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Topic: Help with a factorisation, Please [Important]
Replies: 5   Last Post: Jan 17, 2014 11:05 PM

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David Bernier

Posts: 3,240
Registered: 12/13/04
Re: Help with a factorisation, Please [Important]
Posted: Jan 17, 2014 4:24 AM
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On 01/17/2014 01:02 AM, Hlauk wrote:
> On Thursday, January 16, 2014 8:13:16 PM UTC-5, Port563 wrote:
>> "Hlauk" <hlauk.h.bogart@gmail.com> wrote...
>>
>> On Tuesday, January 14, 2014 1:37:57 AM UTC-5, Hlauk wrote:
>>

>>> --------------------------------------------------------------------
>>
>>>
>>
>>> Your largest composite with 1699 digits ---
>>
>>> 25078099047327723946350190226140745685367118558583345025289348176974163010411395316011792952658538429334151523870332354891166103276440801557659131336083939591996638761324223372906247841901475408686293744445242419144759693733122427112722955818350152682398386303846871903580749249506780978156931841817426878977420521922905061599669777155838220730017253990874874170295034250992001811785387558791545027473296324797838675186778250165112060417154208246732765007577018933269476071405746514663467109377080413521009195905694267772462994043436332067349693160166509285030356626859800062443851229274235662348321472177100809054075649802106597348299839668776704804633241746472679913178420604117172712671611173752279103915487141077768915131480547442474918808593690153223918309056138299857348410690989835944663530286997918228041295150335602763768908249515443799991914420368928629781195542195897517395817082279286799740494854106478197676325813004404575151410928827092637285183626617155565045057569440547183405982
45325602511255244250207904704121983163309894950931458712809580594862337802500418398431405480893860157182072480132320585828889914529391213150639488900110069086872461727290576704598789739555487214162249774467649923114928283271931634840282117207293377910135595752147127557251622558326421157857414637362020364123648587909877696219093586729789995137470194719156642385692199285919411113718990106768290110675682960300014511068937681320680843298237148951823441428351943405242753083069015608592612389742057748503592682293924554770508834146124744108391100201797587401627078021473436036153227723256430759847253758147206996347241255869426284445333291765102703844080551306884674006671923619625045833456199941934043201
>>
>>
>>
>>
>>
>> OK, I verified this is the same as (from my post):
>>
>>
>>
>> 2507809904732772394635019022614074568536711855858334502528934
>>
>> 817697416301041139531601179295265853842933415152387033235489116610
>>
>> 327644080155765913133608393959199663876132422337290624784190147540
>>
>> 868629374444524241914475969373312242711272295581835015268239838630
>>
>> 384687190358074924950678097815693184181742687897742052192290506159
>>
>> 966977715583822073001725399087487417029503425099200181178538755879
>>
>> 154502747329632479783867518677825016511206041715420824673276500757
>>
>> 701893326947607140574651466346710937708041352100919590569426777246
>>
>> 299404343633206734969316016650928503035662685980006244385122927423
>>
>> 566234832147217710080905407564980210659734829983966877670480463324
>>
>> 174647267991317842060411717271267161117375227910391548714107776891
>>
>> 513148054744247491880859369015322391830905613829985734841069098983
>>
>> 594466353028699791822804129515033560276376890824951544379999191442
>>
>> 036892862978119554219589751739581708227928679974049485410647819767
>>
>> 632581300440457515141092882709263728518362661715556504505756944054
>>
>> 718340598245325602511255244250207904704121983163309894950931458712
>>
>> 809580594862337802500418398431405480893860157182072480132320585828
>>
>> 889914529391213150639488900110069086872461727290576704598789739555
>>
>> 487214162249774467649923114928283271931634840282117207293377910135
>>
>> 595752147127557251622558326421157857414637362020364123648587909877
>>
>> 696219093586729789995137470194719156642385692199285919411113718990
>>
>> 106768290110675682960300014511068937681320680843298237148951823441
>>
>> 428351943405242753083069015608592612389742057748503592682293924554
>>
>> 770508834146124744108391100201797587401627078021473436036153227723
>>
>> 256430759847253758147206996347241255869426284445333291765102703844
>>
>> 080551306884674006671923619625045833456199941934043201
>>
>>
>>

>>> It's two equal length prime divisors (almost divisors ;-) that pass the
>>
>>> Rabin\ECM prime check routine.
>>
>>> Each have 850 digits.
>>
>>> 1416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807585172453732300928844166217497513339011609552754760919432516613342515934207128898323163882601844369732916068174529024861277545319487121471318760920620239340889401489931662342201511922351961295458131967951747811556037841876042724783551666212991970250914479154838122415284133433810002231952457010874393873648375474828092884155484954150665010040649757624512486556720828457365151063247005373731824240405315952807
>>
>>> *
>>
>>>
>>
>> 1770526018141491614407259702105100764853320993433918854323143311828504559932281568403417164774129727767937973233258960531417869154791449718757166263352175084205902906411830149870782190611494853119159500850627921132314967534864622853968281878451296339249552996839810639903047368948159314728787456210040455400379720320243257005672775658799271345813421955207157078958926088995352495877342479165872567062949759368661982825535716205490253234626149634159872442901796752029687494869421240126801755023653346857933983845087465422809197335301053824637618390891818836163045387587785956197009705433494726504331049316206029065561153849895620105141406304181518409968975495678735572997371533634997521094733104882238457865188663927542646216364735063353512995572269827640397882974905439707087911820392750572584187225969534614037631650962011453192523104153734860286207
>>
>>
>>
>>
>>
>> OK, their product is:
>>
>>
>>
>> 2507809904732772394635019022614074568536711855858334502528934
>>
>> 817697416301041139531601179295265853842933415152387033235489116610
>>
>> 327644080155765913133608393959199663876132422337290624784190147540
>>
>> 868629374444524241914475969373312242711272295581835015268239838630
>>
>> 384687190358074924950678097815693184181742687897742052192290506159
>>
>> 966977715583822073001725399087487417029503425099200181178538755879
>>
>> 154502747329632479783867518677825016511206041715420824673276500757
>>
>> 701893326947607140574651466346710937708041352100919590569426777246
>>
>> 299404343633206734969316016650928503035662685980006244385122927423
>>
>> 566234832147217710080905407564980210659734829983966877670480463324
>>
>> 174647267991317842060411717271267161117375227910391548714107776891
>>
>> 513148054744247491880859369015322391830905613829985734841069098983
>>
>> 594466353028699791822804129515033560276376890824951544379999191442
>>
>> 036892862978119554219589751739581708227928679974049485410647819767
>>
>> 632581300440457515141092882709263728518362661715556504505756944054
>>
>> 718340598245325602511255244250207904704121983163309894950931458712
>>
>> 809580594862337802500418398431405480893860157182072480132320585828
>>
>> 889914529391213150639488900110069086872461727290576704598789739555
>>
>> 487214162249774467649923114928283271931634840282117207293377910135
>>
>> 595752147127557250844728429820046612378504904853272516739292947225
>>
>> 066807891058876965460238429071539770341127021354286561627703549774
>>
>> 034055911674583516166539973954550875737988869787406172532125894957
>>
>> 799322449958952254336458637651616571353096259583644577791390792684
>>
>> 136681347381041272054978946086366240554307431354448479560112832384
>>
>> 391699305252179237024194426842152350253762960181356406133002579643
>>
>> 628445350566157023487860488049102964562235014925033049
>>
>>
>>
>> and the difference between it and "my" number is:
>>
>>
>>
>> 777829896601111245036132457167091606909294962652629411202527
>>
>> 852824534899041123179386301258670844999357783410169216072712378436
>>
>> 092166793760040556518061943331811055892064616825928483629029493446
>>
>> 290498746610377956976041036645798164859014890903131870633827486765
>>
>> 083472053412154115431346847319646667024956476040395338864731454595
>>
>> 074521123012569505088905615663324263976885632100124200452105956318
>>
>> 516983184063131575942868893964927009010152
>>
>>
>>
>> which is impressively small. :-)
>>
>>
>>
>>
>>

>>> = a semi prime composite with the first 1267 high order digits the same
>>
>>> as your composite.
>>
>>> 25078099047327723946350190226140745685367118558583345025289348176974163010411395316011792952658538429334151523870332354891166103276440801557659131336083939591996638761324223372906247841901475408686293744445242419144759693733122427112722955818350152682398386303846871903580749249506780978156931841817426878977420521922905061599669777155838220730017253990874874170295034250992001811785387558791545027473296324797838675186778250165112060417154208246732765007577018933269476071405746514663467109377080413521009195905694267772462994043436332067349693160166509285030356626859800062443851229274235662348321472177100809054075649802106597348299839668776704804633241746472679913178420604117172712671611173752279103915487141077768915131480547442474918808593690153223918309056138299857348410690989835944663530286997918228041295150335602763768908249515443799991914420368928629781195542195897517395817082279286799740494854106478197676325813004404575151410928827092637285183626617155565045057569440547183405982
45325602511255244250207904704121983163309894950931458712809580594862337802500418398431405480893860157182072480132320585828889914529391213150639488900110069086872461727290576704598789739555487214162249774467649923114928283271931634840282117207293377910135595752147127557250844728429820046612378504904853272516739292947225066807891058876965460238429071539770341127021354286561627703549774034055911674583516166539973954550875737988869787406172532125894957799322449958952254336458637651616571353096259583644577791390792684136681347381041272054978946086366240554307431354448479560112832384391699305252179237024194426842152350253762960181356406133002579643628445350566157023487860488049102964562235014925033049
>>
>>>
>>
>>> This was derived from your composite!
>>
>>>
>>
>>> Hint--- Look at the ratio between the two primes and tie that in with ---
>>
>>>
>>
>>> (sqrt(of your composite))/(sqrt(ratio)) = start of smallest divisor.
>>
>>
>>
>>
>>
>> Well, the ratio is very, very close to 1.25.
>>
>>
>>
>> So you computed SQRT(my composite)/SQRT(1.25) and then hunted for the first
>>
>> prime you encountered (via sieve? which program?) that was (say) larger than
>>
>> that.
>>
>>
>>
>> Then divided it into my composite, found the quotient and found the first
>>
>> prime that was (say) smaller than it.
>>
>>
>>
>> I'd know which way around it was if I knew whether the ratio was more or
>>
>> less than 1.25.
>>
>>
>>
>>
>>

>>> This would make an interesting encryption scheme but oh-oh,
>>
>>
>>
>>
>>
>> Not too wise a one, if people know private keys will be close to
>>
>> 1.118033988749894848204586834365 and 1.118033988749894848204586834365^(-1)
>>
>> times the square root of the public key! :-)
>>
>>
>>
>> Yes, "randomly" deciding that ratio would be the key, and various posts by
>>
>> me
>>
>> in the discussion with pubkeybreaker brought that up.
>>
>>
>>
>> The ratio would need to be generated intelligently, as it otherwise would
>>
>> prove a vulnerability. For example, using a pseudorandom number generator
>>
>> that
>>
>> generated a 128 bit binary number from 0 to 2^128 - 1 and which was then
>>
>> used to get our ratio would significantly weaken a 512 bit encryption
>>
>> system.
>>
>>
>>
>> But I'm certain you meant that.
>>
>>
>>
>>
>>

>>> now the cat's out of the bag.
>>
>>> You would have your composite as the public key where the logical step
>>
>>> would be to find it's prime factors to access the private key but
>>
>>> this would not be so.
>>
>>> Instead your composite would be the public key and unbeknownst to
>>
>>> the one decrypting my semi-prime composite would be the mid level private
>>
>>> key which then would have to be factored for the final private key.
>>
>>> So a 5 or 6 fold whammy of deception is possible.
>>
>>
>>
>>
>>
>> Yes, I was sure that was the path you were going to suggest. Interesting.
>>
>>
>>
>> I think it adds only one layer of complexity; "my" composite needs to have
>>
>> its smallest factor very large, else it will become obvious that it is not
>>
>> an "almost" prime (semi-prime).
>>
>>
>>
>>
>>

>>> #1 Your composite need not be factored but if it was, it would be a dead
>>
>>> end.
>>
>>
>>
>>
>>
>> Not quite. If it was even part-factored and that could be seen not to be
>>
>> the
>>
>> private key (or too small, so obviously not the private key), then they know
>>
>> what to do next and the one extra layer is guessing the ratio.
>>
>>
>>
>> Not sure how the actual mechanism of public key cryptography would work,
>>
>> though.
>>
>> How would that which is encrypted via the public key be able to be
>>
>> decrypted,
>>
>> exactly?
>>
>>
>>
>> pubkeybreaker may well be along soon; he knows much more about this area
>>
>> than I.
>>
>>
>>
>>
>>
>>
>>
>>
>>

>>> #2 On the other hand if the one decrypting is using your composite as a
>>
>>> catalyst
>>
>>>
>>
>>> to iterate the smallest divisor to a predetermined stopping point which
>>
>>>
>>
>>> in-itself ---
>>
>>>
>>
>>>
>>
>>>
>>
>>> #3 creates possibilities almost uncountable because that is how many
>>
>>> discrete
>>
>>>
>>
>>> semi - primes are possible with a composite of this size .
>>
>>>
>>
>>>
>>
>>>
>>
>>> #4 Finding, in this case, a compatible smaller prime divisor match with
>>
>>> your
>>
>>>
>>
>>> composite in smallest next_prime look-up with largest next_prime look-up
>>
>>>
>>
>>> dividing into your composite. It is what I did to achieve my two factors
>>
>>> above.
>>
>>>
>>
>>>
>>
>>>
>>
>>> #5 Many easy or complex ratio's can be used to construct these 2 degree
>>
>>>
>>
>>> monic polynomials.
>>
>>>
>>
>>> if a complex ratio is used and the composites is very large like yours the
>>
>>>
>>
>>> look-up process can take days. I was lucky because I used a simple ratio
>>
>>>
>>
>>> that created a 4 cycle (gap) polynomial that only took a day or two to
>>
>>> find the
>>
>>>
>>
>>> two factors. I worked with some that had 1000 cycle but very hard
>>
>>>
>>
>>> to find two matching factors. The 4 cycle gap that I used here is for
>>
>>>
>>
>>> the small divisor only. The large divisor has a 5 cycle gap. This comes
>>
>>>
>>
>>> automatically with the iterative process.
>>
>>>
>>
>>>
>>
>>>
>>
>>> Some ratios will not work, these are rare, because of 0(mod 3) or
>>
>>>
>>
>>> 0(mod 5) or even integers alternating between small and large divisors.
>>
>>>
>>
>>> In the polynomial cycle where a small and large prime match is not
>>
>>> possible
>>
>>>
>>
>>> because one or the other will be a 0(mod 3or5) or an even integer
>>
>>>
>>
>>> thus no prime pair match is ever possible because of this alternating
>>
>>> pattern
>>
>>>
>>
>>> between the two divisors.
>>
>>>
>>
>>>
>>
>>>
>>
>>>
>>
>>>
>>
>>> These residuals can be checked when first starting the iterative process
>>
>>>
>>
>>> for any particular ratio and not much work has to be done to either rule
>>
>>>
>>
>>> in or rule out a certain ratio.
>>
>>>
>>
>>>
>>
>>>
>>
>>>
>>
>>>
>>
>>> Cheers
>>
>>>
>>
>>>
>>
>>>
>>
>>> Dan
>>
>> Again, using Port563 1699 digit composite to produce another prime pair from
>>
>> the
>>
>> same polynomial but skipping down stream from the last semi-prime created
>>
>> with a
>>
>> gap that equals 805 digits and then conducting the probable prime search.
>>
>>
>>
>> gap =
>>
>>
>>
>> 2948602489324412628623949272393032074218793597790851845907327495026951131224194303203769964432571608251437251915411344242143959789812682523963954733972734669439197534906904112638208297009020407674421784458363420492449624523729161237704579371478537032780524187352664035667746086081038223365135829513283821227257315862017581919027211620909597263746901425097912401124193536464431552919968175146853859356886326538473880803166476350009972032457606829512595370401394508556838129116040181602448065712022505781729402748223521490142792454740633168221695922335657821374674969447393473222910395298402961973825046535416294674701529835525980988112169739641630367348266642330490652511339380504473424419543361083468300369433804477287189477423100758558261597650956345122052311503113850013209424497000018755752361789408078
>>
>>
>>
>> Just imagine how long it would take to find all the semi-primes within that
>>
>> gap!
>>
>>
>>
>> The two new Rabin\ECM certified primes =
>>
>>
>>
>> 1416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807585172453732300928844166217546704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696608543983
>>
>> *
>>
>> 1770526018141491614407259702105100764853320993433918854323143311828504559932281568403417164774129727767937973233258960531417869154791449718757166263352175084205902906411830149870782190611494853119159500850627921132314967534864622853968281878451296339249552996839810639903047368948159314728787456210040455400379720320243257005672775658799271345813421955207157078958926088995352495877342479165872567062949759368661982825535716205490253234626149634159872442901796752029687433380547502319371349237401318751850330995002612166966050645077153065049892267988518498722060992259717657549424426942322750901753580534034221821744452869899280028380364484570729047366710271490540313201675486608881880528773373990591002759103884547167454359577337817308552993957165148491966499540049897949888121541471781384807413917082717824804669044895535104954843942227120744547237
>>
>>
>>
>> = semi-prime composite =
>>
>>
>>
>> 250780990473277239463501902261407456853671185585833450252893481769741630104113953160117929526585384293341515238703323548911661032764408015576591313360839395919966387613242233729062478419014754086862937444452424191447596937331224271127229558183501526823983863038468719035807492495067809781569318418174268789774205219229050615996697771558382207300172539908748741702950342509920018117853875587915450274732963247978386751867782501651120604171542082467327650075770189332694760714057465146634671093770804135210091959056942677724629940434363320673496931601665092850303566268598000624438512292742356623483214721771008090540756498021065973482998396687767048046332417464726799131784206041171727126716111737522791039154871410777689151314805474424749188085936901532239183090561382998573484106909898359446635302869979182280412951503356027637689082495154437999919144203689286297811955421958975173958170822792573137156016099801919581836018826836623872154330203086335640102333571058433231020255317409028576898999

3888350595843900008063944914309300639345940216958724043370383059955433689862210101422385073219438372723709059639870961305160753291686633779160951867329544899519797691622830618517751516190351384648965953240392607252910701352904423219372519943546476485037683351022934020786413175509851871465524645547966945978265412144058590457881086844507853408916476169368946618464516157445587521947325968343889168801786763791750433060732945534129154237950836203559299977947774989504860863235727256318168391122434537109161496716091154301155366392929102315337315719017973911976940701937140179608358959972155944168710936654760622364865160776339859422798144535351623298506393039063043173474651063552102945806482653135624971e+1698
>>
>>
>>
>> Dividing each prime into the 1699 digit composite, the trailing zeros in the
>>
>> quotients = only 44
>>
>> This semi-prime is still well in range of the same triangle # index as the
>>
>> 1699
>>
>> digit composite.
>>
>> The previous primes when divided into 1699 digit composite had 417 trailing
>>
>> zeros.
>>
>> Where the best number of trailing zeros for any ratio could possibly be 424
>>
>> but
>>
>> highly unlikely.
>>
>> For each new high order digit that is attained in the search for these
>>
>> prime pairs the semi-primes that are found grow exponentially.
>>
>>
>>
>> Dan
>>
>>
>>
>> ---------------
>>
>>
>>
>>
>>
>>
>>
>> Thank you for the entertaining idea!

>
> The trick in iterating the smallest divisor of your composite or any large
> prime or composite is, first step ---(sqrt(your c))/(sqrt( r )).
> (r) represents the selected ratio and in this case r = 1.25
>
> That produces the smallest of your smallest(int)divisor of your composite.
> The next step is starting to build the polynomial with the smallest divisor.
>
> smallest divisor =
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428572988368918293191363646250175843311183122894983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
>
> If int(small divisor) is even add 1 which I did here.
>
> Largest divisor below without remainder just to keep these numbers small
> in this explanation.
> The remainder of the largest divisor is important because this determines
> what action to take on the smallest divisor for each following iteration.
> There are many iterations especially with a composite and divisor of this size.
>
> 1.770526018141491614407259702105100764853320993433918854323143311828504559932281568403417164774129727767937973233258960531417869154791449718757166263352175084205902906411830149870782190611494853119159500850627921132314967534864622853968281878451296339249552996839810639903047368948159314728787456210040455400379720320243257005672775658799271345813421955207157078958926088995352495877342479165872567062949759368661982825535716235461147866489204557812719804138978903618729683380547502319371349237401318751850330995002612166966050645077153065049892267988518498722060992259717657549424426942322750901753580534034221821744452869899280028380364484570729047366710271490540313201675486608881880528773373990591002759103884547167454359577337817308552993957165148491966499540049897949888121541471781384807413917082717824804669044895535104954843942227120752613607e+849
>
>
> There is no known algorithm for what is done next and many more steps
> following to build this polynomial.
>
> Step#1
> Add 10^424 as a trial to the smallest divisor.
> All the next iteration will be - 10^423*n,- 10^422*n,-10^421*n, -10^420*n
> where n could =[0,1,2,3,4,5,6,7,8,9] or skipping a -10^# because all 9's
> are showing instead of zeros in the remainder. At times you have to add back the
> last 10^# because of the nine's showing up in the remainder.
> Keep dividing into the composite and take note of the remainder.
> The key from here on is, building zeros starting with 1 zero and going
> from there all the way to 424 trailing zeros in the quotient remainder.
> Skipping the first few iterations just divide into the composite at the
> pause point - e+416 or -10^416*n these examples below represent just my pause
> points. You can divide each one and observe how the quotient remainder keeps
> adding more and more zeros as you minus 10^#.
> e+416 or - 10^416 limit for pause point below.
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345634193191363646250175843311183122894983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
>
> next = e+412
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998691363646250175843311183122894983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
>
> e+408 or - 10^408 is the limit for smallest divisor below.
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807646250175843311183122894983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
> e+407 or - 10^407 was the limit for the smallest divisor below.
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807646250175843311183122894983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
> e+405 - etc..
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807585250175843311183122894983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
> e+400
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807585172455843311183122894983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
> e+386
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807585172453732300928844194983746704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
> e+380
> 1.416420814513193291525807761684080611882656794747135083458514649462803647945825254722733731819303782214350378586607168425134295323833159775005733010681740067364722325129464119896625752489195882495327600680502336905851974027891698283174625502761037071399642397471848511922437895158527451783029964968032364320303776256194605604538220527039417076650737564165725663167140871196281996701873983332698053650359807494929586260428573012345633998681807585172453732300928844166217546704438001855497079389921055001480264796002089733572840516061722452039913814390814798977648793807774126039539541553858200721402864427227377457395562295919424022704291587656583237893368217192432250561340389287105504423018699192472802207283107637733963487661870253846842395165732118793573199632039918359910497233177425107845931133666174259843735235916428083963875153781696602090887e+849
>
> etc...Note the progression of zeros in the remainder in each pause point above.
>
> As you will note I am just showing my pause points and not showing all the
> iterations in between. I have become very used to doing this and will tackle
> any large composite.


How about the 130-digit composite number:

47485133560063190433548259679204355907477810307482663990525310023918\
21388435849831808696657092215642611599144927466443608010614619 ?

dave



> All the above calculations for building this polynomial are taken from my first
> post.
> The last post,(there are 2 but hit enter twice and got a repeat entry), just
> show a smaller and larger prime searched in the area of 44 trailing zeros
> in the quotient. The first post starts the search for the two primes with the
> limit of 424 trailing zeros in the quotient which is the last step. When the
> two primes were found there was 417 trailing zeros in the quotient.
> There is an exponential growth of these semi-primes for each one less trailing
> zero.
>
> I am sure there are many questions and what if any good can be derived from
> this process. I do not know, but it is interesting.
>
>
> Cheers,
>
> Dan
>



--
http://www.bibliotecapleyades.net/sociopolitica/last_circle/1.htm



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