That fact explains any growth in the Collatz sequence, and it also explains why the Collatz sequence will always converge to one. A Proof of Collatz Conjecture ...
Is the Collatz Conjecture solvable? David Cole, +10 years experience in applied and pure mathematics Updated Oct 8
Yes! Here’s why: Probability (Collatz sequence does not converge to one) is
∴ Probability (Collatz sequence does not converge to one) is zero. Notes:
We conclude the Collatz Conjecture is true. An Important Note: Hmm. The value, l_m =1, will occur as often as the values, l_m > 1, according to the Law of Large Numbers. That fact explains any growth in the Collatz sequence, and it also explains why the Collatz sequence will always converge to one. A Proof of Collatz Conjecture links: Proof of Collatz Conjecture 1; Proof of Collatz Conjecture 2. ***** Mathematica 11 software function, f[], for simulation of data: f[m_] := ( n = m; Print[n]; prob = 1.; While[n > 1, n = 3*n + 1; x = n; i = 0;cnt = 0; icnt = 0; While[ EvenQ[x], ++i; x = x/2]; s = 1 2^(i); cnt = cnt + 1;
If[i == 1, icnt = icnt + 1]; Print[x]; prob = prob * s; n = x; ]; Return[{m, prob, cnt, icnt, 100. * icnt/cnt}] ); Example 1: f[n_0 = 1367] yielded a probability (prob) equal to 0.0245548 before the Collatz sequence of odd integers, {1367, 2051, 3077, 577, 433, 325, 61, 23, 35, 53, 5, 1}, converged to one; Example 2: {n_0 =1917424378915227003508542412844973284190406292736261788098013, prob = 2.14803*10^104, cnt = number of generated odd integers of Collatz Sequence =541, icnt =number of times l_m or i equals one = 269, icnt/cnt = 49.7227%}; Example 3: {n_0 =1134322358892035783221922112828664036984286481192427929865901, prob =1.30642*10^113, cnt = number of generated odd integers of Collatz Sequence =573, icnt =number of times l_m or i equals one = 301, icnt/cnt = 52.5305%}; Example 4: {n_0=213462941702491294842676105674313654405811909 58189557170758121139151539071578979902206895286197 85736017138776077084475141785860712679923126183263 337224705889777568364923077634920833936889569371693845, prob =5.118883814495941*10^344, cnt = number of generated odd integers of Collatz Sequence =1772, icnt =number of times l_m or i equals one = 902, icnt/cnt = 50.9029%}; Example 5: {n_0=512709863063541744157676500687820140602351966 85228248206814417094197778849935907952658996100931 28705190376501117808672822442559762109162154795813 45143225719293515382043035432292644579316426326043 27959309047296117852464116451552006938140911375159 71220827245093623536901266542538168823297737870198 13374226655258178640629301324445245691260252991300 27489769054257219611688952506485293912653506715662 97565983934897155998449353970226052506798233687116 66952432259540982600273716847824779862125123955378
96476946053465518317699027713874756594350363746915 4282367557173000697411352547275127041561294264392881037, prob =8.737445781552508*10^906, cnt = number of generated odd integers of Collatz Sequence =4745, icnt =number of times l_m or i equals one = 2361, icnt/cnt =49.7576%}; Example 6: {1406327762451259405799550612954203042243355642416 53342042062534292543873720400915659831616183786803 36250976929977554115988656742407236537804672742504 34655762428217882162731590544249813923132375395279 11003562403536606167626428101307928207420696623679 97245418924968008683962288000442223706728841146025 39221451613965103942130078956037214634287307961934 29993553670389134447532955950823373462373177095138 28325820988051111078494941251932744193885881700139 19903432752657241801594580375643826393276566853395 19800776029335674818165601701638530343410476813866 95513987894517162144180387973588110498638039833406 94507850474865252528608142730563850176633425673622 93758712355626888648865202554555045680130195591369 39848249439267597194100125965409514825013795365309 86079617422073273365156478090003328931030996278566 16490757226391430744718122294715501769791758878940 29081602985328973329913929587125576895302455974725 69650744351569492714126454007914507975728918169288 92312581098906142128613507567842323880999851490736 01732683739129576489115827533586287256506422857231 04736144528746265715288840490434387926431579131207 42327499451018554718382329532651014310583830233379 16565921229598915084628052843710864261292843087672 22694321354597000202120938472169382526057272776156 95751259859147329522703464752617915804408079606280 67000989493125920838698849505407564500303198859451 21048373774789645965057646162181095996854818696682 70395862589911971110146470524184641616752257055648 45031249673700776231104043246339586899159060915793 18924612042268619417144979568262122520977402628017 01769222808488386035039737777825916626059163021079 77477680121700762450106759038379219829317393277415 66841539391464929784221781874119900022994842964774 91929640986353911913333101498400081780191897633848 18976967327417574720936892098208038575911003526362 08435028747738646520619097226811087286229491638080 27528477864850239758758279614304332195951174939369 08404122634282699383907516940648663656336210503231 496938770261232735159529795490355396958247378256473, prob =7.060409409804900*10^3120, Ask Question Ask or Search Quora Read cnt = number of generated odd integers of Collatz Sequence =16257,
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icnt =number of times l_m or i equals one = 8155, icnt/cnt =50.163%}. We expected icnt/cnt to be 50% according to the Law of Large Numbers when l_m = 1. The six examples are typical empirical affirmations of the theory proving the Collatz Conjecture. 545 Views · View Upvotes Upvoters
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