Timeline for Convergence of rivers of numbers
Current License: CC BY-SA 4.0
19 events
| when toggle format | what | by | license | comment | |
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| Aug 10, 2022 at 21:55 | history | edited | butter-imbiber | CC BY-SA 4.0 |
typo: meant "rightmost" not "leftmost"
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| Mar 8, 2021 at 12:55 | comment | added | Henry | @LSpice River 9 is OEIS A016096 - your search had spurious 3,6,108 | |
| Mar 8, 2021 at 3:56 | review | Close votes | |||
| Mar 9, 2021 at 11:32 | |||||
| Mar 7, 2021 at 17:58 | comment | added | butter-imbiber | @JoshuaZ Apologies, there is a typo in my first reply to you that I cannot now correct. I should of course have defined equivalence by saying "$2^u x \equiv 2^v y \pmod{b-1}$ for some $u,v \geq 0$", rather than reusing $b$. Facepalm! | |
| Mar 7, 2021 at 17:56 | comment | added | butter-imbiber | A small but probably worthwhile point, I think, is I'd probably rather frame the general claim for (integer) base $b>1$ as "$\operatorname{river}{x}$ meets $\operatorname{river}{y}$ iff $2^u x \equiv 2^v y \pmod{b-1}$ for some $u,v$" and say that meeting forms an equivalence relation with finitely many classes, as discussed in my previous comment, rather than necessarily focussing on the representative rivers with "smallest" first entry. | |
| Mar 7, 2021 at 17:51 | comment | added | butter-imbiber | @JoshuaZ For general (integer) $b>1$, let's say $x,y\in\{1,2,\dots,b-1\}$ are equivalent if $2^a x \equiv 2^b y \pmod{b-1}$ for some $a,b\geq 0$. Take as representative of each resulting equivalence classes the smallest member. I conjecture the main digital rivers in base $b$ are those given by those representatives. These are $k \in \{1,3,9\}$ for $b=10$; $k\in\{1\}$ for $b=9$; $k\in\{1,5\}$ for $b=11$; $k\in\{1,3,7\}$ for $b=15$, for example. (Some calculations I've run support my conjecture.) | |
| Mar 7, 2021 at 15:00 | comment | added | JoshuaZ | The base b generalization here is obvious. Related question that I don't see an immediate proof of : Given a base $b>1$ there is a finite list of rivers where any other river eventually meets them. In fact, I don't even see immediately how to prove there even exists such a $b$. | |
| Mar 7, 2021 at 12:11 | comment | added | butter-imbiber | @GerryMyerson I've picked through those OEIS entries and their linked pages as best I could over a protracted Sunday tea break. I can find the claim conjectured in several places. I can find an estimated value for the $m$th term of an arbitrary binary digital river. I can find many case studies demonstrating the existence and abundance of self-numbers (those numbers not of the form $f(n)$ for any $n$). But no breakthroughs yet </3 | |
| Mar 7, 2021 at 12:00 | history | edited | butter-imbiber | CC BY-SA 4.0 |
Forgot coefficient of $\operatorname{ls} n$ equivalent formula for $\operatorname{tr} n$
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| Mar 7, 2021 at 10:30 | comment | added | Gerry Myerson | There are many references given at the links found by @LSpice, perhaps the question is answered at one of them. | |
| Mar 7, 2021 at 10:24 | comment | added | butter-imbiber | @LSpice No it was a mistake haha, thanks for spotting! I've edited the post to clarify that I only want remainders in {1, 2, ..., 9}. (I've also corrected the second formulation of $ \operatorname{tr}$.) | |
| Mar 7, 2021 at 10:21 | history | edited | butter-imbiber | CC BY-SA 4.0 |
Addressed typos (0 and 9 are congruent modulo 9)
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| Mar 6, 2021 at 22:53 | comment | added | LSpice | Related to self numbers. | |
| Mar 6, 2021 at 22:43 | comment | added | Henry | The slightly surprising thing is that river 1 and river 5 (OEIS A007618) do not meet until $620$ despite the small digitsum additions | |
| Mar 6, 2021 at 22:03 | history | edited | LSpice | CC BY-SA 4.0 |
Proofreading
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| Mar 6, 2021 at 21:56 | comment | added | LSpice | Is it intentional that you include both 0 and 9 among the results of reduction modulo 9? | |
| Mar 6, 2021 at 21:48 | comment | added | LSpice | River 1 and river 3; OEIS does not know river 9, although it matches A008591 (multiples of 9) for a while, and, as you remark, is a subsequence of it, for obvious reasons. | |
| Mar 6, 2021 at 21:27 | review | First posts | |||
| Mar 6, 2021 at 21:47 | |||||
| Mar 6, 2021 at 21:25 | history | asked | butter-imbiber | CC BY-SA 4.0 |