1406 846.4963478 423.2481739 363.2061841 60.04198976
CF 9538352369...91<300> 1410 848.9045878 424.4522939 299.9794734 124.4728205
CF 5200022845...79<336> 1414 851.3128277 425.6564139 335.7160053 89.94040862
CF 7551498865...41<411> 1418 853.7210677 426.8605339 410.8780332 15.98250069
CF 5571909275...19<412> 1422 856.1293077 428.0646538 411.746004 16.3186498
CF 1903575173...13<397> 1424 857.3334277 428.6667138 396.27957 32.38714379
CF 1568735374...93<342> 1426 858.5375476 429.2687738 341.1955497 88.07322413
CF 1722241391...11<264> 1430 860.9457876 430.4728938 263.236094 167.2367998
CF 1612642333...97<392> 1432 862.1499076 431.0749538 391.2075381 39.86741573
CF 3600801087...37<414> 1434 863.3540276 431.6770138 413.5563991 18.12061465
CF 1367995743...09<419> 1438 865.7622675 432.8811338 418.1360847 14.74504902
CF 2347224163...61<281> 1440 866.9663875 433.4831938 280.3705546 153.1126392
CF 1107612281...53<328> 1442 868.1705075 434.0852537 327.0443878 107.040866
CF 3415850654...01<434> 1446 870.5787475 435.2893737 433.5334989 1.755874856
CF 6615381289...61<407> 1448 871.7828674 435.8914337 406.8205549 29.07087884
CF 2591726152...01<294> 1450 872.9869874 436.4934937 293.4135891 143.0799046
CF 7099731154...57<393> 1454 875.3952274 437.6976137 392.8512419 44.84637179
CF 1096038891...69<313> 1456 876.5993474 438.2996737 312.039826 126.2598477
CF 1912682319...67<402> 1458 877.8034674 438.9017337 401.2816428 37.62009083
CF 2204425028...41<399> 1462 880.2117073 440.1058537 398.3432953 41.76255833
CF 2480158224...53<399> 1464 881.4158273 440.7079137 398.3944794 42.31343426
CF 1635529654...03<406> 1466 882.6199473 441.3099736 405.2136584 36.09631522
CF 2743063342...61<304> 1470 885.0281873 442.5140936 303.4382358 139.0758578
CF 4767838195...17<347> 1472 886.2323072 443.1161536 346.6783215 96.43783211
CF 1792424794...73<433> 1478 889.8446672 444.9223336 432.2534409 12.66889265
CF 1436876825...01<323> 1480 891.0487872 445.5243936 322.1574195 123.366974
CF 2241810551...11<364> 1482 892.2529071 446.1264536 363.3505989 82.77585467
CF 6608760454...89<423> 1486 894.6611471 447.3305736 422.82012 24.51045355
CF 4360186443...69<425> 1488 895.8652671 447.9326335 424.6395051 23.29312849
CF 3471265886...71<310> 1490 897.0693871 448.5346935 309.5404879 138.9942057
CF 3966933815...03<426> 1494 899.477627 449.7388135 425.598455 24.14035857
CF 5844199157...21<346> 1496 900.681747 450.3408735 345.766725 104.5741485
CF 1411843282...71<384> 1498 901.885867 450.9429335 383.1497865 67.79314701
CF 1037635148...17<449> 1502 904.294107 452.1470535 448.0160447 4.131008813
CF 4360149882...91<445> 1506 906.7023469 453.3511735 444.6395014 8.711672051
CF 1559542307...21<355> 1510 909.1105869 454.5552935 354.1929972 100.3622963
CF 1352797426...17<348> 1512 910.3147069 455.1573534 347.1312328 108.0261207
CF 2309910350...17<444> 1514 911.5188269 455.7594134 443.3635951 12.39581831
CF 1435997219...89<360> 1518 913.9270668 456.9635334 359.1571536 97.80637982
CF 5028404989...41<281> 1520 915.1311868 457.5655934 280.7014302 176.8641632

yangchuanju · 发表于 2023-5-9 05:20

本帖最后由 yangchuanju 于 2023-5-9 05:26 编辑

（接上楼）
分解情况复合因子 a 4^a+2^a+1对数平方根对数复合因子对数对数差
CF 4903793186...51<458> 1522 916.3353068 458.1676534 457.6905321 0.477121255
CF 2486139636...01<355> 1526 918.7435468 459.3717734 354.3955255 104.9762479
CF 9792363949...57<439> 1528 919.9476667 459.9738334 438.9908875 20.98294583
CF 2227370108...31<333> 1530 921.1517867 460.5758934 332.3477924 128.228101
CF 6470160238...39<407> 1534 923.5600267 461.7800133 406.810915 54.96909831
CF 9512628585...21<441> 1536 924.7641467 462.3820733 440.9783005 21.4037728
CF 1377321386...51<463> 1538 925.9682667 462.9841333 462.1390353 0.84509804
CF 2182920908...21<456> 1542 928.3765066 464.1882533 455.339038 8.849215313
CF 8259853672...33<394> 1544 929.5806266 464.7903133 393.9169724 70.87334095
CF 2038298947...77<420> 1546 930.7847466 465.3923733 419.3092679 46.08310542
CF 8560761973...01<342> 1550 933.1929866 466.5964933 341.9325124 124.6639809
CF 1141686079...69<406> 1552 934.3971065 467.1985533 405.0575467 62.14100656
CF 3700075844...61<373> 1554 935.6012265 467.8006133 372.5682106 95.23240264
CF 2281685843...51<434> 1558 938.0094665 469.0047332 433.3582558 35.6464774
CF 6115739648...61<347> 1560 939.2135865 469.6067932 346.786449 122.8203442
CF 4695135121...57<389> 1562 940.4177065 470.2088532 388.6716481 81.53720513
CF 2624944121...77<428> 1566 942.8259464 471.4129732 427.4191201 43.99385315
CF 5783267250...01<271> 1568 944.0300664 472.0150332 270.7621733 201.2528599
CF 1001935128...71<361> 1570 945.2341864 472.6170932 360.0008396 112.6162536
CF 4963626864...59<413> 1574 947.6424264 473.8212132 412.6957991 61.12541405
CF 5037612187...49<426> 1576 948.8465463 474.4232732 425.7022247 48.72104844
CF 2192310626...79<444> 1578 950.0506663 475.0253332 443.3409021 31.68443107
CF 9463555638...51<384> 1582 952.4589063 476.2294531 383.9760543 92.2533988
CF 3042241988...01<434> 1584 953.6630263 476.8315131 433.4831938 43.34831938
CF 1162209988...97<416> 1586 954.8671462 477.4335731 415.0652846 62.36828852
CF 4316609117...01<376> 1590 957.2753862 478.6376931 375.6351427 103.0025504
CF 7117630629...17<471> 1592 958.4795062 479.2397531 470.8523354 8.387417651
CF 2074983595...97<476> 1594 959.6836262 479.8418131 475.3170147 4.524798421
CF 1344817886...13<425> 1598 962.0918661 481.0459331 424.1286635 56.91726959
CF 2081586438...01<386> 1600 963.2959861 481.6479931 385.3183944 96.32959861
CF 5951170545...47<469> 1602 964.5001061 482.2500531 468.7746024 13.47545066
CF 1199811646...17<427> 1606 966.9083461 483.454173 426.0791131 57.37505996
CF 1029139381...31<278> 1610 969.316586 484.658293 277.0124742 207.6458188
CF 3513819075...93<464> 1614 971.724826 485.862413 463.5457794 22.31663361
CF 4093013964...69<440> 1616 972.928946 486.464473 439.6120432 46.85242977
CF 2107250443...97<450> 1618 974.133066 487.066533 449.3237162 37.74281683
CF 5426250350...17<473> 1622 976.5413059 488.270653 472.7344998 15.53615314
CF 2772050929...61<347> 1624 977.7454259 488.872713 346.4428012 142.4299118
CF 6224863379...11<451> 1626 978.9495459 489.4747729 450.7941298 38.68064313
CF 1840983115...91<384> 1630 981.3577859 490.6788929 383.2650498 107.4138431
CF 2410312426...81<463> 1632 982.5619058 491.2809529 462.3820733 28.89887958
CF 3887413667...83<415> 1634 983.7660258 491.8830129 414.5896608 77.29335216
CF 5802718824...63<341> 1638 986.1742658 493.0871329 340.7636315 152.3235014
CF 1959620352...01<386> 1640 987.3783858 493.6891929 385.2921719 108.3970209
CF 4470609478...71<470> 1642 988.5825058 494.2912529 469.6503667 24.64088615
CF 4469663443...39<495> 1646 990.9907457 495.4953729 494.6502748 0.84509804
CF 1414780619...77<459> 1648 992.1948657 496.0974329 458.1506891 37.94674375
CF 1154980698...01<338> 1650 993.3989857 496.6994928 337.0625747 159.6369181
CF 2669878963...19<498> 1654 995.8072257 497.9036128 497.4264916 0.477121255
CF 1610263559...33<356> 1656 997.0113456 498.5056728 355.206897 143.2987759
CF 1902286108...31<484> 1658 998.2154656 499.1077328 483.2792758 15.82845697
CF 1893078061...57<479> 1662 1000.623706 500.3118528 478.2771685 22.03468427
CF 3870082418...61<405> 1666 1003.031946 501.5159728 404.5877202 96.92825256
CF 1035681189...71<379> 1670 1005.440186 502.7200928 378.0152261 124.7048667
CF 2863985622...01<434> 1672 1006.644306 503.3221528 433.4569708 69.86518192
CF 2423126783...11<468> 1674 1007.848425 503.9242127 467.3843761 36.5398366
CF 2682675472...57<480> 1678 1010.256665 505.1283327 479.4285681 25.69976459
CF 1700062508...21<278> 1680 1011.460785 505.7303927 277.2304649 228.4999278
CF 3811803796...69<458> 1682 1012.664905 506.3324527 457.5811305 48.75132217
CF 3139931659...31<365> 1690 1017.481385 508.7406927 364.4969202 144.2437725
CF 1874116365...21<381> 1694 1019.889625 509.9448127 380.2727966 129.6720161
CF 5353615857...29<449> 1696 1021.093745 510.5468726 448.7286472 61.81822544
CF 8901617844...61<497> 1698 1022.297865 511.1489326 496.9494689 14.19946369
CF 5481223886...09<467> 1702 1024.706105 512.3530526 466.7388775 45.61417508
CF 1281436594...59<490> 1706 1027.114345 513.5571726 489.1076971 24.44947548
CF 1729554573...41<365> 1710 1029.522585 514.7612926 364.2379343 150.5233583
CF 3442456730...97<507> 1712 1030.726705 515.3633526 506.5368685 8.826484087
CF 1821562615...99<507> 1714 1031.930825 515.9654126 506.2604441 9.704968464
CF 3026444195...51<477> 1718 1034.339065 517.1695326 476.4809327 40.68859988
CF 6158279435...21<353> 1720 1035.543185 517.7715925 352.7894594 164.9821332
CF 5403491542...39<519> 1726 1039.155545 519.5777725 518.7326745 0.84509804
CF 9960947970...01<418> 1728 1040.359665 520.1798325 417.9983007 102.1815318
CF 3195151596...41<363> 1730 1041.563785 520.7818925 362.5044915 158.277401
CF 9177020744...69<487> 1734 1043.972025 521.9860125 486.9627017 35.02331077
CF 2863985632...01<434> 1736 1045.176145 522.5880725 433.4569708 89.13110164
CF 3034355811...51<470> 1738 1046.380265 523.1901325 469.4820665 53.70806596
CF 9779294988...89<452> 1742 1048.788505 524.3942524 451.9903075 72.4039449
CF 3093185025...17<473> 1744 1049.992625 524.9963124 472.4904059 52.50590654
CF 1515932654...81<521> 1746 1051.196745 525.5983724 520.1806799 5.417692521
CF 1721847894...01<362> 1750 1053.604985 526.8024924 361.2359948 165.5664976
CF 6130040033...33<502> 1752 1054.809105 527.4045524 501.7874633 25.61708909
CF 3384473670...71<528> 1754 1056.013225 528.0066124 527.5294911 0.477121255
CF 1550218233...69<264> 1756 1057.217345 528.6086724 263.1903928 265.4182795
CF 1540797883...79<467> 1758 1058.421465 529.2107324 466.1877457 63.0229867
CF 8672436486...53<524> 1762 1060.829705 530.4148524 523.9381411 6.476711232
CF 1945558759...81<506> 1766 1063.237945 531.6189723 505.2890444 26.32992799
CF 7840038869...41<424> 1768 1064.442065 532.2210323 423.8943182 108.3267141
CF 1126231727...01<392> 1770 1065.646185 532.8230923 391.0516278 141.7714646
CF 3968558677...49<266> 1772 1066.850305 533.4251523 265.5986328 267.8265195
CF 8031701136...91<517> 1774 1068.054425 534.0272123 516.9048075 17.12240477
CF 5744871785...53<474> 1776 1069.258545 534.6292723 473.7592803 60.86999196
CF 8966821244...07<413> 1778 1070.462665 535.2313323 412.9526385 122.2786938
CF 1104843004...21<478> 1782 1072.870905 536.4354523 477.0433006 59.3921517
CF 6802525638...29<470> 1784 1074.075025 537.0375123 469.8326702 67.20484208
CF 4389933040...33<474> 1786 1075.279145 537.6395723 473.6424579 63.99711436
CF 1356994393...11<424> 1790 1077.687384 538.8436922 423.1325781 115.7111142
CF 6042379711...61<477> 1794 1080.095624 540.0478122 476.781208 63.26660421
CF 9342262034...71<500> 1798 1082.503864 541.2519322 499.970452 41.28148016
CF 8153898176...01<323> 1800 1083.707984 541.8539922 322.9113653 218.9426269
CF 5508594428...11<490> 1802 1084.912104 542.4560522 489.7410408 52.71501139
CF 1615700599...01<241> 1804 1086.116224 543.0581122 240.2083609 302.8497513
CF 7560962637...77<451> 1806 1087.320344 543.6601722 450.8785771 92.78159508
CF 4404364240...29<521> 1808 1088.524464 544.2622322 520.6438832 23.61834893
CF 9496792988...21<430> 1810 1089.728584 544.8642922 429.977577 114.8867152
CF 7553648088...17<519> 1814 1092.136824 546.0684121 518.8781567 27.19025539
CF 1043154920...53<509> 1816 1093.340944 546.6704721 508.0183488 38.65212332
CF 8016336363...69<486> 1818 1094.545064 547.2725321 485.9039759 61.36855619
CF 1598485138...73<511> 1824 1098.157424 549.0787121 510.2037086 38.87500349
CF 1675893944...49<484> 1826 1099.361544 549.6807721 483.2242465 66.45652555
CF 2105550828...37<261> 1828 1100.565664 550.2828321 260.3233657 289.9594663
CF 9254486827...31<397> 1830 1101.769784 550.8848921 396.9663523 153.9185397
CF 1273312533...21<550> 1832 1102.973904 551.4869521 549.104935 2.382017043
CF 3056567793...11<470> 1834 1104.178024 552.089012 469.485234 82.60377802
CF 4383357824...23<520> 1838 1106.586264 553.293132 519.6418069 33.65132511
CF 2332896080...61<368> 1840 1107.790384 553.895192 367.3678954 186.5272966
CF 1382338480...63<527> 1842 1108.994504 554.497252 526.1406144 28.35663762
CF 6089820301...21<261> 1844 1110.198624 555.099312 260.7846045 294.3147075
CF 5890274095...19<499> 1846 1111.402744 555.701372 498.7701355 56.93123649
CF 1805930414...01<415> 1850 1113.810984 556.905492 414.256701 142.648791
CF 1020788075...37<276> 1852 1115.015104 557.507552 275.0089356 282.4986164
CF 2184558708...13<530> 1854 1116.219224 558.109612 529.3393637 28.77024824
CF 5316491572...97<452> 1856 1117.423344 558.711672 451.7256251 106.9860468
CF 2362393392...67<520> 1858 1118.627464 559.3137319 519.3733522 39.94037972
CF 5468595671...41<424> 1862 1121.035704 560.5178519 423.7378758 136.7799761
CF 1117049961...29<281> 1868 1124.648064 562.3240319 280.0480726 282.2759593
CF 2690092160...71<386> 1870 1125.852184 562.9260919 385.4297672 177.4963247
CF 2022646841...31<551> 1874 1128.260424 564.1302119 550.3059201 13.82429181
CF 8855989742...51<539> 1878 1130.668664 565.3343319 538.9472371 26.38709475
CF 1111972932...81<400> 1880 1131.872784 565.9363918 399.0460942 166.8902976
CF 8740494660...33<562> 1882 1133.076904 566.5384518 561.941536 4.596915828
CF 1318165452...73<510> 1886 1135.485144 567.7425718 509.1199699 58.6226019
CF 1595824772...81<531> 1888 1136.689264 568.3446318 530.2029852 38.14164661
CF 1846158975...71<339> 1890 1137.893384 568.9466918 338.2662691 230.6804227
CF 2021686204...59<570> 1894 1140.301624 570.1508118 569.3057137 0.84509804
CF 2371680257...97<555> 1896 1141.505744 570.7528718 554.3750561 16.37781564
CF 2647387648...51<521> 1898 1142.709864 571.3549318 520.4228175 50.93211423
CF 6355897681...01<571> 1902 1145.118104 572.5590518 570.8031769 1.755874856
CF 6534514260...61<414> 1904 1146.322223 573.1611117 413.8152133 159.3458984
CF 3204861806...73<564> 1906 1147.526343 573.7631717 563.5058093 10.25736243
CF 1848463339...01<430> 1910 1149.934583 574.9672917 429.2668108 145.7004809
CF 1539340397...21<574> 1912 1151.138703 575.5693517 573.1873347 2.382017043
CF 1154801265...69<497> 1914 1152.342823 576.1714117 496.0625073 80.10890445
CF 2129178416...37<277> 1916 1153.546943 576.7734717 276.3282121 300.4452596
CF 7133876546...83<483> 1918 1154.751063 577.3755317 482.8533256 94.52220609
CF 1053782258...81<410> 1920 1155.955183 577.9775917 409.0227509 168.9548408
CF 2757501397...63<515> 1922 1157.159303 578.5796517 514.4405157 64.13913593
CF 3026152849...13<242> 1924 1158.363423 579.1817117 241.4808909 337.7008208
CF 2942256025...57<541> 1926 1159.567543 579.7837716 540.4686805 39.31509119
CF 7598203694...69<541> 1928 1160.771663 580.3858316 540.8807109 39.50512071
CF 1113413585...91<454> 1930 1161.975783 580.9878916 453.0466565 127.9412351
CF 1059781721...89<564> 1934 1164.384023 582.1920116 563.0252164 19.16679519
CF 1843939810...71<507> 1938 1166.792263 583.3961316 506.2657467 77.13038486
CF 3200368000...27<561> 1942 1169.200503 584.6002516 560.5051999 24.09505166
CF 2844433406...73<529> 1944 1170.404623 585.2023116 528.4539958 56.7483158
CF 9527223453...29<495> 1946 1171.608743 585.8043716 494.9789664 90.82540521
CF 5045528884...33<287> 1948 1172.812863 586.4064316 286.7029067 299.7035249
CF 5108812787...01<391> 1950 1174.016983 587.0084915 390.70832 196.3001716
CF 1808766217...53<550> 1952 1175.221103 587.6105515 549.2573824 38.3531691
CF 2330845500...51<588> 1954 1176.425223 588.2126115 587.3675135 0.84509804
CF 6836980430...27<493> 1958 1178.833463 589.4167315 492.8348643 96.58186717
CF 1530552073...01<335> 1960 1180.037583 590.0187915 334.1848481 255.8339434
CF 7685301528...33<569> 1962 1181.241703 590.6208515 568.8856609 21.73519058
CF 4087480953...89<296> 1964 1182.445823 591.2229115 295.6114557 295.6114557
CF 3337139103...31<575> 1966 1183.649943 591.8249715 574.5233743 17.30159717
CF 4796795197...53<536> 1968 1184.854063 592.4270315 535.6809512 56.74608029
CF 1426102593...11<455> 1970 1186.058183 593.0290915 454.1541508 138.8749407
CF 1120446755...77<266> 1972 1187.262303 593.6311514 265.0493912 328.5817602
CF 4817549388...07<495> 1974 1188.466423 594.2332114 494.6828262 99.55038527
CF 8428393180...41<451> 1976 1189.670543 594.8352714 450.9257448 143.9095266
CF 2304227226...47<544> 1978 1190.874663 595.4373314 543.3625253 52.07480612
CF 9565972676...97<591> 1982 1193.282903 596.6414514 590.9807291 5.66072227
CF 3301213880...13<524> 1984 1194.487023 597.2435114 523.5186737 73.72483773
CF 6743250094...29<560> 1986 1195.691143 597.8455714 559.8288693 38.01670212
CF 1213227803...21<469> 1990 1198.099383 599.0496914 468.0839424 130.965749
CF 1232086227...53<568> 1992 1199.303503 599.6517514 567.0906411 32.56111026
CF 2006806187...39<586> 1994 1200.507623 600.2538114 585.3025054 14.95130592
CF 7715284392...31<567> 1998 1202.915863 601.4579313 566.8873519 34.5705794
CF 3076566531...01<473> 2000 1204.119983 602.0599913 472.4880663 129.571925

342个未完全分解的复合因子对数都小于相应平方根对数（平方根对数-复合因子对数都大于0），亦即复合因子都小于平方根，尚若完全分解后的最大素因子肯定小于相应的平方根了！

yangchuanju · 发表于 2023-5-9 05:35

先生要找大素数，可到4^p+2^p+1之中找一找看看，这里p是素数，不必限定这条件、那条件！

4^a+2^a+1中的那5个数能说明什么？
（a=3c, t>a^2*m^2, 4^a+2^a+1不含素因子3,7,73， y>(4^a+2^a+1)/mt的平方根）

yangchuanju · 发表于 2023-5-9 05:46

本帖最后由 yangchuanju 于 2023-5-9 05:49 编辑

梅森数中的最大素因子
各行数字中的第1个数字为素数号，1——素数3， 2——素数5， 100——素数541
A003260
Largest prime factor of n-th Mersenne number (A001348(n)).
1 3
2 7
3 31
4 127
5 89
6 8191
7 131071
8 524287
9 178481
10 2089
11 2147483647
12 616318177
13 164511353
14 2099863
15 13264529
16 20394401
17 3203431780337
18 2305843009213693951
19 761838257287
20 212885833
21 9361973132609
22 1113491139767
23 57912614113275649087721
24 618970019642690137449562111
25 13842607235828485645766393
26 341117531003194129
27 3976656429941438590393
28 162259276829213363391578010288127
29 870035986098720987332873
30 1066818132868207
31 170141183460469231731687303715884105727
32 10350794431055162386718619237468234569
33 5439042183600204290159
34 123876132205208335762278423601
35 8235109336690846723986161
36 7289088383388253664437433
37 2134387368610417
38 36230454570129675721
39 79638304766856507377778616296087448490695649
40 155285743288572277679887
41 1489459109360039866456940197095433721664951999121
42 7923871097285295625344647665764672671
43 87274497124602996457
44 14732265321145317331353282383
45 26828803997912886929710867041891989490486893845712448833
46 4884164093883941177660049098586324302977543600799
47 3593875704495823757388199894268773153439
48 596242599987116128415063
49 7992177738205979626491506950867720953545660121688631
50 467795120187583723534280000348743236593
51 116868129879077600270344856324766260085066532853492178431
52 7110008717824458123105014279253754096863768062879
53 160619474372352289412737508720216839225805656328990879953332340439
54 12070396178249893039969681
55 374550598501810936581776630096313181393
56 383725126655170964501315730676446647
57 68625988504811774259364670661552948915363901845035416371912463477873783063
58 248927757868131890277330541567820045256364273970773286542188386932989391
59 6955979459776540052280934851589652278783
60 48009215293052652841860443273079338843737271906291675944391068955229998769420319
61 23579543011798993222850893929565870383844167873851502677311057483194673
62 396645227028138890415611220710757921643910743103031701971222447
63 112177476474470525577861298937835338545723093134076373561
64 294803681348959296477194164064643062187559537539328375831
65 26693012026551688286164949958620483258358551879
66 9815639231755686605031317440031161584572466128599
67 298542624980197463613767215333569428005686468835821253721796682625551919
68 78778047326466742993612420842416198311394008068822475527239136925369
69 20270345302545987116040069442814496729341666112096057885992643120463337596490211193
70 18555393648630683868229284313709360336855095474246691696225599
71 6725414756111955781503880188940925566051960039574573675843402666863
72 100361196281293745682520861860411315001
73 19755740081951910036006278827509875120092863638283602681
74 752440346497356983142327449546457327748644897934114291899411428982990336039662496766303354959577078458241
75 6809649408891001685768937590233308625949604176033855796938978177320539702698633946720428389517879894953
76 240522700235167893496900256599634325263
77 4684435266636161232578932847604331726884269415306219621279642876954933236537677535849040755779223719
78 6597485910270326519900042655193
79 4743358775443804666040010704534780418840545962266329593
80 3881196575913244673719425770871246487895686937951690944453838586764072695131586617955811936945129
81 29919490848598531825060153417921002916701815927
82 8819779591697258388298117725624832271141577326602771028307143781815455970700534027206522451123308835472505327249
83 1807482391092819529831423005040763105191863029850140579776353298087457
84 270704078857734344240200528055897595961633472511075822112527121
85 873880146833642190373525520936770796845382029997855219402285283144955696825577908510162169
86 123219439267346362049744425289349676468781136823956005602631224069302162695430546376768705960936201429580820215522273
87 89396693392545895668303801721053258296972567022618106513441
88 2475539419689929784935319344449409898291165097323714578650943035813830300993611462717419801770460539016610145009605554380104535919
89 846134609236527432935428641453947808692744612842997575850108349114305165850593069285923876628410633
90 15148264043785111348665069495360623752616947973471649653354617
91 2727318928892040009397199815225412029789790574177544409019502626344207905841
92 10869186607773319558730501484452015470992764113369214593828236615596881
93 82033219963138371097689272308258116841679442057301643873942124991182012434598644913857356023840478815121709542915222280972560231358838127531337
94 14797326616665978116353515926860025681383
95 39075504626391841678304934944805852280404731716385642050296152320994438836806257083337312828162589099799400566633
96 357801561527383951750371336247776228772287580084037416747290336593974702826921943012497755232377
97 10751168288022146026259956092344345159681167846281852853736470411971627982051345994439553634116438524169
98 6864797660130609714981900799081393217269435300143305409394463459185543183397656052122559640661454554977296311391480858037121987999716643812574028291115057151
99 171417691861249198128317096534322116476165056718630345094896620367860006486977101859504089
100 3913112911835820709255943886069525362580351752909309688547154382752707879

yangchuanju · 发表于 2023-5-9 05:54

(2^p+1)/3最大最大素数
A000979
Wagstaff primes: primes of form (2^p + 1)/3.
1 3
2 11
3 43
4 683
5 2731
6 43691
7 174763
8 2796203
9 715827883
10 2932031007403
11 768614336404564651
12 201487636602438195784363
13 845100400152152934331135470251
14 56713727820156410577229101238628035243
15 62357403192785191176690552862561408838653121833643
16 1046183622564446793972631570534611069350392574077339085483
17 267823007376498379256993682056860433753700498963798805883563
18 5562466239377370006237035693149875298444543026970449921737087520370363869220418099018130434731
19 95562442332919646317117537304253622533190207882011713489066201641121786503686867002917439712921903606443
20 3506757267698915671493993255253419110366893201882115906332187268712488102864720548075777690851725634151321979237452145142012954833455869242085209847101253899970149114085629732127775838589548478180862167823854251

太阳 · 发表于 2023-5-9 06:44

4^418+2^418+1，复合因子(2^836+2^418+1)/800594313431043275064149904996914078513728979970328751414181，192位合数
如何判断复合因子小于\(\sqrt{4^{418}+2^{418}+1}\)

yangchuanju · 发表于 2023-5-9 06:55

本帖最后由 yangchuanju 于 2023-5-9 10:51 编辑

yangchuanju 发表于 2023-5-9 05:35
先生要找大素数，可到4^p+2^p+1之中找一找看看，这里p是素数，不必限定这条件、那条件！

4^a+2^a+1中的 ...

p 最大素因子 4^p+2^p+1 平方根最大素因子与平方根
3 73 73 8.544003745 大于
5 151 1057 32.51153641 大于
7 337 16513 128.5029183 大于
11 599479 4196353 2048.500183 大于
13 121369 67117057 8192.500046 大于
17 11119 17180000257 131072.5 小于
19 1212847 2.74878E+11 524288.5 大于
23 10052678938039<14> 7.03688E+13 8388608.5 大于
29 9857737155463<13> 2.8823E+17 536870912.5 大于
31 658812288653553079<18> 4.61169E+18 2147483649 大于
37 616318177 1.88895E+22 1.37439E+11 小于
41 177722253954175633<18> 4.8357E+24 2.19902E+12 大于
43 11053036065049294753459639<26> 7.73713E+25 8.79609E+12 大于
47 646675035253258729<18> 1.9807E+28 1.40737E+14 大于
53 229890275929<12> 8.11296E+31 9.0072E+15 小于
59 9213624084535989031<19> 3.32307E+35 5.76461E+17 大于
61 37201708625305146303973352041<29> 5.31691E+36 2.30584E+18 大于
67 87449423397425857942678833145441<32> 2.17781E+40 1.47574E+20 大于
71 4205268574191396793<19> 5.57519E+42 2.36118E+21 小于
73 4815314615204347717321<22> 8.9203E+43 9.44473E+21 大于
79 31357373417090093431<20> 3.65375E+47 6.04463E+23 大于
83 8241594690167137359552274418432855740327<40> 9.35361E+49 9.67141E+24 小于
89 24124332437713924084267316537353<32> 3.83124E+53 6.1897E+26 大于
97 1170711644777651877659556633665719<34> 2.51084E+58 1.58456E+29 大于

除p=17时，4^p+2^p+1的最大素因子小于4^p+2^p+1的平方根外，其余的素指数的最大素因子都大于4^p+2^p+1的平方根！

p=17,37,53,71,79都是最大素因子小于平方根的！
特此改正！

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4^a+2^a+1的最大素因子

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