Supposedly attached image
Some papers on GNFS:
http://www.ams.org/notices/199612/pomerance.pdf
An explanation of how it works is underway. It uses smooth numbers, though. In the 4th paper there is a part on smooth numbers
We were analyzing 2d today. Below are some tasks I think are worth looking at. Probably for rsa100 it should be done, so irrelevant patterns go away.
sqrt0 = floor_sqrt
analyze sequence f(x) = sqrt0(x*d), starting from x=1
analyze sequence f(x) = sqrt0(x*e), starting from x=1
analyze sequence f(x) = sqrt0(x*f), starting from x=1
hypothesis 1
the first and second sequence always repeat
if they don't repeat or repeat in too many terms, you can tack on another sqrt (just like the tree) and they will repeat.
my idea for analyzing these sequences comes from looking at sqrt(2d) and asking are 3d and 4d significant as well? combined with the remainder tree and the fact that finding where a sequence repeats solves the number in Shor's algorithm.
hypothesis 2
if 1 in f=2d+1-e is actually A in (e, N, T) then maybe f2=2d+a-e is significant too.
hypothesis 3
if the 2nd hypothesis is correct, then maybe 1 in (n-1) is actually (n-A) and therefore (n-a) is significant too
f2 - f = a - A
I've been testing triangle numbers on rsa100.
If you calculate T-1(N) and take the difference of T(T-1(N)) and T(T-1(N)+1) you get a number that is extremely close to T-t on the order of same amount of digits (the difference in elements between na transform and solution in e,1)
Also if you recursively calculate a summation like so
T(1)+T(2)+T(4)+T(11)+T(68)+T(2407)+T(2899431)+T(4203353390611)+T(8834089863183560067072041)+T(39020571855401265512289573339484371018905006900193)+T(N)
and
T(1)+T(2)+T(4)+T(11)+T(68)+T(2407)+T(2899431)+T(4203353390611)+T(8834089863183560067072041)+T(39020571855401265512289573339484371018905006900193)+T(N-1)
where each preceding term from N and N-1 are the triangle inverses of the last term, then the difference between those sequences is N.
if you remove N and N-1 as terms and take the difference with N like so you also get a number that is extremely close to T-t on the order of same amount of digits
T(1)+T(2)+T(4)+T(11)+T(68)+T(2407)+T(2899431)+T(4203353390611)+T(8834089863183560067072041)+T(39020571855401265512289573339484371018905006900193) - N
-
N - T(1)+T(2)+T(4)+T(11)+T(68)+T(2407)+T(2899431)+T(4203353390611)+T(8834089863183560067072041)+T(39020571855401265512289573339484371018905006900192)
the two numbers close to T-t in question being
16822699634989797327123095165092932420211999031884
and
16822699634989797327123104264258038130140741377251
where T-t (if you can calculate this from c you win) =
18987613968471836961404436377722813927282768319099
Where T-1 is the inverse triangle function.
This is false. I made a mistake there, N for c6107 is way larger. N = 2976
However I have another observation to make up for that mistake in this triangle number analysis. x of (-f, 1, T) = T-1(N)
7461
{65:3645:86:85:1:7461} (65, 3645, 43) (e, N, T)
{65:1:3730:85:3645:3817} (65, 1, 43) (e, 1, T)
{-108:1:3558:84:3474:3644} (-108, 1, 43) (-f, 1, T)
∆-1(N) = 84
6107
{23:2976:78:77:1:6107} (23, 2976, 39) (e, N, T)
{23:1:3053:77:2976:3132} (23, 1, 39) (e, 1, T)
{-134:1:2897:76:2821:2975} (-134, 1, 39) (-f, 1, T)
∆-1(N) = 76
93801
{165:46595:306:305:1:93801} (165, 46595, 153) (e, N, T)
{165:1:46900:305:46595:47207} (165, 1, 153) (e, 1, T)
{-448:1:46288:304:45984:46594} (-448, 1, 153) (-f, 1, T)
∆-1(N) = 304
145
{1:61:12:11:1:145} (1, 61, 6) (e, N, T)
{1:1:72:11:61:85} (1, 1, 6) (e, 1, T)
{-24:1:48:10:38:60} (-24, 1, 6) (-f, 1, T)
∆-1(N) = 10
287
{31:128:16:15:1:287} (31, 128, 8) (e, N, T)
{31:1:143:15:128:160} (31, 1, 8) (e, 1, T)
{-2:1:111:14:97:127} (-2, 1, 8) (-f, 1, T)
∆-1(N) = 15
rsa100c
{61218444075812733697456051513875809617598014768503:761302513961266680267809189066318714859034057480651323757098846024549192056136964455981270339102876:39020571855401265512289573339484371018905006900194:39020571855401265512289573339484371018905006900193:1:1522605027922533360535618378132637429718068114961380688657908494580122963258952897654000350692006139} (61218444075812733697456051513875809617598014768503, 761302513961266680267809189066318714859034057480651323757098846024549192056136964455981270339102876, 19510285927700632756144786669742185509452503450097) (e, N, T)
{61218444075812733697456051513875809617598014768503:1:761302513961266680267809189066318714859034057480690344328954247290061481629476448827000175346003069:39020571855401265512289573339484371018905006900193:761302513961266680267809189066318714859034057480651323757098846024549192056136964455981270339102876:761302513961266680267809189066318714859034057480729364900809648555573771202815933198019080352903264} (61218444075812733697456051513875809617598014768503, 1, 19510285927700632756144786669742185509452503450097) (e, 1, T)
{-16822699634989797327123095165092932420211999031886:1:761302513961266680267809189066318714859034057480612303185243444759036902482797480084962365332202681:39020571855401265512289573339484371018905006900192:761302513961266680267809189066318714859034057480573282613388043493524612909457995713943460325302489:761302513961266680267809189066318714859034057480651323757098846024549192056136964455981270339102875} (-16822699634989797327123095165092932420211999031886, 1, 19510285927700632756144786669742185509452503450097) (-f, 1, T)
∆-1(N) = 39020571855401265512289573339484371018905006900193
Same observation holds except it is the x of the e,1 record.
T-1(N) is either x{-f,1,T} or x{e,1,T}
But there is an asymmetry here.
T-1(n) or T-1(n-1) is not equal (or even close to) x{-f, 1, t}, or x{e, 1, t} (solution records)
rsa110 analysis
{7251398426599644794623954759043454469676218891789649338:17897117089862934387495903916284227701501889012114113090783625819274031122082020851638527741916458186435175261:5982828275968304004100317854118230313685793843723609073:5982828275968304004100317854118230313685793843723609072:1:35794234179725868774991807832568455403003778024228226193532908190484670252364677411513516111204504060317568667} (7251398426599644794623954759043454469676218891789649338, 17897117089862934387495903916284227701501889012114113090783625819274031122082020851638527741916458186435175261, 2991414137984152002050158927059115156842896921861804537) (e, N, T)
{7251398426599644794623954759043454469676218891789649338:1:17897117089862934387495903916284227701501889012114113096766454095242335126182338705756758055602252030158784333:5982828275968304004100317854118230313685793843723609072:17897117089862934387495903916284227701501889012114113090783625819274031122082020851638527741916458186435175261:17897117089862934387495903916284227701501889012114113102749282371210639130282656559874988369288045873882393407} (7251398426599644794623954759043454469676218891789649338, 1, 2991414137984152002050158927059115156842896921861804537) (e, 1, T)
{-4714258125336963213576680949193006157695368795657568809:1:17897117089862934387495903916284227701501889012114113096766454095242335126182338705756758055602252030158784333:5982828275968304004100317854118230313685793843723609073:17897117089862934387495903916284227701501889012114113090783625819274031122082020851638527741916458186435175260:17897117089862934387495903916284227701501889012114113102749282371210639130282656559874988369288045873882393408} (-4714258125336963213576680949193006157695368795657568809, 1, 2991414137984152002050158927059115156842896921861804537) (-f, 1, T)
∆-1(N) = 5982828275968304004100317854118230313685793843723609072
∆-1(N-1) = 5982828275968304004100317854118230313685793843723609072
∆-1(n) = 56415892823691513151002588
∆-1(n-1) = 56415892823691513151002588
{7251398426599644794623954759043454469676218891789649338:1:9303852447694684408344911826978140118950296432398266895204888436423619628647716470410476187019755759685209:136410061562149325263764671139067929487183338122546740:9303852447694684408344911826978140118950296432398266758794826874274294364883045331342546699836417637138469:9303852447694684408344911826978140118950296432398267031614949998572944892412387609478405674203093882231951} (7251398426599644794623954759043454469676218891789649338, 1, 68205030781074662631882335569533964743591669061273371) (e, 1, t)
{-4714258125336963213576680949193006157695368795657568809:1:9303852447694684408344911826978140118950296432398261048786674030268940792094533491248091988409250158622877:136410061562149325263764671139067929487183338122546741:9303852447694684408344911826978140118950296432398260912376612468119615528329862352180162501225912036076136:9303852447694684408344911826978140118950296432398261185196735592418266055859204630316021475592588281169620} (-4714258125336963213576680949193006157695368795657568809, 1, 68205030781074662631882335569533964743591669061273371) (-f, 1, t)
{7251398426599644794623954759043454469676218891789649338:1591376481547123786477396152060299026390686203475593:5982828275968304004100317854118230313685793843723609073:136410061562149325263764671139067929487183338122546740:5846418214406154678836553182979162384198610505601062333:6122421090493547576937037317561418841225758554253106999} (7251398426599644794623954759043454469676218891789649338, 1591376481547123786477396152060299026390686203475593, 68205030781074662631882335569533964743591669061273371) (e, n, t)
T-1(N) = x{e, 1, T} or x{-f, 1, T} observation holds.
triangleSummation(T-1(N)) = ts
ts = T(1)+T(2)+T(4)+T(13)+T(100)+T(5078)+T(12897767)+T(83176210704686)+T(3459141013595226072509109016)+T(5982828275968304004100317854118230313685793843723609072)
ts = 17897117089862934387495903916284227701501889012114113096132169019926664730920520256395609944257205578415628169
ts N
ts - N = 5348543200652633608838499404757082202340747391980452908
triangleSummation(T-1(N) - 1) = ts2
ts2 = T(1)+T(2)+T(4)+T(13)+T(100)+T(5078)+T(12897767)+T(83176210704686)+T(3459141013595226072509109016)+T(5982828275968304004100317854118230313685793843723609071)
ts2 = 17897117089862934387495903916284227701501889012114113090149340743958360726820202402277379630571411734692019097
N ts2
N - ts2 = 634285075315670395261818449361148111345046451743156164
num1 = ts - N
num2 = N - ts2
(num1 - num2)/2 =
2991414137984152002050158927059115156842896921861804536
T-t =
2923209107203077339418276591489581192099305252800531166
Approximation of T-t reached from c.
In cases where our approximation is too far away to be an approximation, 2 other approximations can be calculated, which work for rsa129 and rsa120 (the approximations are close, but the search space is so massive that iteration is still computationally unfeasible, so the approximation needs to be improved or learned from to get closer.)
rsa120 analysis
{276798643817788533350132260296625874116995171400783089588379:113505240647718681667129980473746834447937668233042390019086152667335248378823675608062518130457849501431800035015938730:476456169332959066192086833057656566733872508443700132335510:476456169332959066192086833057656566733872508443700132335509:1:227010481295437363334259960947493668895875336466084780038173258247009162675779735389791151574049166747880487470296548479} (276798643817788533350132260296625874116995171400783089588379, 113505240647718681667129980473746834447937668233042390019086152667335248378823675608062518130457849501431800035015938730, 238228084666479533096043416528828283366936254221850066167755) (e, N, T)
{276798643817788533350132260296625874116995171400783089588379:1:113505240647718681667129980473746834447937668233042390019086629123504581337889867694895575787024583373940243735148274239:476456169332959066192086833057656566733872508443700132335509:113505240647718681667129980473746834447937668233042390019086152667335248378823675608062518130457849501431800035015938730:113505240647718681667129980473746834447937668233042390019087105579673914296956059781728633443591317246448687435280609750} (276798643817788533350132260296625874116995171400783089588379, 1, 238228084666479533096043416528828283366936254221850066167755) (e, 1, T)
{-676113694848129599034041405818687259350749845486617175082642:1:113505240647718681667129980473746834447937668233042390019085676211165915419757483521229460473891115628923356334883603219:476456169332959066192086833057656566733872508443700132335508:113505240647718681667129980473746834447937668233042390019085199754996582460691291434396402817324381756414912634751267711:113505240647718681667129980473746834447937668233042390019086152667335248378823675608062518130457849501431800035015938729} (-676113694848129599034041405818687259350749845486617175082642, 1, 238228084666479533096043416528828283366936254221850066167755) (-f, 1, T)
e%4 = 3
∆-1(N) = 476456169332959066192086833057656566733872508443700132335508
triangleSummation(T-1(N)) =
T(1)+T(2)+T(5)+T(16)+T(142)+T(10281)+T(52863280)+T(1397263246090468)+T(976172289437637198279784066271)+T(476456169332959066192086833057656566733872508443700132335508)
113505240647718681667129980473746834447937668233042390019086252496098005964090096585348899032777024250358969739065683157
triangleSummation(T-1(N) - 1) =
T(1)+T(2)+T(5)+T(16)+T(142)+T(10281)+T(52863280)+T(1397263246090468)+T(976172289437637198279784066271)+T(476456169332959066192086833057656566733872508443700132335507)
113505240647718681667129980473746834447937668233042390019085776039928673005023904498515841376210290377850526038933347649
T-t approximation = 238228084666479533096043416528828283366936254221850066167754, 60 digits
T-t approximation 2 = 138399321908894266675066130147925964192187327052146016423327, 60 digits
T-t approximation 3 = 476456169332959066192086833057656566733872508443700132335509, 60 digits
T-t = 163707277846749007875573151874570744031821201620085731703441, 60 digits
Closest approximation: approximation 2, 60 digits
Closest approximation difference: 25307955937854741200507021726644779839633874567939715280114, 59 digits
∆-1(n) = 260470505313768807462186889446
[1/2]
{276798643817788533350132260296625874116995171400783089588379:1:11106701298127191112847355240730405916226091301428275464843399758107464187140818760660545833177599981455232730223775381:149041613639461050440940529308515078670230105203528668928627:11106701298127191112847355240730405916226091301428275464843250716493824726090377820131237318098929751350029201554846754:11106701298127191112847355240730405916226091301428275464843548799721103648191259701189854348256270211560436258892704010} (276798643817788533350132260296625874116995171400783089588379, 1, 74520806819730525220470264654257539335115052601764334464314) (e, 1, t)
{-676113694848129599034041405818687259350749845486617175082642:1:11106701298127191112847355240730405916226091301428275464842774260324491767024185733298179661532195878841585501422511243:149041613639461050440940529308515078670230105203528668928626:11106701298127191112847355240730405916226091301428275464842625218710852305973744792768871146453525648736381972753582617:11106701298127191112847355240730405916226091301428275464842923301938131228074626673827488176610866108946789030091439871} (-676113694848129599034041405818687259350749845486617175082642, 1, 74520806819730525220470264654257539335115052601764334464314) (-f, 1, t)
{276798643817788533350132260296625874116995171400783089588379:33922442069205032282149019766764497978931624740053939704438:476456169332959066192086833057656566733872508443700132335510:149041613639461050440940529308515078670230105203528668928627:327414555693498015751146303749141488063642403240171463406883:693342667110830181197325401899700641361965863127336680673013} (276798643817788533350132260296625874116995171400783089588379, 33922442069205032282149019766764497978931624740053939704438, 74520806819730525220470264654257539335115052601764334464314) (e, n, t)
To brainstorm that missing piece in the approximation I would look at the closest value and its difference.
[2/2]
rsa 129
{10127537895024395905251173100883802246370188433498376141602081316:57190812878944433834617889988073306005109148360621181281280921457158533038780477389984354088670452233171543980194516501823035186:10694934584086471525314207693308900296322993593605128511616736585:10694934584086471525314207693308900296322993593605128511616736584:1:114381625757888867669235779976146612010218296721242362562561842935706935245733897830597123563958705058989075147599290026879543541} (10127537895024395905251173100883802246370188433498376141602081316, 57190812878944433834617889988073306005109148360621181281280921457158533038780477389984354088670452233171543980194516501823035186, 5347467292043235762657103846654450148161496796802564255808368293) (e, N, T)
{10127537895024395905251173100883802246370188433498376141602081316:1:57190812878944433834617889988073306005109148360621181281280921467853467622866948915298561781979352529494537573799645013439771770:10694934584086471525314207693308900296322993593605128511616736584:57190812878944433834617889988073306005109148360621181281280921457158533038780477389984354088670452233171543980194516501823035186:57190812878944433834617889988073306005109148360621181281280921478548402206953420440612769475288252825817531167404773525056508356} (10127537895024395905251173100883802246370188433498376141602081316, 1, 5347467292043235762657103846654450148161496796802564255808368293) (e, 1, T)
{-11262331273148547145377242285733998346275798753711880881631391855:1:57190812878944433834617889988073306005109148360621181281280921467853467622866948915298561781979352529494537573799645013439771770:10694934584086471525314207693308900296322993593605128511616736585:57190812878944433834617889988073306005109148360621181281280921457158533038780477389984354088670452233171543980194516501823035185:57190812878944433834617889988073306005109148360621181281280921478548402206953420440612769475288252825817531167404773525056508357} (-11262331273148547145377242285733998346275798753711880881631391855, 1, 5347467292043235762657103846654450148161496796802564255808368293) (-f, 1, T)
e%4 = 0
∆-1(N) = 10694934584086471525314207693308900296322993593605128511616736583
∆-1(N-1) = 10694934584086471525314207693308900296322993593605128511616736583
triangleSummation(T-1(N)) =
T(1)+T(2)+T(5)+T(19)+T(195)+T(19232)+T(184947519)+T(17102792626580353)+T(146252757813905659871375936577926)+T(10694934584086471525314207693308900296322993593605128511616736583)
57190812878944433834617889988073306005109148360621181281280921457442231383311515200015871384883089959790751131906158453282803705
triangleSummation(T-1(N) - 1) =
T(1)+T(2)+T(5)+T(19)+T(195)+T(19232)+T(184947519)+T(17102792626580353)+T(146252757813905659871375936577926)+T(10694934584086471525314207693308900296322993593605128511616736582)
57190812878944433834617889988073306005109148360621181281280921446747296799225043674701663691574189663467757538301029941666067122
T-t approximation = 5347467292043235762657103846654450148161496796802564255808368291, 64 digits
T-t approximation 2 = 5063768947512197952625586550441812421542289645090922304348599772, 64 digits
T-t approximation 3 = 10694934584086471525314207693308900296322993593605128511616736584, 65 digits
T-t = 1028759532095353253114134196990247228746418031676057044411877127, 64 digits
Closest approximation: 5063768947512197952625586550441812421542289645090922304348599772, 64 digits
Closest approximation difference: 4035009415416844699511452353451565192795871613414865259936722645, 64 digits
∆-1(n) = 190419700934841194543373870443698
∆-1(n-1) = 190419700934841194543373870443698
{10127537895024395905251173100883802246370188433498376141602081316:1:37302473431668114358131420783420175468441264382066034437953513370074778641178128081332420460685355563492329655197776241440137438:8637415519895765019085939299328405838830157530253014422792982330:37302473431668114358131420783420175468441264382066034437953513361437363121282363062246481161356949724662172124944761818647155108:37302473431668114358131420783420175468441264382066034437953513378712194161073893100418359760013761402322487185450790664233119770} (10127537895024395905251173100883802246370188433498376141602081316, 1, 4318707759947882509542969649664202919415078765126507211396491166) (e, 1, t)
{-11262331273148547145377242285733998346275798753711880881631391855:1:37302473431668114358131420783420175468441264382066034437953513368017259576987421575104152066704861105999493591845662152616383184:8637415519895765019085939299328405838830157530253014422792982331:37302473431668114358131420783420175468441264382066034437953513359379844057091656556018212767376455267169336061592647729823400853:37302473431668114358131420783420175468441264382066034437953513376654675096883186594190091366033266944829651122098676575409365517} (-11262331273148547145377242285733998346275798753711880881631391855, 1, 4318707759947882509542969649664202919415078765126507211396491166) (-f, 1, t)
{10127537895024395905251173100883802246370188433498376141602081316:18129831252057180249554918905369072823951862938527912049817621466:10694934584086471525314207693308900296322993593605128511616736582:8637415519895765019085939299328405838830157530253014422792982330:2057519064190706506228268393980494457492836063352114088823754252:55592012608096597043509984803375451783056877000913967034044961844} (10127537895024395905251173100883802246370188433498376141602081316, 18129831252057180249554918905369072823951862938527912049817621466, 4318707759947882509542969649664202919415078765126507211396491166) (e, n, t)
[2/2]
Approximation didn't work as well for rsa129. Still carving out a methodology here.
Elaborating on this, he gave me polynomial for even and odd e
For even e:
nd = 2t^2 + 2nt + (e/2)
For odd e:
nd = 2t^2 + 2(n-1)t - (n - ((e+1)/2))
And when graphed the points which are whole integers will be elements, each point being (t, n). So two elements exist for semiprime c in this curve.
Pics are the points for c6107.
A way to find the preceeding integer point on this curve would be a solution. The problem is known in mathematical circles as non-trivial but not computationally infeasible.
I'm still going to be working on the approximations if anyone has thoughts on those let me know. If we can calculate more accurate approximations we can get within the realm of solving these numbers (and maybe a solution like Chris described, since this approximation comes from a recursive triangle some of input T-1(N)).
Like Chris said there are different solutions just like a 4d object viewed in 3d space looks like different 3d objects. So do not be discouraged if it seems like things are going in different directions/different paths.
rsa2048 approximations
https://pastebin.com/kLbpDdge
c6107 is special because n is a triangle number.
6107
{23:2976:78:77:1:6107} (23, 2976, 39) (e, N, T)
{23:1:3053:77:2976:3132} (23, 1, 39) (e, 1, T)
{-134:1:2897:76:2821:2975} (-134, 1, 39) (-f, 1, T)
e%4 = 3
2d = 156
f = 134
∆-1(N) = 76
∆-1(N-1) = 76
triangleSummation(T-1(N)) =
T(1)+T(2)+T(4)+T(11)+T(76)
3006
triangleSummation(T-1(N) - 1) =
T(1)+T(2)+T(4)+T(11)+T(75)
2930
T-t approximation = 38, 2 digits
T-t approximation 2 = 8, 1 digits
T-t approximation 3 = 77, 2 digits
T-t = 15, 2 digits
Closest approximation: 8, 1 digits
Closest approximation difference: 7, 1 digits
∆-1(n) = 8
∆-1(n-1) = 7
{23:1:1163:47:1116:1212} (23, 1, 24) (e, 1, t)
{-134:1:1037:46:991:1085} (-134, 1, 24) (-f, 1, t)
{23:36:78:47:31:197} (23, 36, 24) (e, n, t)
Some experimental data for safekeeping.
c = rsa100
N = 761302513961266680267809189066318714859034057480651323757098846024549192056136964455981270339102876
∆-1(N) = 39020571855401265512289573339484371018905006900193
n = 14387588531011964456730684619177102985211280936
∆-1(n) = 169632476436630285119505
triangleSummation(T-1(N)) =
T(1)+T(2)+T(4)+T(11)+T(68)+T(2407)+T(2899431)+T(4203353390611)+T(8834089863183560067072041)+T(39020571855401265512289573339484371018905006900193)
triangleSummation(T-1(N) - 1) =
T(1)+T(3)+T(10)+T(67)+T(2406)+T(2899430)+T(4203353390610)+T(8834089863183560067072040)+T(39020571855401265512289573339484371018905006900192)
triangleSummation(T-1(N) - sqrt(d)) =
T(1)+T(2)+T(8)+T(63)+T(2372)+T(2898174)+T(4203351809684)+T(8834089863181060738192547)+T(39020571855401265512289567092839523150469841441251)
triangleSummation(T-1(N) - sqrt(2d)) =
T(1)+T(2)+T(8)+T(62)+T(2371)+T(2898118)+T(4203351666598)+T(8834089863180587847385815)+T(39020571855401265512289564505394507835344939828152)
∆1(∆-1(n)) = 14387588531011964456730676802388333031208282265
∆2(∆-2(n)) = 14387588530975214951661888938891904772729775106
∆3(∆-3(n)) = 14387555083370601538478896472570573635945074246
∆4(∆-4(n)) = 14273631718948863438208830586582694349507198856
∆5(∆-5(n)) = 9488836738160187798300677759237642920019716506
∆6(∆-6(n)) = 53327858858072239299138189326373798809103885
∆7(∆-7(n)) = 2076895351339769460477611370186681
c = rsa110
N = 17897117089862934387495903916284227701501889012114113090783625819274031122082020851638527741916458186435175261
∆-1(N) = 5982828275968304004100317854118230313685793843723609072
n = 1591376481547123786477396152060299026390686203475593
∆-1(n) = 56415892823691513151002588
triangleSummation(T-1(N)) =
T(1)+T(2)+T(4)+T(13)+T(100)+T(5078)+T(12897767)+T(83176210704686)+T(3459141013595226072509109016)+T(5982828275968304004100317854118230313685793843723609072)
triangleSummation(T-1(N) - 1) =
T(1)+T(3)+T(12)+T(99)+T(5077)+T(12897766)+T(83176210704685)+T(3459141013595226072509109015)+T(5982828275968304004100317854118230313685793843723609071)
triangleSummation(T-1(N) - sqrt(d)) =
T(1)+T(3)+T(11)+T(92)+T(5026)+T(12895116)+T(83176203672128)+T(3459141013595176615638324526)+T(5982828275968304004100317851672248245892102105984803468)
triangleSummation(T-1(N) - sqrt(2d)) =
T(1)+T(3)+T(11)+T(92)+T(5025)+T(12894997)+T(83176203035627)+T(3459141013595167258046486330)+T(5982828275968304004100317850659089300090567771214500056)
∆1(∆-1(n)) = 1591376481547123786477396143030577099882291366850166
∆2(∆-2(n)) = 1591376481546920421154689382900985625636153296142100
∆3(∆-3(n)) = 1591373857158754367809961369378063355314169683653515
∆4(∆-4(n)) = 1585747893594404099564413340729515765120866343772760
∆5(∆-5(n)) = 1338420359857405075273316351571726222383145912185046
∆6(∆-6(n)) = 9989452752508151923071579773982579506854290142986
∆7(∆-7(n)) = 30700002123226936025189367747945843590228731690
I created a new tree algorithm that uses T-1(input) as the d values and input - T(T-1(input)) as the e values.
Triangle Tree {
145 (c)
145 (c)
| 9 (e)
| | 3 (e)
| | | 2 (d)
| | | | 1 (e)
| | | | 1 (d)
| | 3 (d)
| | | 2 (d)
| | | | 1 (e)
| | | | 1 (d)
| 16 (d)
| | 1 (e)
| | 5 (d)
| | | 2 (e)
| | | | 1 (e)
| | | | 1 (d)
| | | 2 (d)
| | | | 1 (e)
| | | | 1 (d)
}
Remainder Tree {
145 (c)
| 1 (e)
| 12 (d)
| | 3 (e)
| | | 2 (e)
| | | | 1 (e)
| | | | 1 (d)
| | | 1 (d)
| | 3 (d)
| | | 2 (e)
| | | | 1 (e)
| | | | 1 (d)
| | | 1 (d)
}
N for 145 = 61
Triangle Tree(N) {
61 (c)
| 6 (e)
| | 3 (d)
| | | 2 (d)
| | | | 1 (e)
| | | | 1 (d)
| 10 (d)
| | 4 (d)
| | | 1 (e)
| | | 2 (d)
| | | | 1 (e)
| | | | 1 (d)
}
Remainder Tree(N) {
61 (c)
| 12 (e)
| | 3 (e)
| | | 2 (e)
| | | | 1 (e)
| | | | 1 (d)
| | | 1 (d)
| | 3 (d)
| | | 2 (e)
| | | | 1 (e)
| | | | 1 (d)
| | | 1 (d)
| 7 (d)
| | 3 (e)
| | | 2 (e)
| | | | 1 (e)
| | | | 1 (d)
| | | 1 (d)
| | 2 (d)
| | | 1 (e)
| | | 1 (d)
}
Take 16^2 + 9 from the 145 triangle tree
Triangle Tree(16^2+9) {
265 (c)
| 12 (e)
| | 2 (e)
| | | 1 (e)
| | | 1 (d)
| | 4 (d)
| | | 1 (e)
| | | 2 (d)
| | | | 1 (e)
| | | | 1 (d)
| 22 (d)
| | 1 (e)
| | 6 (d)
| | | 3 (d)
| | | | 2 (d)
| | | | | 1 (e)
| | | | | 1 (d)
}
22^2 + 12 = 31st triangle number = 496
Triangle Tree(T(31)) {
496 (c)
| 31 (d)
| | 3 (e)
| | | 2 (d)
| | | | 1 (e)
| | | | 1 (d)
| | 7 (d)
| | | 1 (e)
| | | 3 (d)
| | | | 2 (d)
| | | | | 1 (e)
| | | | | 1 (d)
}
The first 3 numbers are familiar for those who have poured over the patterns of 145