The Collatz function as an automorphic Cayley colour graph:decidability of $an+b$ conjectures, proof of the $3n + 1$ conjecture
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2020
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866911854783102976 |
|---|---|
| author | Kleinnijenhuis, Jan Kleinnijenhuis, Alissa M. Aydogan, Mustafa G. |
| author_facet | Kleinnijenhuis, Jan Kleinnijenhuis, Alissa M. Aydogan, Mustafa G. |
| contents | The Collatz conjecture states that repeated steps of $n\mathrm{\to }\mathrm{3}n\mathrm{+1}$ at odd numbers and $n\mathrm{\to }n\mathrm{/2}$ at even numbers amount to walks over root paths to the branching number $c=4$ in the `trivial' cyclic root $4\to 2\to 1\to 4\to \dots $ of one connected Collatz graph. The Collatz graph with reverse arrows $n \to 2n$ and $n \to (n-1)/3$ can be transformed to a 3-regular automorphic Cayley color graph $T_{\ge 0}$ with as nodes the branching numbers with a remainder of $4$ or $16$ when divided by $18$, building the congruence classes $[4,16]_{18}$. Labeling the $2^k$ breadth-first ordered root paths with $2^k$ binary numbers on the binary number line, for $k=1,2,3,\dots$, and pairing them with the $2^k$ output numbers of these root paths, gives $2^k$ paired numbers. The 3-regular Cayley graph of these paired branching numbers can be transformed to a 4-regular Middle Pages graph. This 4-regular graph offers to all paired branching numbers from the congruence classes $[4,16]_{18}$ a unique Eulerian tour to and from the trivial root number pair {0,c=4}. This proves Collatz's $3n+1$ conjecture. Whether a specific $an+b$ conjecture offers a Eulerian tour to all its paired branching numbers can be decided by whether it offers such a tour to paired branching numbers lower than $2a^3$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2008_13643 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Collatz function as an automorphic Cayley colour graph:decidability of $an+b$ conjectures, proof of the $3n + 1$ conjecture Kleinnijenhuis, Jan Kleinnijenhuis, Alissa M. Aydogan, Mustafa G. General Mathematics 05C05, 05C20, 05C60, 05C76, 11B50, 11F03 G.2.2 The Collatz conjecture states that repeated steps of $n\mathrm{\to }\mathrm{3}n\mathrm{+1}$ at odd numbers and $n\mathrm{\to }n\mathrm{/2}$ at even numbers amount to walks over root paths to the branching number $c=4$ in the `trivial' cyclic root $4\to 2\to 1\to 4\to \dots $ of one connected Collatz graph. The Collatz graph with reverse arrows $n \to 2n$ and $n \to (n-1)/3$ can be transformed to a 3-regular automorphic Cayley color graph $T_{\ge 0}$ with as nodes the branching numbers with a remainder of $4$ or $16$ when divided by $18$, building the congruence classes $[4,16]_{18}$. Labeling the $2^k$ breadth-first ordered root paths with $2^k$ binary numbers on the binary number line, for $k=1,2,3,\dots$, and pairing them with the $2^k$ output numbers of these root paths, gives $2^k$ paired numbers. The 3-regular Cayley graph of these paired branching numbers can be transformed to a 4-regular Middle Pages graph. This 4-regular graph offers to all paired branching numbers from the congruence classes $[4,16]_{18}$ a unique Eulerian tour to and from the trivial root number pair {0,c=4}. This proves Collatz's $3n+1$ conjecture. Whether a specific $an+b$ conjecture offers a Eulerian tour to all its paired branching numbers can be decided by whether it offers such a tour to paired branching numbers lower than $2a^3$. |
| title | The Collatz function as an automorphic Cayley colour graph:decidability of $an+b$ conjectures, proof of the $3n + 1$ conjecture |
| topic | General Mathematics 05C05, 05C20, 05C60, 05C76, 11B50, 11F03 G.2.2 |
| url | https://arxiv.org/abs/2008.13643 |