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Paolo Dolce

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Rony A. Bitan

The Geometric Gauss–Dedekind

Martí Oller

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Clément Rieutord

Carlos Gustavo Moreira

Harold Erazo

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Eryk Tadeusz Lipka

Eryk Tadeusz LIPKA
Corrigendum to “Automaticity of the sequence of last nonzero digits of
factorial in a fixed base”
Tome 37, no 2 (2025), p. 711-714.
https://doi.org/10.5802/jtnb.1338
© Les auteurs, 2025.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 4.0 FRANCE.
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Journal de Théorie des Nombres
de Bordeaux 37 (2025), 711–714
Corrigendum to “Automaticity of the sequence of
last nonzero digits of factorial in a fixed base”
J. Théor. Nombres Bordeaux 31 (2019), no. 1, p. 283-291
par Eryk Tadeusz LIPKA
Résumé. Nous corrigeons notre preuve précédente du fait que la suite des
derniers chiffres non nuls de n! est automatique lorsque la base de numération
est une puissance d’un nombre premier. L’erreur a été récemment remarquée
par J.-M. Deshouillers.
Abstract. We correct our previous proof of the fact that the sequence of the
last nonzero digits of a factorial is automatic when considered in a base being
a power of a prime. The error was recently noticed by J.-M. Deshouillers.
Let ℓb(x) denote the last (least-significant) nonzero digit of x in a base
b. In our previous paper we proposed a complete characterization of such
bases b, that (ℓb (n!))n∈N is automatic. This was a continuation of previous
work by Deshouillers and Ruzsa [2] who considered case b = 12. We used
the following lemma to show that for some b, (ℓb (n!))n∈N is automatic.
Lemma 3.6. If b = pa, p ∈ P, a ∈ N then the sequence (ℓb (n!))n∈N is
b-automatic.
We claimed that the above fact is true and was generally known before.
Because we couldn’t find a concrete proof of it, we decided to include one
in our paper for the sake of completeness. Proofs of special cases of this
fact could be found elsewhere [3]. The work on the topic was continued
by Sobolewski [5], who generalized the problem by looking at sequences of
k last nonzero digits of n! for k ≥ 1. It is worth noting that his results
imply Lemma 3.6. The most significant difference is that Sobolewski’s ap-
proach is using fixed points of morphisms, while we focused on an automata
construction.
However, as recently pointed out by J.-M. Deshouillers in private com-
munications, the proof we provided is actually wrong. We start the proof
Manuscrit reçu le 22 avril 2024, révisé le 19 mai 2024, accepté le 17 juillet 2024.
2010 Mathematics Subject Classification. 11B85, 11A63, 68Q45, 68R15.
Mots-clefs. Automatic sequence, factorial, the last nonzero digit.
712 Eryk Tadeusz Lipka
with “notice that ℓb(xy) = ℓb(ℓb(x)ℓb(y))”, which is false for b = pa, a ≥ 2.
A simple counterexample is
3 = ℓ4(12) = ℓ4(6 · 2) ̸= ℓ4(ℓ4(6)ℓ4(2)) = ℓ4(2 · 2) = ℓ4(4) = 1
and it is easy to construct similar counterexamples for other b. A few sen-
tences later, we use Euler’s totient theorem without ensuring that the base
of exponentiation is coprime with the modulus. Both of those errors are the
result of (obviously wrongly) assuming that ℓb(x) for a positive integer x
is coprime with b. This makes our proof incorrect for cases where b is not
a prime but a power of a prime. Below we present a corrected proof where
lb is, for the most part, replaced with another sequence that actually has
the required properties. As this is only a correction, we assume the reader
is familiar with basic definitions and facts regarding automatic sequences
and the automata theory that we used in the original paper. For in-depth
details, we recommend the book by Allouche and Shallit [1].
Proof. Fix a base b = pa. Instead of directly constructing an automaton
computing lb(n!), we will construct smaller automata, the output of which
can be easily combined to obtain lb(n!).
Let v(x) = max{k ∈ N : pk|x}, f(x) = x
pv(x)
mod b, i.e., the value of f(x)
is the number one obtains by writing x in the base p, removing the trailing
zeros and taking last a of the remaining digits. We have
xy
pv(xy)
= xy
pv(x)+v(y)
= x
pv(x)
· y
pv(y)
,
f(xy) =
(
x
pv(x)
· y
pv(y)
)
mod b = (f(x)f(y)) mod b = f(f(x)f(y)).
For any positive integer x the value of f(x) is coprime with b, so from
Euler’s totient theorem f(x)φ(b) ≡ 1 mod b. Let Ay(n) = #{i ∈ N : 1 ≤ i ≤
n, f(i) = y} mod φ(b). Value of f(n!) can be computed if we know Ay(n)
for all 1 ≤ y < b using the formula
(0.1)
f(n!) =
∏
1≤i≤n
f(i) mod b
=
∏
1≤y<b
y#{i∈N:1≤i≤n,f(i)=y} mod b
=
∏
1≤y<b
yAy(n) mod b.
Now we will construct an automaton computing Ay(n) for given 1 ≤ y <
b. If p|y then Ay(n) = 0, so from now on assume p ∤ y. The set {i ∈ N :
1 ≤ i ≤ n, f(i) = y} can be written as a disjoint sum
⊔
0≤k By,k(n) where
By,k(n) = {i ∈ N : 1 ≤ i ≤ n, f(i) = y, v(i) = k}. Because plogp(n)+1 > n,
for k > logp(n) we have By,k = ∅.
Corrigendum to “Automaticity of the sequence. . . ” 713
Notice, that By,k = {(0 ·b+y)pk, (1 ·b+y)pk, (2 ·b+y)pk, . . .}∩{1, . . . , n},
so By,k is an arithmetic progression with a difference of bpk. The number
of elements in By,k is
⌊
n
bpk
⌋
when
⌊
n
bpk
⌋
bpk + ypk > n or
⌈
n
bpk
⌉
otherwise.
That last condition can be rewritten as ypk ≤ (n mod (bpk)). If we look
at the digits of n = ngng−1 . . . n1n0 then the condition is equivalent to
checking whether nk+a−1 . . . nk+1nk ≥ y, and the number
⌊
n
bpk
⌋
is just
ng . . . nk+a+1nk+a.
Our automaton will read a number from left to right and store internally
three values. After reading the digits ng . . . nh, the memory will consist
of: (ng . . . nh+a) mod φ(b), (nh+a−1 . . . nh), (
∑logb n
k=h By,k(n)) mod φ(b). For-
mally, we define an automaton (Q, Σ, ρ, q0, ∆, τ) by:
• the input alphabet Σ = {0, 1, 2, . . . , p − 1};
• the output alphabet ∆ = {0, 1, . . . φ(b) − 1};
• the set of states Q = ∆ × Σa × ∆;
• the initial state q0 = (0, (0 . . . 0), 0);
• the output function τ (u, v, w) = w;
• the transition function ρ : Q × Σ 7→ Q, which is defined as follows:
for a given state (u, (va−1 . . . v0), w) and an input digit s the transi-
tion function returns the state with the following three components
pu + va−1 mod φ(b),
(va−2 . . . v0s),(
w + pu + va−1 +
{
1 if (va−2 . . . v0s) ≥ y
0 otherwise
)
mod φ(b).
It is easy to check that after reading the digit nh−1 the internal state of
the automaton is exactly as we stated: the transition function replaces
(ng . . . nh+a) mod φ(b) by (ng . . . nh+anh+a−1) mod φ(b), (nh+a−1 . . . nh) by
(nh+a−2 . . . nh−1) and (
∑logb n
k=h By,k(n)) mod φ(b) by (
∑logb n
k=h−1 By,k(n)) mod
φ(b). The output of this automaton is (
∑logb n
k=0 By,k(n)) mod φ(b) = Ay(n).
This construction can be repeated for all possible y, and by taking the
product of those automata, we obtain an automaton that computes Ay(n)
for each 1 ≤ y < n, p ∤ y. Change of the output function according to (0.1)
yields an automaton that computes f(n!).
To recover the last nonzero base-b digit we only need to know the “align-
ment”, or what blocks of base-p digits correspond to a single base-b digit.
We have lb(n!) = f(n!)p(v(n!) mod a) mod b. By Legendre’s formula v(n!) =
n−sp(n)
p−1 , where sp(n) is the sum of base p digits of n. As we need v(n!) mod a,
we can compute it from n mod (a · (p−1)) and sp(n) mod (a · (p−1)), both
of which are classic examples of p-automatic sequences.
714 Eryk Tadeusz Lipka
Combining the p-automata computing f(n!), v(n!) mod a we obtain a
p-automaton, and thus a b-automaton computing lb(n!). □
References
[1] J.-P. Allouche & J. Shallit, Automatic Sequences. Theory, Applications, Generalizations,
Cambridge University Press, 2003, xvi+571 pages.
[2] J.-M. Deshouillers & I. Z. Ruzsa, “The least nonzero digit of n! in base 12”, Publ. Math.
Debr. 79 (2011), no. 3-4, p. 395-400.
[3] G. P. Dresden, “Three transcendental numbers from the last non-zero digits of nn, Fn, and
n!”, Math. Mag. 81 (2008), no. 2, p. 96-105.
[4] E. T. Lipka, “Automaticity of the sequence of last nonzero digits of factorial in a fixed base”,
J. Théor. Nombres Bordeaux 31 (2019), no. 1, p. 283-291.
[5] B. Sobolewski, “On the last nonzero digits of n! in a given base”, Acta Arith. 191 (2019),
no. 2, p. 173-189.
Eryk Tadeusz Lipka
University of the National Education Comission
Podchorążych 2
30-084 Kraków, Poland
E-mail: eryklipka0@gmail.com


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The $3$ -adic representations arising from elliptic curves over $\mathbb{Q}_3$ with potential good reduction

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