in reply to [OT] math fulguration

Use List::Util::sum for summation: 
#! /usr/bin/perl
use warnings;
use strict;
use feature qw{ say };

use List::Util qw{ sum };
for my $A (2..20) {
    for my $n (1..10) {
        die "$A, $n\n"
            unless $A ** $n == 1 + ($A - 1) * sum(map $A ** $_, 0 .. $
+n - 1)
    }
}

[download]

The proof can be compiled in TeX: 

\documentclass{article}
\title{Summation Fulguration}
\newtheorem{thm}{Theorem}
\newtheorem{prf}{Proof}[thm]
\begin{document}
\begin{thm}
$$a^n = 1 + (a - 1) \sum_{i=0}^{n-1} a^i$$
\end{thm}
\begin{prf}
$$1 + (a - 1) \sum_{i=0}^{n-1} a^i$$
$$= 1 + \sum_{i=0}^{n-1}(a-1)a^i$$
$$= 1 + \sum_{i=0}^{n-1} a^{i+1} - a^i$$
$$= 1 + \sum_{i=0}^{n-1}a^{i+1} - \sum_{i=0}^{n-1}a^i$$
$$= 1 + (\sum_{i=1}^{n-1}a^i) + a^n - (a^0 + \sum_{i=1}^{n-1}a^i)$$
$$= 1 + (\sum_{i=1}^{n-1}a^i) + a^n - a^0 - \sum_{i=1}^{n-1}a^i$$
$$= 1 + a^n - a^0$$
$$= 1 + a^n - 1$$
$$= a^n$$
Q.E.D.
\end{prf}
\end{document}

[download]

The result can be checked here.

I have no idea how broadly it is known.

map{substr$_->[0],$_->[1]||0,1}[\*||{},3],[[]],[ref qr-1,-,-1],[{}],[sub{}^*ARGV,3]

Replies are listed 'Best First'.
Re^2: [OT] math fulguration
by LanX (Saint) on Apr 06, 2021 at 21:20 UTC
    > I have no idea how broadly it is known.

    Pretty much, what Salva pointed out is discussed at university in CS in the context of numeral systems to basis n, before introducing binary system.

    By intuition I'd say it's also connected to some school math, like the theorem that every periodic fraction can be expressed in the form of

    n+(m/9...9) with n,m in N

    like

      DB<3> p 3+7834/9999
3.78347834783478

    [download]

    This works in any number system.

    What I learned was the word "fulguration". ;-)

    Cheers Rolf
    (addicted to the Perl Programming Language :)
    Wikisyntax for the Monastery

[reply]
[d/l]
[select]
      ... learned ... the word "fulguration".

      Related: fulgurite. Also: Is there a link to a definition of "fulguration" in a mathematical context? I seem to see it only in the medical context of "radiofrequency ablation" or coagulation.

      Give a man a fish:  <%-{-{-{-<

[reply]
[d/l]
        > Is there a link to a definition of "fulguration" in a mathematical context?

        I'm pretty sure not, google would find it, never heard it.

        I understood "lightning" and saw the parallel to German Geistesblitz (literally ghost/spirit + flash)

        • flash of inspiration
        • flash of wit
        • flash of genius
        • scintillation
        • sudden inspiration
        • brain wave (colloquial?)
        • sudden inspirations

        Cheers Rolf
        (addicted to the Perl Programming Language :)
        Wikisyntax for the Monastery

[reply]
        I think Discipulus meant "enlightenment" in the spiritual sense, the bodhi or satori.

        map{substr$_->[0],$_->[1]||0,1}[\*||{},3],[[]],[ref qr-1,-,-1],[{}],[sub{}^*ARGV,3]
[reply]
[d/l]
      "fulguration"

      I don't recall that word at all from any crufty discussion in number theory or logic design with the faculties of a handful of american universities where I might have been a fly on the wall. Others' mileage may vary. The usage that is common for us of a certain age is the one that is a synonym for 'ablation'.

[reply]

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