太阳先生莫再漫天抛谬论！

yangchuanju

太阳先生莫再漫天抛谬论！
两天来太阳黑斑突然爆发——太阳黑子普天遮地的抛向地球！
y>aaa，y>bbb，y>ccc……，搞的人心惶惶！

太阳先生应该明晰，合数可以分解成两个或多个素数的乘积，
其中的最小素因子一定小于该合数的平方根，但最大素因子是否大于它的平方根就不能确定了；
现捕获一个太阳黑子进行剖析：y＞√【[(2^a+1)^2+3]/[(2^m+1)^2+3]】

已知：偶数a＞0，c＞0，m＞0，t＞0，a＝c^2，a≠m
方程[(2^a+1)^2+3]/[(2^m+1)^2+3]＝t，有唯一的正整数解，√a＝m
[(2^a+1)^2+3]/[(2^m+1)^2+3]的最大质因数是y
求证:y＞√【[(2^a+1)^2+3]/[(2^m+1)^2+3]】

令m=6, a=36
2^36+1=68719476737
2^6+1=65
68719476737^2+3=4722366483007084167172
65^2+3=4228
相除等于1116926793521070049，整除，合数；
1116926793521070049=7*337*29191*106681*152041。
最大素因子152041不大于1116926793521070049的平方根。

yangchuanju

令m=2, a=4
2^4+1=17
2^2+1=5
17^2+3=292
5^2+3=28
292/28=10.42857143
不整除。

m=4, a=16
2^16+1=65537
2^4+1=17
65537^2+3=42950983372
17^2+3=292
42950983372/292=14709241
整除=631*23311。

太阳

a＝36，m＝6，有三个解，2，6，8，它没有具备唯一的正整数解，\(\sqrt{a}＝m\)，舍去删除，假反例

太阳

已知：奇数\(a＞0\)，\(c＞0\)，\(a＝3c\)，\(k＞0\)，\(\frac{4^a+2^a+1}{73}\ne k\)，\(t＞a^2m^2\)
\(\left( 4^a+2^a+1\right)\)的最小质因数是\(m\)，\(\frac{4^a+2^a+1}{m}\)的最小质因数是\(t\)
\(\frac{4^a+2^a+1}{mt}\)的最大质因数是\(y\)
求证:\(y＞\sqrt{\frac{4^a+2^a+1}{mt}}\)
已知：整数\(a＞0\)，\(c＞0\)，\(m＞0\)，\(t＞0\)，\(a＝c^2\)，\(a\ne m\)
方程\(\frac{\left( 2^a+1\right)^2+3}{\left( 2^m+1\right)^2+3}＝t\)，有唯一的正整数解，\(\sqrt{a}＝m\)
\(\frac{\left( 2^a+1\right)^2+3}{\left( 2^m+1\right)^2+3}\)的最大质因数是\(y\)
求证:\(y＞\sqrt{\frac{\left( 2^a+1\right)^2+3}{\left( 2^m+1\right)^2+3}}\)
已知:整数\(a＞0\)，\(c＞0\)，\(m＞0\)，\(t＞0\)
\(a＝\left( 3^c+1\right)^2\)，方程\(\frac{\left( 2^a+1\right)^2+3}{\left( 2^m+1\right)+3}＝t\)
求证:方程\(\frac{\left( 2^a+1\right)^2+3}{\left( 2^m+1\right)+3}＝t\)，最小正整数解，\(m＝3^c+1\)
例1：\(a＝81\)，\(\frac{4^{81}+2^{81}+1}{73}\ne k\)
\(\left( 4^{81}+2^{81}+1\right)\)的最小质因数是487
\(\frac{4^{81}+2^{81}+1}{487}\)的最小质因数是16753783618801
\(\frac{4^{81}+2^{81}+1}{487\times16753783618801}\)的最大质因数是3712990163251158343＞\(\sqrt{\frac{4^{81}+2^{81}+1}{487\times3712990163251158343}}\)
例2：\(a＝100\)，\(m\ne100\)，方程\(\frac{\left( 2^a+1\right)^2+3}{\left( 2^m+1\right)^2+3}＝t\)，有唯一的正整数解，\(\sqrt{a}＝m＝10\)
\(\frac{\left( 2^{100}+1\right)^2+3}{\left( 2^{10}+1\right)^2+3}\)的最大质因数是170886618823141738081830950807292771648313599433＞\(\sqrt{\frac{\left( 2^{100}+1\right)^2+3}{\left( 2^{10}+1\right)^2+3}}\)
\(a＝16，100，6724，532900\)，具备条件\(a＝\left( 3^c+1\right)^2\)

太阳

\(a＝16，100，6724，532900\)，具备条件\(a＝\left( 3^c+1\right)^2\)

yangchuanju

太阳 发表于 2023-5-5 12:29
已知：奇数\(a＞0\)，\(c＞0\)，\(a＝3c\)，\(k＞0\)，\(\frac{4^a+2^a+1}{73}\ne k\)，\(t＞a^2m^2\)
\(\ ...

只想告诉先生一个事实，合数的最大素因子不一定都大于它的平方根！

先生不要认为加上这样那样的限制条件，合数的最大素因子就一定大于它的平方根了！

太阳先生的各个求证命题不会总是成立的！

yangchuanju

太阳 发表于 2023-5-5 12:29
已知：奇数\(a＞0\)，\(c＞0\)，\(a＝3c\)，\(k＞0\)，\(\frac{4^a+2^a+1}{73}\ne k\)，\(t＞a^2m^2\)
\(\ ...

太阳先生不就是要找几个大素数吗？

明确告诉您，梅森数中含有多个已知的最大素数，今后世界上出现的更大素数可能还都是梅森素数！
梅森数的余因子中、双梅森数中也一定含有相当多的大素数；
清一色数111……11，10的幂数加1即100……01等整数中都含有相当多大素数；
在这类整数中耐心找一找就是了。

建议太阳先生多学一些素数判断知识、因式分解知识。
不要再寄希望于加上几个条件，就能找到亿位大素数；
在您不知道如何判断它是不是素数时，你也根本不能确认它是素数还是合数！

yangchuanju

太阳 发表于 2023-5-5 12:29
已知：奇数\(a＞0\)，\(c＞0\)，\(a＝3c\)，\(k＞0\)，\(\frac{4^a+2^a+1}{73}\ne k\)，\(t＞a^2m^2\)
\(\ ...

太阳命题
已知：奇数a＞0，c＞0，a＝3c，k＞0，(4^a+2^a+1)/73≠k，t＞a^2*m^2；
(4^a+2^a+1)的最小质因数是m，(4^a+2^a+1)/m的最小质因数是t；
(4^a+2^a+1)/mt的最大质因数是y；
求证：y>√[(4^a+2^a+1)/mt]。

(2^3a-1)/(2^a-1)之中含有1个或多个素因子，最小素因子一定是小于其平方根的；
但最大素因子不一定大于其平方根吧！
太阳先生规定去掉两个小素因子，认为它的最大素因子大于不含那两个小素因子的平方根，
同样是不对的！
一旦代数式的指数a确定以后，它的最小素因子m、第2小素因子t就都确定了，
太阳先生凭什么硬性规定第2素因子t必须大于a^2*m^2呢？

yangchuanju

(2^27-1)/(2^9-1)=262657
(2^54-1)/(2^18-1)=3×87211×262657
(2^81-1)/(2^27-1)=2593×71119×97685839
(2^108-1)/(2^36-1)=3×87211×246241×262657×279073
(2^135-1)/(2^45-1)=271×262657×348031×49971617830801<14>
(2^162-1)/(2^54-1)
        =3×163×2593×71119×135433×97685839×272010961
2^189-1<57>/(2^63-1)
        =262657×1560007×207617485544258392970753527<27>
2^216-1<66>/(2^72-1)
        =3×87211×246241×262657×279073×33975937×138991501037953<15>
2^243-1<74>/(2^81-1)
        =487×16753783618801<14>×192971705688577<15>×3712990163251158343<19>

yangchuanju

(2^3a-1)/(2^a-1)      分解式       （下表各行中的“商”前的数字270等即为指数3a）
270        商
3×271×811×15121×87211×262657×348031×49971617830801<14>×385838642647891<15>
297        商
262657×8950393×170886618823141738081830950807292771648313599433<48>
324        商
3×163×2593×71119×135433×3618757×97685839×106979941×168410989×272010961×4977454861<10>
351        商
262657×446473×29121769×571890896913727<15>×93715008807883087<17>×150832426800173710177<21>
378        商3×379×87211×119827×262657×1560007×127391413339<12>×56202143607667<14>×207617485544258392970753527<27>
405        商2593×71119×537841×97685839×11096527935003481<17>×17645665556213400107370602081155737281406841<44>
432        商3×87211×246241×262657×279073×33975937×209924353×4261383649<10>×138991501037953<15>×24929060818265360451708193<26>
459        商262657×407770693450231393<18>×24418671951944649151<20>×49848448234572624009465371493197779785120970152607<50>
486        商3×487×1459×139483×16753783618801<14>×192971705688577<15>×3712990163251158343<19>×10429407431911334611<20>×918125051602568899753<21>
513        商
262657×524287×35473416481<11>×121323854647<12>×2237717449946593<16>×6164134759...47<56>
540        商3×271×541×811×15121×30241×49681×87211×246241×262657×279073×348031×49971617830801<14>×165041853060421<15>×385838642647891<15>×166242935471754241<18>
567        商
2593×71119×97685839×3417579232...89<98>
594        商3×23761×87211×262657×694387×8950393×6215074747201<13>×14973866897175265228063698945547<32>×170886618823141738081830950807292771648313599433<48>
621        商
262657×634569679×93747988411543<14>×2751471927...49<97>
648        商3×163×1297×2593×3889×71119×135433×3618757×30433969×97685839×106979941×168410989×272010961×1164777409<10>×4977454861<10>×3718266498433<13>×138991501037953<15>×1174029487714513<16>
675        商271×262657×348031×1605151×49971617830801<14>×1094270085398478390395590841401<31>×2842496263188647640089794561760551<34>×470390038503476855180627941942761032401<39>
702        商3×87211×262657×446473×29121769×571890896913727<15>×93715008807883087<17>×150832426800173710177<21>×4247713303224552237738169<25>×24841125429051585062538961751269988364169<41>
729        商
80191×97687×379081×664728004346558283448724389870269691211809<42>×1012137457...61<90>
756        商3×379×757×87211×119827×246241×262657×279073×1560007×127391413339<12>×56202143607667<14>×207617485544258392970753527<27>×456376431053626339473533320957<30>×304832756195865229284807891468769<33>
783        商1567×262657×602999957032279571271592229136193<33>×1345967414522954303164296915490846371433<40>×4110057838...87<77>
810        商3×163×2593×6481×9721×71119×135433×537841×74967931×97685839×272010961×2437880491<10>×448217524891<12>×10360573664851<14>××1969543281137041<16>×11096527935003481<17>×17645665556213400107370602081155737281406841<44>
837        商
262657×1127316245518063<16>×3186418650...63<148>
864        商3×87211×246241×262657×279073×33975937×209924353×4261383649<10>×68016300334849<14>×138991501037953<15>×24929060818265360451708193<26>×7311824282...09<73>
891        商
1783×2593×71119×97685839×12430037668834128259094186647<29>×1623954538...89<132>
918        商3×3673×87211×98227×262657×33204917536003441<17>×407770693450231393<18>×572404735191016891<18>×24418671951944649151<20>×72665575884323021522460727660186327381229881<44>×49848448234572624009465371493197779785120970152607<50>
945        商271×262657×348031×1560007×49971617830801<14>×207617485544258392970753527<27>×124339521078546949914304521499392241<36>×8937128331...01<95>
972        商3×487×1459×2917×4861×139483×3333950193493<13>×16753783618801<14>×26129603777437<14>×192971705688577<15>×3712990163251158343<19>×10429407431911334611<20>×918125051602568899753<21>×15778453094691989880197773477<29>×1753477469677913202190537606674204157<37>
999        商
7993×262657×473589937×610750585993378533918571366551132613003293697<45>×5042105883...13<138>
1026        商3×57457×87211×144667×262657×1465129×35473416481<11>×121323854647<12>×2237717449946593<16>×6164134759...47<56>×1615550938...03<87>
1053        商
2593×71119×44207047×97685839×24295813738965169951447<23>×1087462356...09<166>
1080        商3×271×541×811×2161×15121×21601×30241×49681×87211×246241×262657×279073×348031×33975937×201519653761<12>×49971617830801<14>×138991501037953<15>×165041853060421<15>×385838642647891<15>×1117180440577441<16>×469775495062434961<18>×4732268694...81<53>
1107        商
262657×7675195491326802969559<22>×472897006006202784083089<24>×1516672492...51<172>
1134        商3×163×2593×71119×135433×97685839×272010961×562873504411<12>×4744655685883<13>×1279685033...13<74>×3417579232...89<98>
1161        商
6967×262657×5979151×60499150497847<14>×2191649124...53<59>×6848918993...47<145>
1188        商3×2377×22573×23761×87211×246241×262657×279073×694387×8950393×155399494141<12>×4712151755917<13>×5292250152949<13>×6215074747201<13>×41523259994275786297957<23>×14973866897175265228063698945547<32>×272007548484389196113813589140615077929661<42>×170886618823141738081830950807292771648313599433<48>
1215        商487×49677578641<11>×16753783618801<14>×192971705688577<15>×3712990163251158343<19>×6606576725...21<57>×3558773006...41<128>
1242        商3×31051×87211×262657×25069771×624456487×1159185064537<13>×93747988411543<14>×5521094269128307<16>×185541553091188485307<21>×55128155016097727617011127633<29>×3173270838240154999061250226081<31>×2751471927...49<97>
1269        商262657×13409665391953<14>×980230389772951<15>×2000007978762572383159<22>×4054639296...27<55>×1675935479...91<146>
1296        商3×163×1297×2593×3889×10369×71119×135433×259201×3618757×30433969×97685839×106979941×168410989×272010961×1164777409<10>××4977454861<10>×3718266498433<13>×134921168163073<15>×1174029487714513<16>×1824045366145909775041<22>×524833094849624730914401<24>×4310500811...09<76>
1323        商2647×262657×1560007×5828257×17072111071<11>×1105196924013081302593<22>×2741672362528725535068727<25>×207617485544258392970753527<27>×1105597066...97<58>×1165947723...67<130>
1350        商3×271×811×15121×87211×262657×348031×1605151×377204851×578348287651<12>×49971617830801<14>×385838642647891<15>×126274316116265155801<21>×635558882028912657106651<24>×1094270085398478390395590841401<31>×2842496263188647640089794561760551<34>×470390038503476855180627941942761032401<39>×134140801860123003022669016950307743899703651<45>
1377        商2593×71119×97685839×246083671×2643524023...03<64>×1890820020...57<189>
1404        商3×9829×87211×246241×262657×279073×446473×29121769×59012929×7011385057<10>×21455171100793<14>×520378545363301<15>×571890896913727<15>×93715008807883087<17>×150832426800173710177<21>×4247713303224552237738169<25>×371434919794191620009861713658005494601<39>×24841125429051585062538961751269988364169<41>×657605388714787226759598226179270662456521<42>
1431        商
25759×262657×973081×1691845504865603430675449959<28>××1367061933...41<245>
1458        商3×80191×97687×379081×227862073×3110690934667<13>×216892513252489863991753<24>×1102099161075964924744009<25>×664728004346558283448724389870269691211809<42>×3930633012...17<78>×1012137457...61<90>
1485        商271×262657×348031×8950393×19694071×49971617830801<14>×106638643913461155414345241<27>×170886618823141738081830950807292771648313599433<48>×3307778274605860019775884877775856144362368451681<49>×3745535208...31<61>×2123956160...01<75>
1512        商3×379×757×87211×119827×246241×262657×279073×1560007×33975937×127391413339<12>×56202143607667<14>×138991501037953<15>×207617485544258392970753527<27>×456376431053626339473533320957<30>×304832756195865229284807891468769<33>×1109067877...61<131>
1539        商
2593×3079×71119×97685839×8092308241<10>×7721537672504812373263<22>×2074769626...77<258>
1566        商3×1567×87211×262657×12745935523<11>×602999957032279571271592229136193<33>×1345967414522954303164296915490846371433<40>××4436770067...01<62>×4110057838...87<77>×9279542732...67<80>
1593        商
22303×262657×59785353215457224743<20>×1410939304...01<291>
1620        商3×163×1621×2593×6481×9721×71119×135433×537841×3618757×6876901×74967931×97685839×106979941×168410989×272010961×1511474581<10>×2437880491<10>×2458695061<10>×3934029061<10>×4977454861<10>×448217524891<12>×9893662806061<13>×10360573664851<14>×1969543281137041<16>×11096527935003481<17>×1583423452213582178911805893942695192421<40>×4343952637722706853771280086533392805261<40>×17645665556213400107370602081155737281406841<44>
1647        商
262657×190183230986460463<18>×6797888520...27<308>合数
1674        商3×87211×262657×1127316245518063<16>×258451922773648099179883<24>×1395292603...63<140>×3186418650...63<148>
1701        商487×2010583×12488743×199870903×16753783618801<14>×192971705688577<15>×3712990163251158343<19>×1183324910403468982373156143<28>×873143258470761326993484965437820755249<39>×7697965562...41<205>合数
1728        商3×87211×246241×262657×279073×33975937×209924353×1718990209<10>×4261383649<10>×278452876033<12>×8148919324033<13>×68016300334849<14>×138991501037953<15>×24929060818265360451708193<26>×672915391135935402269916482777538049<36>×7311824282...09<73>×2623877886...77<116>
1755        商271×3511^2×262657×348031×446473×1969111×29121769×49971617830801<14>×571890896913727<15>×93715008807883087<17>×4242734772486358591<19>×85488365519409100951<20>×150832426800173710177<21>×5597910681...51<63>×2500659682...81<146>
1782        商3×163×1783×2593×7129×71119×135433×97685839×272010961×352246478761<12>×620617419930817<15>×1409033313878253109224688819<28>×12430037668834128259094186647<29>×236552624101029630306294196057<30>×6928662288...51<76>×1623954538...89<132>
1809        商262657×11270071×347092831×1740474132911148803713034775690007<34>×648168721774409511378116151410898673195879<42>×6162343774...27<309>合数
1836        商3×3673×27541×87211×98227×246241×262657×279073×67135177×1426465062181<13>×48752128660789<14>×2152204823142217<16>××33204917536003441<17>×407770693450231393<18>×572404735191016891<18>×24418671951944649151<20>×72665575884323021522460727660186327381229881<44>×49848448234572624009465371493197779785120970152607<50>×6261723151...53<55>×1427306129...37<66>
1863        商
2593×3727×71119×27695359×97685839××218797132447<12>×1861344789...59<336>合数
1890        商3×271×379×811×15121×69931×87211×119827×262657×348031×1560007×127391413339<12>×49971617830801<14>×56202143607667<14>×385838642647891<15>×207617485544258392970753527<27>×535027074087349862035936711342201<33>×124339521078546949914304521499392241<36>××2958456773...51<93>×8937128331...01<95>
1917        商
262657×4205268574191396793<19>×69466156512528108241<20>×2852148991...09<360>
1944        商3×487×1459×2917×4861×139483×26232337×13655624113<11>×3333950193493<13>×16753783618801<14>×26129603777437<14>××192971705688577<15>×3712990163251158343<19>×10429407431911334611<20>×918125051602568899753<21>×15778453094691989880197773477<29>×1753477469677913202190537606674204157<37>×1581214773543289355763694808184205062516817<43>×2062043345...33<136>
1971        商
262657×1406791071629857<16>×22068362846714807160397927912339216441<38>×4385565518...73<338>合数
1998        商3×7993×23977×87211×262657×473589937×5814899281<10>×97377658867<11>×241947707873451279499<21>×610750585993378533918571366551132613003293697<45>×5042105883...13<138>×3562597098...61<150>
2025        商2593×71119×81001×537841×97685839×429004351×11096527935003481<17>×2029839982282855554442383177052070534551<40>×17645665556213400107370602081155737281406841<44>×1836459846...01<273>合数
2052        商3×2053×8209×16417×57457×87211×144667×246241×262657×279073×1465129×3311929×68897953×35473416481<11>×121323854647<12>×××2237717449946593<16>×64510639132991228987740813<26>×1227944524001822172393353063497<31>×4683836497...29<54>×6164134759...47<56>×4986165496...33<60>×1615550938...03<87>
2079        商4159×16633×262657×1560007×8950393×80932047967<11>×207617485544258392970753527<27>×170886618823141738081830950807292771648313599433<48>×2318271715...89<307>合数
2106        商3×163×2593×71119×135433×44207047×97685839×272010961×10166476786129<14>×93715008807883087<17>×93961711110587347<17>×24295813738965169951447<23>×1087462356...09<166>×1222688431...27<166>
2133        商12799×262657×566577025549417<15>×31357373417090093431<20>×2264185782428196872809<22>×52701137729911178439185017<26>×2189026924529636243180163130111<31>×2332634540...61<327>合数
2160        商3×271×541×577×811×2161×15121×21601×30241×49681×87211×246241×262657×279073×348031×6281281×33975937×148759201×209924353×4261383649<10>×4518776161<10>×201519653761<12>×49971617830801<14>×138991501037953<15>×165041853060421<15>×385838642647891<15>×1117180440577441<16>×166242935471754241<18>×24929060818265360451708193<26>×30263419164953868175425499201<29>×4732268694...81<53>×1935559921...41<121>
2187        商
39367×7606246033<10>×263196614521<12>×529063556041<12>×1912682319...67<402>合数