Math::N...gitRadix(3)

💖 Math::NumSeq::RepdigitRadix 💖

okay so this is a page from the User Contributed Perl Documentation — it's Math::NumSeq::RepdigitRadix and it's about to tell you which radix makes a number go full repdigit ✨ buckle up bestie

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🌸 NAME 🌸

Math::NumSeq::RepdigitRadix — it's giving radix in which i is a repdigit 💅

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🦋 SYNOPSIS 🦋

use Math::NumSeq::RepdigitRadix;
my $seq = Math::NumSeq::RepdigitRadix->new;
my ($i, $value) = $seq->next;

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🎀 DESCRIPTION 🎀

So the value this sequence spits out is the radix in which i is a repdigit, and the vibe goes:

2, 0, 0, 2, 3, 4, 5, 2, 3, 8, 4, 10, etc
starting i=0

okay so let's break it down fr fr:

• i=0 is just taken to be a repdigit "00" in base 2. iconic of it honestly.
• i=1 and i=2 are not repdigits in any radix. not even one. tragic.
• then i=3 is repdigit "11" in base 2 👑
• any i>=3 is at worst a repdigit "11" in base i-1, but it may be a repdigit in a smaller base. for example i=8 is "22" in base 3 🍓

Is this behaviour for i=0,1,2 any good? Perhaps it will change. (yes bestie, the author is literally asking. we love an honest king.)

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👑 FUNCTIONS 👑

See "FUNCTIONS" in Math::NumSeq for behaviour common to all sequence classes. (iykyk 💜)

• "$seq = Math::NumSeq::RepdigitRadix->new ()" — Create and return a new sequence object. boom, you have a sequence 💖

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🎲 Random Access 🎲

• "$value = $seq->ith($i)" — Return the radix in which $i is a repdigit.

⚠️ The current code relies on factorizing $i, and a hard limit of 2**32 is placed on $i in the interests of not going into a near-infinite loop. so like, do not send it a big number, it will simply not have a good time 😭

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🧮 FORMULAS 🧮

ith() Value

ith() looks for the smallest radix r for which there's a digit d and length len satisfying

i = d * repunit(len)
i = d * (r^(len-1) + r^(len-2) + ... + r^2 + r + 1)

The current approach is to consider repdigit lengths successively from log2(i) downwards and candidate digits d from among the divisors of i.

for len=log2(i) down to 2
  for d each divisor of i, descending
    r = nthroot(i/d, len-1)
    if r >= r_found then next len
    if r <= d then next divisor
    if (r^len-1)/(r-1) == i/d then r_found=r, next len

if no r_found then r_found = i-1

For a given d the radix r to give i would be

i/d = r^(len-1) + ... + r + 1

but it's enough to calculate

i/d = r^(len-1)
r = floor nthroot(i/d, len-1)

and then power up to see if it gives the desired i/d.

repunit(len) = r^(len-1) + ... + r + 1
             = (r^len - 1) / (r-1)
check if equals i/d

floor(nthroot()) is never too small, since an r+1 from it would give

(r+1)^(len-1) = r^(len-1) + binomial*r^(len-2) + ... + 1
              > r^(len-1) +          r^(len-2) + ... + 1

Divisors are taken in descending order so the radices r are in increasing order. So if a repdigit is found in a given len then it's the smallest of that length and can go on to other lengths. slay 🫶

The lengths can be considered in any order but the current code goes from high to low since a bigger length means a smaller maximum radix within that length (occurring when d=1, ie. a repunit), so it might establish a smaller "r_found" and a smaller r_found limits the number of divisors to be tried in subsequent lengths. But does that actually happen often enough to make any difference? 🤔 (asking the real questions)

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ith() Other Possibilities

When len is even the repunit part r^(len-1)+...+1 is a multiple of r+1. Can that cut the search? For a given divisor the r is found easily enough by nthroot, but maybe i with only two prime factors can never be an even length>=4 repdigit, or something like that. ✨ open research question bestie ✨

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🌈 SEE ALSO 🌈

• Math::NumSeq
• Math::NumSeq::RepdigitAny

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🏠 HOME PAGE 🏠

<http://user42.tuxfamily.org/math-numseq/index.html>

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📜 LICENSE 📜

Copyright 2010, 2011, 2012, 2013, 2014, 2016, 2019, 2020 Kevin Ryde

Math-NumSeq is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 3, or (at your option) any later version.

Math-NumSeq is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with Math-NumSeq. If not, see <http://www.gnu.org/licenses/>.

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and that's the whole thing bestie, no cap, every last detail preserved 💖✨🎀

perl v5.42.0 · 2025-07-25 — mwah, go read it again if you need to, ily 🫶💅