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Generating the Prouhet-Thue-Morse sequence in brainfuck

2019-12-28, post № 223

brainfuck, mathematics, programming, #recursive, #Thue-Morse sequence

[-]<<<[-]<<<[-]<<<[-]<<<[-]<<<[-]>>>>>>>>>>>>>>>>>[-]++++++++++++++++++++++++++++++++++++++++++++++++.[-]<[-]>>[-]<[-<+>>+<]>[-<+>]<<<<<[-]>>>[<<<+>>>-]<<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<[<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<]<[-]<[>+<-]>>+<<<<<<<<<[-][-]>>>>>>>>>[>>>>>>>>>]>>[-]+[>[-]+[<<[-]<<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[-]>>>>>>>[-]<<<<<<<<[>>>>>>>>+<<<<<<<<-]>>>>>>>>>>[>>>>>>>[-]<<<<<<<<<[>>>>>>>>>+<<<<<<<<<-]>>>>>>>>>>>]<<[>>>+<<<-]>>>>>>[-]<<<[->>>+<<<][-]>>>>[-]<[-<<<+>>>>+<]>[-<+>]<<<<<<<<<<[-]>>>>>>[<<<<<<+>>>>>>-]<<<<<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<[<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<]<[-]<[>+<-]>>+<<<<<<<<<[-][-]>>>>>>>>>[>>>>>>>>>]>>>>>>>>[-]+<[->[-]<]>[-<+>]<<<<[-]>>>>[-]<[-<<<+>>>>+<]>[-<+>]<<<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<[<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<]<[-]<[>+<-]>>+<<<<<<<<<[-][-]>>>>>>>>>[>>>>>>>>>]>>>>>>>[-]<<<<<<<<<<<<[-]>>[<<+>>>>>>>>>>>>+<<<<<<<<<<-]<<[>>+<<-]>>>>>>>>>>>>>>[-]<<[->>+<<]>>][-]+[<<[-]<<<<<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[-]>>>>>>>[-]<<<<<<<<[>>>>>>>>+<<<<<<<<-]>>>>>>>>>>[>>>>>>>[-]<<<<<<<<<[>>>>>>>>>+<<<<<<<<<-]>>>>>>>>>>>]<<[>>>>>>+<<<<<<-]>>>>>>>>>[-]<<<[->>>+<<<][-]>>>>[-]<[-<<<+>>>>+<]>[-<+>]<<<<<<<[-]>>>[<<<+>>>-]<<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<[<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<]<[-]<[>+<-]>>+<<<<<<<<<[-][-]>>>>>>>>>[>>>>>>>>>]>[-]<<<<<<<<<<<<<<<[-]>>[<<+>>>>>>>>>>>>>>>+<<<<<<<<<<<<<-]<<[>>+<<-]>>>>>>>>>>>>>>>>>[-]<<[->>+<<]>>][-]+[<<[-]<<<<<<<<<<<<<<<<[<<<<<<<<<]>>>>>>>>>[-]>>>>>>>[-]<<<<<<<<[>>>>>>>>+<<<<<<<<-]>>>>>>>>>>[>>>>>>>[-]<<<<<<<<<[>>>>>>>>>+<<<<<<<<<-]>>>>>>>>>>>]<<[>>>>>>>>>+<<<<<<<<<-]>>>>>>>>>>>>[-]<<<[->>>+<<<]>>>>[-]++++++++++++++++++++++++++++++++++++++++++++++++>[-]<<[->+>+<<]>>[-<<+>>]<.<<<<[-]>>>>[-]<[-<<<+>>>>+<]>[-<+>]<<<<<<<[-]>>>[<<<+>>>-]<<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<[<<<<<<<<<<<[-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<<<]<[-]<[>+<-]>>+<<<<<<<<<[-][-]>>>>>>>>>[>>>>>>>>>]>[-]<<<<<<<<<<<<<<<<<<[-]>>[<<+>>>>>>>>>>>>>>>>>>+<<<<<<<<<<<<<<<<-]<<[>>+<<-]>>>>>>>>>>>>>>>>>>>>[-]<<[->>+<<]>>]<]

Try it online.

Factoids #1

2019-11-30, post № 222

mathematics, #F2, #OEIS

IV) Commutative, non-associative operations

For any natural number 𝑛, let \mathrm{Op}_n:=\left\{\star:\mathbb{Z}^2_n\to\mathbb{Z}_n\right\} denote the set of all operations on a set of that order. An operation shall be called commutative iff \mathrm{commut}(\star):\Leftrightarrow\forall\,x,y\in\mathbb{Z}_n:x\star y=y\star x and be called associative iff \mathrm{assoc}(\star):\Leftrightarrow\forall\,x,y,z\in\mathbb{Z}_n:x\star(y\star z)=(x\star y)\star z holds.

With the above defined, one may study \mathrm{CnA}_n:=\{\star\in\mathrm{Op}_n:\mathrm{commut}(\star)\land\lnot\mathrm{assoc}(\star)\}. For 𝑛 = 𝟤, this set is nonempty for the first time, containing a manageable two elements, by name

\mathrm{CnA}_2=\Big\{\mathrm{nor}:(x,y)\mapsto 1+xy,\quad\mathrm{nand}:(x,y)\mapsto (1+x)\cdot(1+y)\Big\}.

However, based on the superexponential nature of \#\mathrm{Op}_n=\#\mathbb{Z}_n^{\mathbb{Z}_n^2}=\#\mathbb{Z}_n^{{\#\mathbb{Z}_n}^2}=n^{n^2}, the sequence \mathrm{A079195}_n:=\#\mathrm{CnA}_n likely also grows rather quickly, OEIS only listing four members;

\mathrm{A079195}=(0,2,666,1\,047\,436,\dots).

Based on this limited numerical evidence, I would suspect the commutative yet non-associative operations to be rather sparse, i. e.

\lim\limits_{n\to\infty}\mathrm{A079195}_n\cdot\left(\#\mathrm{Op_n}\right)^{-1}=0\mod\square.

Analysis source: factoids-1_operations.hs

(Non-)commutative and (non-)associative operations have also been studied nearly twenty years ago by Christian van den Bosch, author of OEIS sequence A079195. Unfortunately, their site appears to be down, which is where they hosted closed binary operations on small sets (resource found on web.archive.org).

V) Arbitrary polynomial extremum difference

Let 𝜀 > 𝟢 be an arbitrary distance, define g:=-4x^4-x^3+8x^2+3x-4. Then f:=\sfrac{\epsilon}{4}\cdot g has two local maxima at - 𝟣 and 𝟣, whose vertical distance is 𝜀.

VI) Digit sum roots

It holds that \mathrm{ds}_{10}(108^{12})=108.

Extending A056154

2019-11-02, post № 221

mathematics, #discovery, #OEIS, #paper, #ternary

Five weeks of work including over six days of dedicated number crunching come to fruition as the thirteenth member of OEIS sequence A056154 is published,

\mathrm{A056154}(13) = 49\,094\,174.

Sequence A056154 is defined as binary exponents which have a ternary representation invariant under endomorphic addition modulo permutation, more formally

\begin{aligned}
a\in\mathrm{A056154}\,:\Longleftrightarrow\,
&a,\log_2(a)\in\mathbb{N}\,\land\,\exists\,\sigma\in\mathrm{Sym}(\{0,\dots,\lfloor\log_3(a+a)\rfloor\}):\\
&\forall\,j\in\mathrm{dom}\,\sigma:\Big\lfloor (a+a)\cdot 3^{-j}\Big\rfloor\equiv\Big\lfloor a\cdot 3^{-\sigma(j)}\Big\rfloor\mod 3.
\end{aligned}

Due to the exponentially defined property, testing a given p\in\mathbb{N} for membership quickly becomes non-trivial, as the trits of 2^p enter the billions.
As an example, 2^{49\,094\,174} requires 30’974’976 trits. Assuming three thousand trits per page and two hundred pages per book, a ternary print-out of said number would require fifty-two books, filling a few book shelves.

For a discussion of the methodology I used to perform the search which lead to the discovery of \mathrm{A056154}(13), I refer to my paper Extending A056154.

A325902

2019-10-05, post № 220

mathematics, #factorization, #integer, #OEIS, #sequence

Fifty is a peculiar integer.
When looking at its neighbors — the largest integer strictly beneath and the smallest strictly above —, more specifically their prime factorization, one finds

49=\underbrace{7^2<50<3\cdot 17}_{7+7+3=17}=51.

Notably, there exists a partition of the neighbor’s factors into two multisets such that both parts’ sums equal another.

Positive integers with the above described property can be found in my most recent addition to the OEIS: sequence A325902.

Digit Sums

2019-09-07, post № 219

mathematics, #base ten, #factoid, #Jack Reacher

Interessant war es auch, drei aufeinanderfolgende Zahlen zu nehmen, von denen die größte durch drei teilbar sein musste, sie zu addieren und aus dem Ergebnis so lange die Quersumme zu bilden, bis eine einstellige Zahl übrig blieb. Diese Zahl war immer sechs.
— Child, Lee: Der Anhalter. München: Blanvalet, 2015; p. 73

Jack Reacher’s at most tangentially to interpreting the sergeant’s reply related base ten factoid’s formal form is

\forall n\in\mathbb{N}^+:\mathrm{fds}_{10}\left(\sum\limits_{j=0}^2 3\,n-j\right)=6,

where \mathrm{fds}_{10} represents the final digit sum in base ten.

A proof of the above claim together with the underlying digit sum results is presented in digit-sums.pdf [1] (source: digit-sums.tex).

Short brainfuck Primes

2019-08-10, post № 218

brainfuck, programming,

>[-]+>>[-]++>[-]<<<[>[-]+>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-][-]+[>>>>[-]<[-]<<<<<[>>>>>>+<+<<<<<-]>>>>>>[<<<<<<+>>>>>>-]<<[-]+>[<->[-]]<[<<<->>>[-]]<<<<->>>>>[-]<[-]<<<<<<[>>>>>>>+<+<<<<<<-]>>>>>>>[<<<<<<<+>>>>>>>-]>[-]<[-]<<<<<<[>>>>>>>+<+<<<<<<-]>>>>>>>[<<<<<<<+>>>>>>>-][-]>[-]>[-]>>[-]<[-]<[>>+<+<-]>>[<<+>>-]>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]>[-]<[-]<<<<<<<<<<<[>>>>>>>>>>>>+<+<<<<<<<<<<<-]>>>>>>>>>>>>[<<<<<<<<<<<<+>>>>>>>>>>>>-]>[-]<[-]<<<<<<<<[>>>>>>>>>+<+<<<<<<<<-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<[-]>>>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<<<[-]+>>>>>[-]]<-<-]>[-]<[-]<<<<<<<<<<<[>>>>>>>>>>>>+<+<<<<<<<<<<<-]>>>>>>>>>>>>[<<<<<<<<<<<<+>>>>>>>>>>>>-]>[-]<[-]<<<<<<<<[>>>>>>>>>+<+<<<<<<<<-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<[-]+>>[>-<-]>>>[-]<[-]<[>>+<+<-]>>[<<+>>-]<[<<<<->>>>[-]]<[-]<[-]<<<<<<<<<<<<[>>>>>>>>>>>>>+<+<<<<<<<<<<<<-]>>>>>>>>>>>>>[<<<<<<<<<<<<<+>>>>>>>>>>>>>-]>[-]<[-]<<<<<<<<<[>>>>>>>>>>+<+<<<<<<<<<-]>>>>>>>>>>[<<<<<<<<<<+>>>>>>>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-][-]+<[<<<<+>>>>>-<[-]]>[>>[-]<[-]<<<<[>>>>>+<+<<<<-]>>>>>[<<<<<+>>>>>-]<[<<<[<<<+>>>-]>>>[-]]<-]<<<<[-]+<[>-<[-]]>[<+>-]<[>>[-]<[-]<<<<<<<<<[>>>>>>>>>>+<+<<<<<<<<<-]>>>>>>>>>>[<<<<<<<<<<+>>>>>>>>>>-]<[>>>[-]<[-]<<<<<<<[>>>>>>>>+<+<<<<<<<-]>>>>>>>>[<<<<<<<<+>>>>>>>>-]<<[-]+>[<->[-]]<[<<<<<->>>>>[-]]<<<<<<->>>>>-]>[-]<[-]<<<<<<<<[>>>>>>>>>+<+<<<<<<<<-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<[<<<<->>>>-]<<<+>>>>>[-]<[-]<<<<[>>>>>+<+<<<<-]>>>>>[<<<<<+>>>>>-]<<[-]+>[<->[-]]<[<<+>>[-]]<[-]>>[-]<[-]<[>>+<+<-]>>[<<+>>-]>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]>[-]<[-]<<<<<<<<<<<[>>>>>>>>>>>>+<+<<<<<<<<<<<-]>>>>>>>>>>>>[<<<<<<<<<<<<+>>>>>>>>>>>>-]>[-]<[-]<<<<<<<<[>>>>>>>>>+<+<<<<<<<<-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<<[-]>>>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<<<[-]+>>>>>[-]]<-<-]>[-]<[-]<<<<<<<<<<<[>>>>>>>>>>>>+<+<<<<<<<<<<<-]>>>>>>>>>>>>[<<<<<<<<<<<<+>>>>>>>>>>>>-]>[-]<[-]<<<<<<<<[>>>>>>>>>+<+<<<<<<<<-]>>>>>>>>>[<<<<<<<<<+>>>>>>>>>-]<<<<[-]+>>[>-<-]>>>[-]<[-]<[>>+<+<-]>>[<<+>>-]<[<<<<->>>>[-]]<[-]<[-]<<<<<<<<<<<<[>>>>>>>>>>>>>+<+<<<<<<<<<<<<-]>>>>>>>>>>>>>[<<<<<<<<<<<<<+>>>>>>>>>>>>>-]>[-]<[-]<<<<<<<<<[>>>>>>>>>>+<+<<<<<<<<<-]>>>>>>>>>>[<<<<<<<<<<+>>>>>>>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-][-]+<[<<<<+>>>>>-<[-]]>[>>[-]<[-]<<<<[>>>>>+<+<<<<-]>>>>>[<<<<<+>>>>>-]<[<<<[<<<+>>>-]>>>[-]]<-]<<<<[-]+<[>-<[-]]>[<+>-]<]<<<<<[-]+>>>>>>[-]<[-]<<<<[>>>>>+<+<<<<-]>>>>>[<<<<<+>>>>>-]<[<<<<<[-]>>>>>[-]]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<[<<<<<[-]>>>>>[-]]<<<<<<[-]+>>>>[-]<[-]<<<<<[>>>>>>+<+<<<<<-]>>>>>>[<<<<<<+>>>>>>-]<<[-]+>[<->[-]]<[>>>[-]<[-]<<<<<[>>>>>>+<+<<<<<-]>>>>>>[<<<<<<+>>>>>>-]<<[-]+>[<->[-]]<[<<<[-]>>>[-]]<[-]]>[-]<[-]<[>>+<+<-]>>[<<+>>-]<[<<->>[-]]<<]<<->>[-]+<<[<<<->>>>>-<<[-]]>>[<[<<<<->>>>[-]]>-]<[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<[>>>[-]++++++++++++++++>>[-]<[-]<<<<<[>>>>>>+<+<<<<<-]>>>>>>[<<<<<<+>>>>>>-]<<<<[-]>>>>>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-][-]+<[>-<[-]]>[<+>-]<[>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<[<<->>-]<<<<<+>>>>>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-][-]+<[>-<[-]]>[<+>-]<]<<<[-]>>[<<+>>-]<[-]++++++++++++++++++++++++++++++++++++++++++++++++>[-]+++++++++>>>[-]<[-]<<<<<[>>>>>>+<+<<<<<-]>>>>>>[<<<<<<+>>>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-]>[-]<[-]<[>>+<+<-]>>[<<+>>-]<[<<<+++++++>>>[-]]<[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<[<+>-]<.[-]++++++++++++++++++++++++++++++++++++++++++++++++>[-]+++++++++>>>[-]<[-]<<<<[>>>>>+<+<<<<-]>>>>>[<<<<<+>>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-]>[-]<[-]<[>>+<+<-]>>[<<+>>-]<[<<<+++++++>>>[-]]<[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<[<+>-]<.[-]++++++++++++++++>>[-]<[-]<<<<<<[>>>>>>>+<+<<<<<<-]>>>>>>>[<<<<<<<+>>>>>>>-]<<<<[-]>>>>>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-][-]+<[>-<[-]]>[<+>-]<[>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<[<<->>-]<<<<<+>>>>>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-][-]+<[>-<[-]]>[<+>-]<]<<<[-]>>[<<+>>-]<[-]++++++++++++++++++++++++++++++++++++++++++++++++>[-]+++++++++>>>[-]<[-]<<<<<[>>>>>>+<+<<<<<-]>>>>>>[<<<<<<+>>>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-]>[-]<[-]<[>>+<+<-]>>[<<+>>-]<[<<<+++++++>>>[-]]<[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<[<+>-]<.[-]++++++++++++++++++++++++++++++++++++++++++++++++>[-]+++++++++>>>[-]<[-]<<<<[>>>>>+<+<<<<-]>>>>>[<<<<<+>>>>>-]>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<<[-]>[>>>>[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<<[-]+>[<->[-]]<[<<<[-]+>>>[-]]<-<-]>[-]<[-]<[>>+<+<-]>>[<<+>>-]<[<<<+++++++>>>[-]]<[-]<[-]<<[>>>+<+<<-]>>>[<<<+>>>-]<[<+>-]<.<<[-]++++++++++.<[-]]<<+>>>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<[-]+>[<->[-]]<[<+>[-]]<<<<[-]+>>>>>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<[-]+>[<->[-]]<[>>>[-]<[-]<<<[>>>>+<+<<<-]>>>>[<<<<+>>>>-]<<[-]+>[<->[-]]<[<<<<<[-]>>>>>[-]]<[-]]<<<<]

Try it online.

Mandelbrot set sketch in Scratch

2019-07-13, post № 217

art, mathematics, programming, #fractal

Despite my personal disbelieve in and dislike of the colored blocks dragging simulator 3, I nevertheless wanted to extract functionality other than the hardcoded cat mascot path tracing from the aforementioned software; one of the most efficient visual result to build effort ratio yields a simple plot of the Mandelbrot set, formally known as

M:=\{z\in\mathbb{C}:|\lim_{n\to\infty}\mathrm{itr}^n(z)|<\infty\}

where the iterator is defined as

\mathrm{itr}^n(z) := \mathrm{itr}^{n-1}(z)^2+z, \\ \mathrm{itr}^0(z) := 0.

The render resolution is kept at a recognizable minimum as not to overburden the machine tasked with creating it.
Source: mandelbrot-set-sketch-in-scratch.sb3

mandelbrot-set-sketch-in-scratch.png

Factoids #0

2019-06-15, post № 216

mathematics, #algebra, #rings

I) Unit polynomials with non-vanishing degree

2t+1\in\mathbb{Z}_4[t] is its own multiplicative inverse, showing that R[t]^*=R^* does not hold in a general commutative Ring with one.

This phenomenon is uniquely characterized by the following equivalence:

R[t]^*=R^*\iff\nexists\,0\neq a,b\in R:a\cdot b=0=a+b
Proof. Negated replication. Let R\not\owns f=\sum_{i=0}^n\alpha_it^i\in R[t]^*,\alpha_n\neq 0 be a unit polynomial of non-vanishing degree n\geq 1. Let g=\sum_{j=0}^m\beta_jt^j\in R[t]^*,\beta_m\neq 0 denote its multiplicative inverse, i. e. f\cdot g=1.
Claim. The polynomial g has non-vanishing degree m\geq 1.
Proof. Suppose g\in R. Since f\cdot g=\sum_{i=0}^n(\alpha_i\cdot g)t^i, it follows from \alpha_n\cdot g=0 that g is a zero divisor. However, at the same time a_0\cdot g=1 implies that g is a unit, arriving at a contradiction.
Since both n,m\geq 1, one concludes \exists 1\leq k\leq m as well as \alpha_n\cdot\beta_m=0.
Existence of the desired ring elements a,b is assured by the following construction.
  • Let k=1\nearrow m rise discretely.
  • If a:=\alpha_n\beta_{m-k}\neq 0, implying b:=\sum_{i=1}^k\alpha_{n-i}\beta_{m-k+i}\neq 0, holds, since the construction arrived at this point, one finds
    a\cdot b=\alpha_n\beta_{m-k}\cdot \sum_{i=1}^k\alpha_{n-i}\beta_{m-k+i}=\sum_{i=1}^k \underbrace{\alpha_n\beta_{m-k+i}}_{=0}\cdot \alpha_{n-i}\beta_{m-k}=0.
  • The above condition is met for at least one 1\leq k\leq m, since otherwise k=m would imply \alpha_n\beta_{m-m}=0, which is impossible since \alpha_n\neq 0 and \beta_0 is a unit element.
By construction, 0\neq a,b as well as a+b=0 are given.
Negated implication. Setting  f:=at+1, g:=bt+1, one calculates
f\cdot g=(at+1)\cdot (bt+1)=abt^2+(a+b)\cdot t+1=0t^2+0t+1=1,
showing R\not\owns f,g\in R[t]^*.
Jonathan Frech's blog; built 2024/05/27 06:43:58 CEST