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Is the Collatz Conjecture solvable?
David Cole, Formulated a mathematical model for primes based on the Goldbach
Conjecture.
Updated April 23, 2018
“Great thinkers have always encountered opposition from mediocre minds.” – Albert
Einstein
.[1]
Yes!! The Collatz Conjecture is quite interesting and solvable! Here’s why:
Consider the following proof of the Collatz Conjecture:
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:
The value, l_m =1, which implies division by two in the Collatz process will occur as often
as all other values, l_m > 1, which imply division either by four or by eight or by sixteen
or by thirty-two, … in the Collatz process according to the our probability calculations
and according to the Law of Large Numbers. Thus, we expect the division by two will
occur 50% of all possible divisions in the Collatz processing. 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 link:
Proof of Collatz Conjecture
.
Reference links: The 3x+ 1 Problem: An Overview
A Case Study of the Collatz Conjecture
;
.
*****
Mathematica 11 software function, f[], for simulation of data:
f[m_] := (
n = m;
Print[n];
prob = 1.;
While[n > 1, n = 3*n + 1;
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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=21346294170249129484267610567431365440581190958189557170758121139151
5390715789799022068952861978573601713877607708447514178586071267992312618
3263337224705889777568364923077634920833936889569371693845,
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%};
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Example 5:
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{n_0=51270986306354174415767650068782014060235196685228248206814417094197
7788499359079526589961009312870519037650111780867282244255976210916215479
5813451432257192935153820430354322926445793164263260432795930904729611785
2464116451552006938140911375159712208272450936235369012665425381688232977
3787019813374226655258178640629301324445245691260252991300274897690542572
1961168895250648529391265350671566297565983934897155998449353970226052506
7982336871166695243225954098260027371684782477986212512395537896476946053
4655183176990277138747565943503637469154282367557173000697411352547275127
041561294264392881037,
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:
{140632776245125940579955061295420304224335564241653342042062534292543873
7204009156598316161837868033625097692997755411598865674240723653780467274
2504346557624282178821627315905442498139231323753952791100356240353660616
7626428101307928207420696623679972454189249680086839622880004422237067288
4114602539221451613965103942130078956037214634287307961934299935536703891
3444753295595082337346237317709513828325820988051111078494941251932744193
8858817001391990343275265724180159458037564382639327656685339519800776029
3356748181656017016385303434104768138669551398789451716214418038797358811
0498638039833406945078504748652525286081427305638501766334256736229375871
2355626888648865202554555045680130195591369398482494392675971941001259654
0951482501379536530986079617422073273365156478090003328931030996278566164
9075722639143074471812229471550176979175887894029081602985328973329913929
5871255768953024559747256965074435156949271412645400791450797572891816928
8923125810989061421286135075678423238809998514907360173268373912957648911
5827533586287256506422857231047361445287462657152888404904343879264315791
3120742327499451018554718382329532651014310583830233379165659212295989150
8462805284371086426129284308767222694321354597000202120938472169382526057
2727761569575125985914732952270346475261791580440807960628067000989493125
9208386988495054075645003031988594512104837377478964596505764616218109599
6854818696682703958625899119711101464705241846416167522570556484503124967
3700776231104043246339586899159060915793189246120422686194171449795682621
2252097740262801701769222808488386035039737777825916626059163021079774776
8012170076245010675903837921982931739327741566841539391464929784221781874
1199000229948429647749192964098635391191333310149840008178019189763384818
9769673274175747209368920982080385759110035263620843502874773864652061909
7226811087286229491638080275284778648502397587582796143043321959511749393
6908404122634282699383907516940648663656336210503231496938770261232735159
529795490355396958247378256473,
prob =7.060409409804900*10^-3120,
cnt = number of generated odd integers of Collatz Sequence =16257,
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.
P.S. There are no exceptions (counterexamples or cycles) that disprove the Collatz
Conjectue. And I challenge anyone to find one. Otherwise, it is an act of pure fiction to
suggest such things without proof.
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Footnotes
[1] David Cole's answer to Are experts always right?
2.9k views · View 3 Upvoters
View 5 Other Answers to this Question
About the Author
David Cole
"What is done is finished. And what is finished is done!"
Independent Research Scientist at ResearchGate.net
Studied Mathematics (college major) & French (college minor)
Lives in Detroit, MI
97.5k content views 2k this month
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