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Selected triangle number factoring using this remarkable constant.

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Danny73

unread,
Jan 2, 2010, 9:10:23 PM1/2/10
to

This ratio constant probably can factor select semi prime composites
that have two equal length factors --->oo

The constant =
1.7716319313702642926933210686094850053842225480939558214253613539850773282389791966696705519923442820932933201674265437101233512108496314705467624966849542731188005323582182177778523673709006609873379369829861555397670809042526919127192172819394923733896803923671261255401777363787347143737810498520022993784

Each iterative process using larger and larger semi-prime
composites continues with a smaller and smaller convergence.

It starts with this unusual odd composite with just 2 prime factors.


88531
t(n) =421 triangle # index
((floor(sqrt(88531))) - 1) =296 + ((floor(sqrt(88531)))/2)
(not even so +1 before division by 2) =297 +1= 298/2 = 149 + 296 =
445
445-421(t(n) index) = 24
t(24) = 300
300+88531 = 88831 = t(421)
421-24 = 397 a prime factor of 88531.

397/223 = 1.780269.. constant correct to one decimal digit.

Continuing from here skipping many that work--
------------------------------------------------------------
The next composite where just the sqrt(174341)-1
= 416 +
sqrt(174341) is odd so +1 then divide by 2. =209+416 = 625.
the third value in the third row below.
Each following example will just show the third
value as being precalculated for the start of the
third row.

174341
t(n) = 590
625 -590 = 35
t(35) = 630
630+174341 = 174971 = t(592) (174971) is on the same
index as t(592).
592-35 = 557 a factor of 174341

557/313 = 1.7795527156.. correct to 2 decimal places
---------------------------------------------------------------------------------


Many are within +/- (1) of having the correct factor
but are not shown above.

Continuing below but with each composite having
(2) factors are 1 digit longer for each factor from previous
composite.
So many solutions are skipped but shows
how this constant converges.
-------------------------------------------------------------------------

The next best composite that continues the convergence of
this constant is ...

179357447
t(n) = 18940
((floor(sqrt(179357447))) - 1) = 13391 + (floor(sqrt(179357447)))/2)
(is even) = 6696 = 20087
20087-18940 = (t(n) index) = 1147
t(1147) = 658378
658378 + 179357447 = 180015825 = t(18974)
18974 - 1147 = 17827 a prime factor of 179357447

17827/10061 = 1.771891... constant correct too 3 decimal places.
Note smaller and smaller trend.
----------------------------------------------------------------------------------------------------
27302014409
t(n) = 233675
((floor(sqrt(27302014409))) - 1) = 165232 +
(floor(sqrt(27302014409)))/2)(is odd so +1 then divide ) = 82617 =
247849
247849 - 233675 = (t(n)index) = 14174
t(14174) = 100458225
100458225 + 27302014409 = 27402472634 = t(234105)
234105 - 14174 = 219931 a factor of 27302014409

What is interesting here is 27402472634 is not a triangle number
but the triangle index it is resident too will still work.

219931/124139 = 1.77165113... constant is correct too 4 decimal
places.
---------------------------------------------------------------------------------------------------------
As each factor increases by one in length so also
does the ratio convergence which always lags about 2 decimal
places behind the length of each factor.
---------------------------------------------------------------------------

4353229083887
t(n) = 2950671
((floor(sqrt(4353229083887))) - 1) = 2086438 +
(floor(sqrt(4353229083887)))/2(is odd, then +1 before division ) =
1043220 = 3129658
3129658 - 2950671 = (t(n)index) = 178987
t(178987) = 16018262578
16018262578 + 4353229083887 = 4369247346465 = t(2956094)
2956094 - 178987 = 2777107 a factor of 4353229083887

2777107/1567541 = 1.7716327675.. correct to 5 decimal places.
-----------------------------------------------------------------------------------------------------
217388918024939
20851327
22116170 - 20851327 = t(1264843) = 799914539746
799914 - 53
9746 + 217388918024939 = 218188832564685 = t(20889654)
20889654 - 1264843 = 19624811 a factor in 217388918024939

19624811/11077249 = 1.771632198.. correct to 5 decimal places.
---------------------------------------------------------------------------------------------------------
435328362428129
29506893
((floor(sqrt(435328362428129))) - 1) = 20864523
(floor(sqrt(435328362428129)))/2(is even, then divide by (2) ) =
10432262 =31296785
31296785 -29506893 = t(1789892)
t(1789892) = 1601857580778
1601857580778+ 435328362428129= 436930220008907= t(29561131)
29561131 - 1789892 = 27771239
27771239/15675511 = 1.7716321337.. correct to 5 decimal places.
------------------------------------------------------------------------------------------------------------
43533031961067529
295069592
312968563 - 295069592 = 17898971
t(17898971) = 160186590378906
160186590378906 + 43533031961067529 = 43693218551446435 = t(295611970)
295611970 - 17898971 = 277712999
277712999/156755471 = 1.771631938766.. correct too 8 decimal places

Many more composites were tested > 43533031961067529 that kept
converging
until I discovered this about the constant --

This can be continued on without knowing the factors of the
composite as in rsa2048.
Where all that is needed is the sqrt(rsa2048) which will lead
too the ((floor(sqrt(rsa2048))) - 1) + (floor(sqrt(rsa2048)))/2(when
even, +1 before division by 2 if not )
creating the psudo target index t(n).
Which in turn creates the secondary psudo target t(n).
From that creating the 2 psudo divisors for rsa2048.

The two psudo divisors for rsa2048 are --

1.19255457110375304567739721244045519941579254744346817088280929933854011082834300274577681154608179553173045095656565789083015235082776646817705422207344776044989145504173977871099772510148796186409336947671245498217694257811610465478701090659083472292485805101567477524002473693868939857193202608023184205395e
+308
and
2.11276775806897918442849641614326613635448531469555943568810129411213449998300630762981603427065139264599500691127244230373643817456280436577289915197802107905854990865841296538543274020440862130972657064639152960641666882749563148799122883598096000763226210525003231524125593166181196906421614852797596202001e
+308
The ratio between terms =
1.7716319313702642926933210686094850053842225480939558214253613539850773282389791966696705519923442820932933201674265437101233512108496314705467624966849542731188005323582182177778523673709006609873379369829861555397670809042526919127192172819394923733896803923671261255401777363787347143737810498520022993784

The above constant is probably correct to 306 decimal digits.

Checking further with a composite that the two prime factors are the
actual ratio of this constant and also of rsa2048 composite length.

c=
25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654634667325978700286196409945277152359652911116220747814834797494124520273481100958873802984024176492225246833573292649968730745633409168700269997784355314427770827350154506462747534358237104269780480177324470358119995052460947987505203719877897352995037853634804560864357696778477226889223362775766444868539694258335216732621
t(n) index of above composite =
224481217368660460915408862554712298245182706157645728389714076402122669037440937713331085448276541778133808334192088523233935894354588832440192398225499679915046958748962829698977599186630094629328094688999361524600369584044761488100895587346975601067822040592783690907334382460447881104660535542147349064389
(floor(sqrt(c))) - 1 =
158732191050391204174482508661063007579358463444809715795726627753579970080749948404278643259568101132671402056190021464753419480472816840646168575222628934671405739213477439533870489791038973166834068736234020361664820266987726919453356824138007381985796493621233035112849373047484148339095287149610832306742
+
(floor(sqrt(c))) + 1(because odd) /2 =
79366095525195602087241254330531503789679231722404857897863313876789985040374974202139321629784050566335701028095010732376709740236408420323084287611314467335702869606738719766935244895519486583417034368117010180832410133493863459726678412069003690992898246810616517556424686523742074169547643574805416153372
=
238098286575586806261723762991594511369037695167214573693589941630369955121124922606417964889352151699007103084285032197130129220709225260969252862833943402007108608820216159300805734686558459750251103104351030542497230400481590379180035236207011072978694740431849552669274059571226222508642930724416248460114
minus index of composite =
224481217368660460915408862554712298245182706157645728389714076402122669037440937713331085448276541778133808334192088523233935894354588832440192398225499679915046958748962829698977599186630094629328094688999361524600369584044761488100895587346975601067822040592783690907334382460447881104660535542147349064389
=
t(n) index =
13617069206926345346314900436882213123854989009568845303875865228247286083683984893086879441075609920873294750092943673896193326354636428529060464608443722092061650071253329601828135499928365120923008415351669017896860816436828891079139648860035471910872699839065861761939677110778341403982395182268899395725
its' triangle # =
92712286893110843907855094168344367390183620871705857718017014197851070846450604376304052754355405476694589094715159555362482970745313446671248545380659694656187557213068810863308275984399873205643525863092034081104830234346834513670152366795649220851770418527179643056779482643638008398421867114141117601942794406292328314490168505987080375842941539670067732570224681500104713635121717577071297751822734259451655529499857948485416851152368436645437139657228885579165138362534988325703497152355147192450945660169117465856959964219181918543351359280532089154414094257822979111750219312409905716096037051769523835675
+ (c)
25195908475657893494027183240048398571429282126204032027777137836043662020707595556264018525880784406918290641249515082189298559149176184502808489120072844992687392807287776735971418347270261896375014971824691165077613379859095700097330459748808428401797429100642458691817195118746121515172654634667325978700286196409945277152359652911116220747814834797494124520273481100958873802984024176492225246833573292649968730745633409168700269997784355314427770827350154506462747534358237104269780480177324470358119995052460947987505203719877897352995037853634804560864357696778477226889223362775766444868539694258335216732621
=
25288620762551004337935038334216742938819465747075737885495154850241513091554046160640322578635139812394985230344230241744661042119921497949479737665453504687343580364500845546834726623254661769580658497687783199158718210093442534611000612115604077622649199519169638334873974601389759523571076501781467096302228990816237605466849821417103301123657776337164192252843705782458978516619145894069296544585396026909420386275133267117185686848936723751073207967007383392041912672720772092595483977329679617550570940712630065453362163684097079271538389212915336650018771791036300206000973582088176350584635731310104740568296
t(n) of above value =
224893845013824263789164542051208826759303520479124788872685994639460736081984615656068482868140749185472795441220187904269837143810916865106350379806245829997916640937094626140371409520369227251895665479990821978538527699186392039878262532458131472674098910364069093286065270276959538310404010045066495457294
-
13617069206926345346314900436882213123854989009568845303875865228247286083683984893086879441075609920873294750092943673896193326354636428529060464608443722092061650071253329601828135499928365120923008415351669017896860816436828891079139648860035471910872699839065861761939677110778341403982395182268899395725
=
211276775806897918442849641614326613635448531469555943568810129411213449998300630762981603427065139264599500691127244230373643817456280436577289915197802107905854990865841296538543274020440862130972657064639152960641666882749563148799122883598096000763226210525003231524125593166181196906421614862797596061569
= A factor of (c)
= The other factor of (c)
=
119255457110375304567739721244045519941579254744346817088280929933854011082834300274577681154608179553173045095656565789083015235082776646817705422207344776044989145504173977871099772510148796186409336947671245498217694257811610465478701090659083472292485805101567477524002473693868939857193202613667697426509
Whos' ratio =
1.7716319313702642926933210686094850053842225480939558214253613539850773282389791966696705519923442820932933201674265437101233512108496314705467624966849542731188005323582182177778523673709006609873379369829861555397670809042526919127192172819394923733896803923671261255401777363787347143737810498520022993784
Constant correct too 307 decimal digits.
Which gives the exact ratio of the two psudo devisors of rsa2048
correct
too 307 decimal digits.

I am not sure this will work for a composite with (1) factor that is
larger by (1) in length then the smaller factor?

Dan


Danny73

unread,
Jan 3, 2010, 3:23:13 PM1/3/10
to
On Jan 2, 9:10 pm, Danny73 <fasttrac...@att.net> wrote:
> This ratio constant probably can factor select semi prime composites
> that have two equal length factors  --->oo
>
> The constant =
> 1.7716319313702642926933210686094850053842225480939558214253613539850773282­389791966696705519923442820932933201674265437101233512108496314705467624966­849542731188005323582182177778523673709006609873379369829861555397670809042­526919127192172819394923733896803923671261255401777363787347143737810498520­022993784
> ---------------------------------------------------------------------------­------

>
> Many are within +/- (1) of having the correct factor
> but are not shown above.
>
> Continuing below but with each composite having
> (2) factors are 1 digit longer for each factor from previous
> composite.
> So many solutions are skipped but shows
> how this constant converges.
> -------------------------------------------------------------------------
>
> The next best composite that continues the convergence of
> this constant is ...
>
> 179357447
> t(n) = 18940
> ((floor(sqrt(179357447))) - 1) = 13391 + (floor(sqrt(179357447)))/2)
> (is even) = 6696 = 20087
> 20087-18940 = (t(n) index) = 1147
> t(1147) = 658378
> 658378 + 179357447 = 180015825 = t(18974)
> 18974 - 1147 = 17827  a prime factor of 179357447
>
> 17827/10061 =  1.771891... constant correct too 3 decimal places.
> Note smaller and smaller trend.
> ---------------------------------------------------------------------------­-------------------------

> 27302014409
> t(n) = 233675
> ((floor(sqrt(27302014409))) - 1) = 165232 +
> (floor(sqrt(27302014409)))/2)(is odd so +1 then divide ) = 82617 =
> 247849
> 247849 -  233675 = (t(n)index) = 14174
> t(14174) = 100458225
> 100458225 + 27302014409 = 27402472634 = t(234105)
> 234105 - 14174 = 219931 a factor of 27302014409
>
> What is interesting here is 27402472634 is not a triangle number
> but the triangle index it is resident too will still work.
>
> 219931/124139 = 1.77165113... constant is correct too 4 decimal
> places.
> ---------------------------------------------------------------------------­------------------------------

> As each factor increases by one in length so also
> does the ratio convergence which always lags about 2 decimal
> places behind the length of each factor.
> ---------------------------------------------------------------------------
>
> 4353229083887
> t(n) = 2950671
> ((floor(sqrt(4353229083887))) - 1) = 2086438  +
> (floor(sqrt(4353229083887)))/2(is odd, then +1 before division ) =
> 1043220 = 3129658
> 3129658 - 2950671 = (t(n)index) = 178987
> t(178987) = 16018262578
> 16018262578 + 4353229083887 = 4369247346465 = t(2956094)
> 2956094 - 178987 = 2777107 a factor of 4353229083887
>
> 2777107/1567541 =  1.7716327675.. correct to 5 decimal places.
> ---------------------------------------------------------------------------­--------------------------

> 217388918024939
> 20851327
> 22116170 - 20851327 = t(1264843) = 799914539746
> 799914 - 53
> 9746 + 217388918024939 = 218188832564685 = t(20889654)
> 20889654 - 1264843 = 19624811 a factor in 217388918024939
>
> 19624811/11077249 = 1.771632198.. correct to 5 decimal places.
> ---------------------------------------------------------------------------­------------------------------

> 435328362428129
> 29506893
> ((floor(sqrt(435328362428129))) - 1) = 20864523
> (floor(sqrt(435328362428129)))/2(is even, then divide by (2) ) =
> 10432262 =31296785
> 31296785 -29506893 = t(1789892)
> t(1789892) = 1601857580778
> 1601857580778+ 435328362428129= 436930220008907= t(29561131)
> 29561131 - 1789892 = 27771239
> 27771239/15675511 = 1.7716321337.. correct to 5 decimal places.
> ---------------------------------------------------------------------------­---------------------------------

> 43533031961067529
> 295069592
> 312968563 - 295069592 = 17898971
> t(17898971) = 160186590378906
> 160186590378906 + 43533031961067529 = 43693218551446435 = t(295611970)
> 295611970 - 17898971 = 277712999
> 277712999/156755471 = 1.771631938766.. correct too 8 decimal places
>
> Many more composites were tested > 43533031961067529  that kept
> converging
> until I discovered this about the constant --
>
> This can be continued on without knowing the factors of the
> composite as in rsa2048.
> Where all that is needed is the sqrt(rsa2048) which will lead
> too the ((floor(sqrt(rsa2048))) - 1) + (floor(sqrt(rsa2048)))/2(when
> even, +1 before division by 2 if not )
> creating the psudo target index t(n).
> Which in turn creates the secondary psudo target t(n).
> From that creating the 2 psudo divisors for rsa2048.
>
> The two psudo divisors for rsa2048 are --
>
> 1.1925545711037530456773972124404551994157925474434681708828092993385401108­283430027457768115460817955317304509565656578908301523508277664681770542220­734477604498914550417397787109977251014879618640933694767124549821769425781­161046547870109065908347229248580510156747752400247369386893985719320260802­3184205395e
> +308
> and
> 2.1127677580689791844284964161432661363544853146955594356881012941121344999­830063076298160342706513926459950069112724423037364381745628043657728991519­780210790585499086584129653854327402044086213097265706463915296064166688274­956314879912288359809600076322621052500323152412559316618119690642161485279­7596202001e

> +308
> The ratio between terms =
> 1.7716319313702642926933210686094850053842225480939558214253613539850773282­389791966696705519923442820932933201674265437101233512108496314705467624966­849542731188005323582182177778523673709006609873379369829861555397670809042­526919127192172819394923733896803923671261255401777363787347143737810498520­022993784

>
> The above constant is  probably correct to 306 decimal digits.
>
> Checking further with a composite that the two prime factors are the
> actual ratio of this constant and also of rsa2048 composite length.
>
> c=
> 251959084756578934940271832400483985714292821262040320277771378360436620207­075955562640185258807844069182906412495150821892985591491761845028084891200­728449926873928072877767359714183472702618963750149718246911650776133798590­957000973304597488084284017974291006424586918171951187461215151726546346673­259787002861964099452771523596529111162207478148347974941245202734811009588­738029840241764922252468335732926499687307456334091687002699977843553144277­708273501545064627475343582371042697804801773244703581199950524609479875052­037198778973529950378536348045608643576967784772268892233627757664448685396­94258335216732621

> t(n) index of above composite =
> 224481217368660460915408862554712298245182706157645728389714076402122669037­440937713331085448276541778133808334192088523233935894354588832440192398225­499679915046958748962829698977599186630094629328094688999361524600369584044­761488100895587346975601067822040592783690907334382460447881104660535542147­349064389
> (floor(sqrt(c))) - 1 =
> 158732191050391204174482508661063007579358463444809715795726627753579970080­749948404278643259568101132671402056190021464753419480472816840646168575222­628934671405739213477439533870489791038973166834068736234020361664820266987­726919453356824138007381985796493621233035112849373047484148339095287149610­832306742

> +
> (floor(sqrt(c))) + 1(because odd) /2 =
> 793660955251956020872412543305315037896792317224048578978633138767899850403­749742021393216297840505663357010280950107323767097402364084203230842876113­144673357028696067387197669352448955194865834170343681170101808324101334938­634597266784120690036909928982468106165175564246865237420741695476435748054­16153372
> =
> 238098286575586806261723762991594511369037695167214573693589941630369955121­124922606417964889352151699007103084285032197130129220709225260969252862833­943402007108608820216159300805734686558459750251103104351030542497230400481­590379180035236207011072978694740431849552669274059571226222508642930724416­248460114

> minus index of composite =
> 224481217368660460915408862554712298245182706157645728389714076402122669037­440937713331085448276541778133808334192088523233935894354588832440192398225­499679915046958748962829698977599186630094629328094688999361524600369584044­761488100895587346975601067822040592783690907334382460447881104660535542147­349064389
> =
> t(n) index =
> 136170692069263453463149004368822131238549890095688453038758652282472860836­839848930868794410756099208732947500929436738961933263546364285290604646084­437220920616500712533296018281354999283651209230084153516690178968608164368­288910791396488600354719108726998390658617619396771107783414039823951822688­99395725
> its' triangle # =
> 927122868931108439078550941683443673901836208717058577180170141978510708464­506043763040527543554054766945890947151595553624829707453134466712485453806­596946561875572130688108633082759843998732056435258630920340811048302343468­345136701523667956492208517704185271796430567794826436380083984218671141411­176019427944062923283144901685059870803758429415396700677325702246815001047­136351217175770712977518227342594516555294998579484854168511523684366454371­396572288855791651383625349883257034971523551471924509456601691174658569599­642191819185433513592805320891544140942578229791117502193124099057160960370­51769523835675
> + (c)
> 251959084756578934940271832400483985714292821262040320277771378360436620207­075955562640185258807844069182906412495150821892985591491761845028084891200­728449926873928072877767359714183472702618963750149718246911650776133798590­957000973304597488084284017974291006424586918171951187461215151726546346673­259787002861964099452771523596529111162207478148347974941245202734811009588­738029840241764922252468335732926499687307456334091687002699977843553144277­708273501545064627475343582371042697804801773244703581199950524609479875052­037198778973529950378536348045608643576967784772268892233627757664448685396­94258335216732621
> =
> 252886207625510043379350383342167429388194657470757378854951548502415130915­540461606403225786351398123949852303442302417446610421199214979494797376654­535046873435803645008455468347266232546617695806584976877831991587182100934­425346110006121156040776226491995191696383348739746013897595235710765017814­670963022289908162376054668498214171033011236577763371641922528437057824589­785166191458940692965445853960269094203862751332671171856868489367237510732­079670073833920419126727207720925954839773296796175505709407126300654533621­636840970792715383892129153366500187717910363002060009735820881763505846357­31310104740568296

> t(n) of above value =
> 224893845013824263789164542051208826759303520479124788872685994639460736081­984615656068482868140749185472795441220187904269837143810916865106350379806­245829997916640937094626140371409520369227251895665479990821978538527699186­392039878262532458131472674098910364069093286065270276959538310404010045066­495457294
> -
> 136170692069263453463149004368822131238549890095688453038758652282472860836­839848930868794410756099208732947500929436738961933263546364285290604646084­437220920616500712533296018281354999283651209230084153516690178968608164368­288910791396488600354719108726998390658617619396771107783414039823951822688­99395725
> =
> 211276775806897918442849641614326613635448531469555943568810129411213449998­300630762981603427065139264599500691127244230373643817456280436577289915197­802107905854990865841296538543274020440862130972657064639152960641666882749­563148799122883598096000763226210525003231524125593166181196906421614862797­596061569

> = A factor of (c)
> = The other factor of (c)
> =
> 119255457110375304567739721244045519941579254744346817088280929933854011082­834300274577681154608179553173045095656565789083015235082776646817705422207­344776044989145504173977871099772510148796186409336947671245498217694257811­610465478701090659083472292485805101567477524002473693868939857193202613667­697426509
> Whos' ratio =
> 1.7716319313702642926933210686094850053842225480939558214253613539850773282­389791966696705519923442820932933201674265437101233512108496314705467624966­849542731188005323582182177778523673709006609873379369829861555397670809042­526919127192172819394923733896803923671261255401777363787347143737810498520­022993784

> Constant correct too 307 decimal digits.
> Which gives the exact ratio of the two psudo devisors of rsa2048
> correct
> too 307 decimal digits.
>
> I am not sure this will work for a composite with (1) factor that is
> larger by (1) in length then the smaller factor?
>
> Dan

I will answer my own question here with one factor
larger by (1) in length --

c = 57401569569077
t(n) =10714623 = triangle index number for (c)
(floor(sqrt(c)))-1 = 7576381 +( floor(sqrt(c))) (is even )so /2 =
3788191+7576381 = 11364572
11364572-10714623=649949= t(649949) = 211217176275 triangle #.
211217176275+57401569569077 = 57612786745352 = t(10734318) index.
10734318- 649949 = 10084369 a factor of 57401569569077

10084369/5692133= 1.7716327078.. correct too 5 decimal places.

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