Journal de Théorie des Nombres de Bordeaux

Julie Desjardins

Root number of twists of an elliptic curve

Guillermo Mantilla-Soler

Guillermo MANTILLA-SOLER
An introduction to oddly tame number fields.
Tome 32, no 3 (2020), p. 711-717.
<http://jtnb.centre-mersenne.org/item?id=JTNB_2020__32_3_711_0>
© Société Arithmétique de Bordeaux, 2020, tous droits réservés.
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cedram
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Journal de Théorie des Nombres
de Bordeaux 32 (2020), 711–717
An introduction to oddly tame number fields.
par Guillermo MANTILLA-SOLER
Résumé. Il résulte des généralités sur les formes quadratiques que la classe
spinorielle de la trace intégrale d’un corps de nombres détermine la signature
et le discriminant du corps. Dans cet article, nous définissons une famille de
corps de nombres, qui contient, entre autres, tous les corps galoisiens de degré
impair modérément ramifiés, pour lesquels la réciproque est vraie. Autrement
dit, pour un corps de nombres K de cette famille, on montre que la classe
spinorielle de la trace intégrale ne contient pas d’informations sur K autres
que celles qui sont fournies par le discriminant et la signature.
Abstract. It follows from generalities of quadratic forms that the spinor
class of the integral trace of a number field determines the signature and the
discriminant of the field. In this paper we define a family of number fields, that
contains among others all odd degree Galois tame number fields, for which
the converse is true. In other words, for a number field K in such family we
prove that the spinor class of the integral trace carries no more information
about K than the discriminant and the signature do.
1. Introduction
Let K be a number field, OK its maximal order and let rK and sK be
the number of real and complex places respectively. The integral trace form
of K is the isometry class of the Z-quadratic module
〈OK , tK〉
associated to the trace pairing
OK ×OK → Z; (x, y) 7→ TrK/Q(xy).
Since (rK+sK , sK) = Sign(〈OK , tK〉) and disc(K) = det(〈OK , tK〉) (see [9])
it follows from generalities of quadratic forms (see [2, IX §4]) that the
spinor genus of 〈OK , tK〉 determines the signature and the discriminant of
K. Besides the trivial case of fields of degree less than 3, there are some
interesting instances where the converse of the above holds:
Mathematics Subject Classification. 11R04, 11S15.
Mots-clefs. Arithmetic invariants, tame fields, arithmetic equivalence, trace forms.
We would like to thank the referee for the careful reading of the paper, and for their helpful
comments.
712 Guillermo Mantilla-Soler
Theorem ([4, Theorem 3.2]). Let K and L be cubic number fields. Then
〈OK , tK〉 and 〈OL, tL〉 belong to the same spinor genus if and only if
disc(K) = disc(L).
Recall that for cubic fields the sign of the discriminant determines the
signature of the field, hence it is not necessary to mention signatures in
the above theorem. There are examples that show that the above theorem
cannot be extended to isometry classes; take for instance (see [7, Theo-
rem 5.2] and [4, Theorem 3.2]) two non isomorphic cubic fields of positive
fundamental discriminant, e.g., any two of the four cubic fields of discrim-
inant 32009. There are some cases in which the isometry class of the trace
is determined by the discriminant and the signature:
Theorem ([1, Theorems 4.2, 4.5]). Let n be a positive integer and let K,L
be two tame Z/nZ-number fields. Then
〈OK , tK〉 ∼= 〈OL, tL〉 if and only if disc(K) = disc(L).
For odd n there is no need to mention the signature since the fields are
totally real. As it turns out, in the above situation, for n even the equality
between discriminants imply the equality of the signatures. The following
example shows that the theorem above cannot be extended to arbitrary
Galois groups, not even abelian ones. All the examples in this paper have
been obtained with the help of John Jones’ tables of number fields [3].
Example 1.1. Let K and L be the number fields defined respectively by
the polynomials fK = x4−41x2 +144 and fL = x4−x3−46x2−115x−35.
Both fields are quartic V4-Galois fields with discriminant equal to 52 · 132 ·
172. Since in the first field p = 5 has one prime factor, while in the second
it has two, we see using [4, Proposition 2.9] that the integral traces of K
and L are not in the same spinor genus, thus they are not isometric.
In this paper we define a class of number fields for which the discriminant
and the signature are necessary and sufficient to determine the spinor class
of the integral trace.
Definition 1.2. Let K be a number field, let p be a rational prime and
let e1, . . . , eg be the ramification indices of p in K. The prime p is called
uniformly ramified in K if e1 = · · · = eg.
The definition of uniformly ramified prime is inspired by the behaviour of
ramification in Galois number fields; however there are non Galois number
fields in which every prime is uniformly ramified, e.g., a field in which every
ramified prime is totally ramified. For instance, the number field defined
by the polynomial x4 − x3 − 7x2 + 11x+ 3 is an S4−quartic field in which
p = 59 is the only ramified prime and such prime is totally ramified.
An introduction to oddly tame number fields. 713
Let K be a number field. A prime p is called an exceptional prime for
K if
• there is an even number of primes above p,
• or p is not uniformly ramified,
• or p is uniformly ramified with common ramification degree e but,
[K:Q]
e is even.
Definition 1.3. A number field K is called oddly tame if the following
conditions hold:
(1) The field K is tame, i.e., every ramified prime p is not wildly
ramified.
(2) There is at most one odd ramified prime that is exceptional for K.
If there is an odd ramified exceptional prime for K, then we call it the
exceptional prime of K.
Remark 1.4. Notice that the collection of oddly tame number fields con-
tains all tame Galois number fields of odd degree and all tame number fields
in which all ramified primes are totally ramified. However, there can be
oddly tame number fields that are of even degree, that are not Galois or in
which no ramified prime is totally ramified. One such example is the sextic
fieldK defined by the polynomial x6−2x5+3x4−9x3+8x2−7x−5. This is a
D12-sextic number field with discriminant 33 · 233 such that 3OK = P20 and
23OK = (P1P2)2 for some prime ideals Pi. This is the unique, up to conju-
gacy, oddly tame number field of signature (2, 2) and discriminant 33 · 233.
However, as verified by Example 1.5, there are pairs of non-isomorphic
oddly tame number fields of the same signature and discriminant.
Our main result states that for oddly tame number fields the signature
and the discriminant are necessary and sufficient to determine the spinor
genus of the integral trace. Moreover, for non totally real fields this can be
improved to isometry class.
Theorem (cf. Theorem 2.4). Let K,L be two oddly tame number fields.
Suppose that the set formed by exceptional primes of K and L contains at
most one element. Then 〈OK , tK〉 and 〈OL, tL〉 belong to the same spinor
genus if and only if disc(K) = disc(L) and Sign(〈OK , tK〉) = Sign(〈OL, tL〉).
Furthermore, If K is not totally real
〈OK , tK〉 ∼= 〈OL, tL〉 if and only if
disc(K) = disc(L) and Sign(〈OK , tK〉) = Sign(〈OL, tL〉)
Example 1.5. Consider the sextic fields K and L defined by the polyno-
mials fK := x6− x5− 2x4 + x3 + 7x2− 6x+ 4 and fL := x6− 3x5 + 10x4−
15x3 + 19x2 − 12x + 3. Both fields have signature (r, s) = (0, 3), discrim-
inant −33 · 1072 and Galois closure with Galois group D12. The fields are
714 Guillermo Mantilla-Soler
not isomorphic; in K the prime p = 3 has only one prime lying over it,
while in L it has three. In both fields the prime p = 3 is uniformly ramified
with ramification index e = 2. In particular, K and L are oddly tame num-
ber fields that are neither Galois or of odd degree or with totally ramified
primes. Moreover, the prime q = 107 is exceptional in both fields. Since K
is not totally real we know by Theorem 2.4 that 〈OK , tK〉 ∼= 〈OL, tL〉.
2. Proofs of our results
One of the main ingredients we use is the set of α-invariants of a number
field; for definitions and properties see [6].
Definition 2.1. Let K be a number and let p be an odd prime. Let g be
the number of prime factors of p in K and let (e1, . . . , eg) and (f1, . . . , fg)
be the ramification and residue degrees of p over K. The first ramification
invariant of p in K is the integer
αKp :=
( g∏
i=1
efii
)
u(F−g)p ,
where F =
∑
fi and up ∈ {1, . . . , p − 1} is the first non quadratic residue
modulo p.
Among the useful properties of such invariants we have that they deter-
mine the genus of the integral trace which in degree at least 3, thanks to [5],
is the same as the spinor genus. The main relation about α-invariants and
the integral trace is the following:
Proposition 2.2 ([4, Proposition 2.9]). Let K,L be tame number fields of
degree n ≥ 3. The forms 〈OK , tK〉 and 〈OL, tL〉 belong to the same spinor
genus if and only if the following conditions hold:
(1) disc(K) = disc(L),
(2) sK = sL,
(3) For every odd prime p that divides the common discriminant of K
and L we have that (
αKp
p
)
=
(
αLp
p
)
.
In oddly tame number fields the αp invariant of a non-exceptional rami-
fied prime can be expressed, up to squares, in terms of the degree and the
discriminant. More explicitly:
Lemma 2.3. Let K be a degree n oddly tame number field of discriminant
d. Let p be an odd non-exceptional ramified prime in K. Then,
αKp =
n
n− vp(d)
mod (Z∗p)2,
where vp denotes the standard p-adic valuation in Q.
An introduction to oddly tame number fields. 715
Proof. Let p 6= 2 be a ramified prime in K, not exceptional. By definition
of α invariants, and since p is uniformly ramified, αKp = eFuF−gp where
e is the ramification invariant of p over K. By hypothesis g is odd, and
since n = eF we have, by hypothesis as well, that F is odd. Thus, αKp = e
mod (Z∗p)2. On the other hand, since p is tame, thanks to [8, Chapter III,
Proposition 13], we have that
vp(d) = (e− 1)F = n− F = n−
n
e
from where the result follows. 
Theorem 2.4. Let Kand L be two oddly tame number fields. Suppose
that the set formed by exceptional primes of K and L contains at most
one element. Then 〈OK , tK〉 and 〈OL, tL〉 belong to the same spinor genus
if and only if disc(K) = disc(L) and Sign(〈OK , tK〉) = Sign(〈OL, tL〉).
Furthermore, If K is not totally real
〈OK , tK〉 ∼= 〈OL, tL〉 if and only if
disc(K) = disc(L) and Sign(〈OK , tK〉) = Sign(〈OL, tL〉)
Proof. We may assume that K and L have degree at least 3. We show
the non trivial implication. Since for number fields of degree at least 3
the genus and the spinor genus of the integral trace form coincide, see [5,
Theorem 2.12], it is enough to show that 〈OK , tK〉 ⊗R ∼= 〈OL, tL〉 ⊗R and
that 〈OK , tK〉 ⊗ Zp ∼= 〈OL, tL〉 ⊗ Zp for all prime p. Since K and L have
the same signature it follows from a result of Taussky [9] that
〈OK , tK〉 ⊗ R ∼= 〈OL, tL〉 ⊗ R.
Since for odd primes p the isometry class of a quadratic forms over Zp, with
invertible discriminant, is determined by the discriminant and dimension
of the form (see [2, Chapter 8, Lemma 3.4]) we have that
〈OK , tK〉 ⊗ Zp ∼= 〈OL, tL〉 ⊗ Zp
for every odd unramified prime p. Given that both fields are tame it follows
from [4, Proposition 2.7] that
〈OK , tK〉 ⊗ Z2 ∼= 〈OL, tL〉 ⊗ Z2.
Thanks to Lemma 2.3 (
αKp
p
)
=
(
αLp
p
)
for every odd prime p that divides the common discriminant and that is
not exceptional.
Hence, by [6, Theorem 3.14]
〈OK , tK〉 ⊗ Zp ∼= 〈OL, tL〉 ⊗ Zp
716 Guillermo Mantilla-Soler
for every prime p with one possible exception; an exceptional prime if it
exists. Since 〈OK , tK〉⊗R ∼= 〈OL, tL〉⊗R it follows from the product formula
for Hilbert symbols that
〈OK , tK〉 ⊗Qp ∼= 〈OL, tL〉 ⊗Qp
for every prime p. Let p be the possible exceptional prime. For α = αKp or
αLp , since p - α one can check that(
α
p
)
= (α, p)p
where (·, ·)p is the p-adic Hilbert symbol. Combining [6, Theorem 2.13]
with [6, Lemma 2.9] and [4, Lemma 2.1] then shows that 〈OK , tK〉 ⊗Qp ∼=
〈OL, tL〉 ⊗Qp implies that (
αKp
p
)
=
(
αLp
p
)
for every prime p. The result then follows from Proposition 2.2 and for the
isometry class in the non totally real case from [4, Theorem 2.13]. 
The following example shows, thanks to Theorem 2.4, that the Galois
structure of number fields with the same integral trace can be different.
Example 2.5. Consider the quartic fields K and L defined by the polyno-
mials fK := x4−x3 +36x2−36x+281 and fL := x4−2x3 +14x2−13x+6.
Both fields have signature (r, s) = (0, 2), discriminant 53 · 292. The field K
is a Z/4Z-number field. In contrast L is not Galois, and its Galois closure
has Galois group D8. In particular, the fields are not isomorphic. In both
fields the prime p = 5 is totally ramified, hence K and L are oddly tame
number fields. The prime q = 29 is exceptional in both fields. Since K is
not totally real 〈OK , tK〉 ∼= 〈OL, tL〉.
References
[1] W. Bolaños & G. Mantilla-Soler, “The trace form over cyclic number fields”, https:
//arxiv.org/abs/1904.10080v2, to appear in Can. J. Math., 2019.
[2] J. W. S. Cassels, Rational quadratic forms, Dover Publications, 2008.
[3] J. Jones & D. Roberts, “A data base of number fields”, LMS J. Comput. Math. 17 (2014),
no. 1, p. 595-618.
[4] G. Mantilla-Soler, “On the arithmetic determination of the trace”, J. Algebra 444 (2015),
p. 272-283.
[5] ———, “The Spinor Genus of the integral trace”, Trans. Am. Math. Soc. 369 (2017), no. 3,
p. 1547-1577.
[6] ———, “The (α, β)-ramification invariants of a number field”, https://arxiv.org/abs/
1906.04254, to appear in Mathematica Slovaca, 2019.
[7] G. Mantilla-Soler & C. Rivera-Guaca, “An introduction to Casimir pairings and some
arithmetic applications”, https://arxiv.org/abs/1812.03133v3, 2019.
[8] J.-P. Serre, Local fields, Graduate Texts in Mathematics, vol. 67, Springer, 1979,
viii+241 pages.
An introduction to oddly tame number fields. 717
[9] O. Taussky, “The discriminant matrices of an algebraic number field”, J. Lond. Math. Soc.
43 (1968), p. 152-154.
Guillermo Mantilla-Soler
Department of Mathematics
Fundación Universitaria Konrad Lorenz
Bogotá, Colombia
Department of Mathematics and Systems Analysis
Aalto University
Helsinki, Finland
E-mail: gmantelia@gmail.com
URL: https://sites.google.com/view/guillermo-mantilla-soler/home


An introduction to oddly tame number fields.

Jesse Jääsaari

On Short Sums Involving Fourier Coefficients of Maass Forms

Alexander D. Rahm

Panagiotis Tsaknias

Genuine Bianchi modular forms of higher level  at varying weight and discriminant

Jędrzej Garnek

On class numbers of division fields of abelian varieties

Eryk Lipka

Eryk LIPKA
Automaticity of the sequence of the last nonzero digits of n! in a fixed base
Tome 31, no 1 (2019), p. 283-291.
<http://jtnb.cedram.org/item?id=JTNB_2019__31_1_283_0>
© Société Arithmétique de Bordeaux, 2019, tous droits réservés.
L’accès aux articles de la revue « Journal de Théorie des Nom-
bres de Bordeaux » (http://jtnb.cedram.org/), implique l’accord
avec les conditions générales d’utilisation (http://jtnb.cedram.
org/legal/). Toute reproduction en tout ou partie de cet article sous
quelque forme que ce soit pour tout usage autre que l’utilisation à
fin strictement personnelle du copiste est constitutive d’une infrac-
tion pénale. Toute copie ou impression de ce fichier doit contenir la
présente mention de copyright.
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Journal de Théorie des Nombres
de Bordeaux 31 (2019), 283–291
Automaticity of the sequence of the last nonzero
digits of n! in a fixed base
par Eryk LIPKA
Résumé. En 2011, Deshouillers et Ruzsa [5] ont donné des arguments en
faveur de la non-automaticité de la suite des derniers chiffres non nuls de
n! en base 12. Cette assertion a été prouvée quelques années plus tard par
Deshoulliers [4]. Dans cet article, nous donnons une preuve alternative qui
nous permet de généraliser le problème et donner une caractérisation com-
plète des bases pour lesquelles la suite des derniers chiffres non nuls de n! est
automatique.
Abstract. In 2011 Deshouillers and Ruzsa [5] tried to argue that the se-
quence of the last nonzero digit of n! in base 12 is not automatic. This state-
ment was proven a few years later by Deshoulliers in [4]. In this paper we
provide an alternate proof that lets us generalize the problem and give an
exact characterization of the bases for which the sequence of the last nonzero
digits of n! is automatic.
1. Introduction
Let (`b(n!))n∈N be the sequence of last nonzero digits of n! in base b.
In this paper we will answer the question for which values of b is this
sequence automatic. It is known that (`b(n!))n∈N is automatic in many
cases including bases being primes or powers of primes. One can also prove
that (`b(n!))n∈N is automatic for some small bases that have more prime
factors, like 6 or 10. In general, for a base of the form b = pa11 p
a2
2 where p1 6=
p2, p1, p2 ∈ P, a1, a2 ∈ N+, it can be shown that (`b(n!))n∈N is automatic
when a1 (p1 − 1) 6= a2 (p2 − 1). The smallest base for which the answer is
unclear is 12. This was the case analysed by Deshouillers and Ruzsa in [5].
They conjectured that (`12(n!))n∈N cannot be automatic, despite the fact
that it is equal to some automatic sequence nearly everywhere. An attempt
to prove that conjecture was made by Deshouliers in his paper [3], and a
few years later he answered the question by proving the following, stronger
result
Manuscrit reçu le 27 août 2018, révisé le 6 novembre 2018, accepté le 16 novembre 2018.
2010 Mathematics Subject Classification. 11B85, 11A63, 68Q45, 68R15.
Mots-clefs. automatic sequence, factorial, the last nonzero digit.
284 Eryk Lipka
Theorem 1.1 (Deshouillers [4]). For a ∈ {3, 6, 9}, the characteristic se-
quence of {n ; `12(n!) = a} is not automatic.
Another way of proving a similar fact (but for a ∈ {4, 8}) was provided
recently by Byszewski and Konieczny in [2]. It seems that both proofs can
be generalized to all cases when a1(p1 − 1) = a2(p2 − 1); however it is
not obvious if, or how, it can be extended to bases with more than two
prime factors. This was our main motivation for writing this paper, and we
provide a complete characterization of the bases for which this sequence is
automatic, including those with many prime factors.
In this paper we will use the following notation: the string of digits of n in
base k will be denoted [n]k, or reversely we use 〈n1n2 . . . nl〉b to describe the
integer with digits n1 . . . nl; by vb(n) we mean the largest integer t such that
bt|n, and sb(n) is the sum of digits of n in base b. This paper is composed of
two main parts: first we recall some basic facts about automatic sequences
for readers not familiar with the topic; in the latter part we present our
results about the automaticity of (`b(n!))n∈N.
Acknowledgements. We would like to thank Piotr Miska and Maciej
Ulas for proofreading and helpful suggestions while preparing this paper
2. Basics of automatic sequences
In this section we will give a short summary of topics from automatic
sequence theory that we will be using later. If the reader is interested in
getting more insight into this topic, we strongly recommend the book of
Allouche and Shallit [1] that covers all important topics in this area.
Definition 2.1. Deterministic finite automaton with output is a 6-tuple
(Q,Σ, ρ, q0,∆, τ) such that
• Q is a finite set of states;
• Σ is an input alphabet;
• ρ : Q× Σ→ Q is a transition function;
• q0 ∈ Q is an initial state;
• ∆ is an output alphabet (finite set);
• τ : Q→ ∆ is an output function.
The transition function can be generalized to take strings of characters
instead of single ones. For a string s1s2s3 . . . we define ρ (q, s1s2s3 . . .) =
ρ (. . . ρ (ρ (q, s1) , s2) . . .).
Definition 2.2. For any finite alphabet Σ, a function f : Σ∗ → ∆ is called
a finite-state function if there exists a deterministic finite automaton with
output (Q,Σ, ρ, q0,∆, τ) such that f(ω) = τ (ρ (q0, ω)).
Automaticity of the sequence of the last nonzero digits of n! in a fixed base 285
Lemma 2.3. If f : Σ∗ → ∆ is a finite-state function then the function
g : Σ∗ → ∆ defined by g(ω) = f(ωR) is also finite-state. ( R denotes taking
the everse of a word).
Proof (Sketch). Let (Q,Σ, ρ, q0,∆, τ) be the automaton that is related to
f , we will define another automaton (Q′,Σ, ρ′, q′0,∆, τ ′). Let Q′ = ∆Q be
all functions from Q to ∆ and q′0 ≡ τ . For all g ∈ Q′ we define τ ′(g) = g(q0),
and for all σ ∈ Σ, q ∈ Q we put ρ′(g, σ)(q) = g(ρ(q, σ)). By induction on
length of the word ω ∈ Σ∗ one can prove that the equation
ρ′(g, ω)(q) = g(ρ(q, ωR))
holds for any g ∈ Q′, q ∈ Q. And finally
g(ω) = τ ′(ρ′(q′0, ω)) = ρ′(q′0, ω)(q0) = g(ρ(q0, ωR)) = f(ωR). 
Definition 2.4. (a(n))n∈N is a k-automatic sequence if the function [n]k →
an is finite-state. By Lemma 2.3 it is not important whether we read the
representation of n from the right or from the left side.
Now we present some simple examples of sequences that are automatic.
Example 2.5. The sequence an = n (mod m) is k-automatic for any k ≥
2,m ∈ Z+. In order to see this, it is enough to take Q = {0, 1, . . . ,m− 1},
ρ(q, σ) = kq + σ (mod m) and read input “from left to right”.
The sequence an = sk(n) (mod m) is k-automatic for any k ≥ 2,m ∈ Z+.
Take Q = {0, 1, . . . ,m− 1} and ρ(q, σ) = q + σ (mod m).
For any k ≥ 2 and x ∈ N, the characteristic sequence an = δx(n) is
k-automatic. The automaton that computes it can be constructed by tak-
ing dlogk(x)e states that counts how many digits were correct plus one
“sinkhole” state that accepts all numbers other than x.
We can also obtain automatic sequences by modifying the existing ones.
Example 2.6. If (a(n))n∈N is a k-automatic sequence then so is bn = f(an)
for any function f taking values from the image of an. The difference will
be only in the output function of the related automaton.
If (a(n))n∈N, (b(n))n∈N are k-automatic sequences, then so is cn =
f(an, bn) for any function f as long as it is well defined on all possible
pairs (an, bn). To obtain such an automaton (Qc,Σ, ρc, qc,∆c, τc) we can
take the “product” of automata (Qa,Σ, ρa, qa,∆a, τa), (Qb,Σ, ρb, qb,∆b, τb)
defined by
• Qc = Qa ×Qb;
• ρc(a, b) = (ρa(a), ρb(b));
• qc = (qa, qb);
• ∆c = f(∆a ×∆b);
• τc(a, b) = f(τa(a), τb(b)).
286 Eryk Lipka
This can be easily generalized to the case with f taking any finite number
of sequences as an input.
By combining the above examples together we can also observe the fol-
lowing.
Lemma 2.7. Let k ∈ N≥2 be fixed, then:
• the characteristic sequence of a finite set is k-automatic;
• if sequence (a(n))n∈N differs from (b(n))n∈N only on finitely many
terms and one of them is k-automatic then so is the other one;
• a periodic sequence is k-automatic;
• an ultimately periodic sequence is k-automatic;
Of course this does not exhaust all possible automatic sequences, but
is enough to give some insight and be useful in our work. We should also
notice what is the relation between automaticity in different bases.
Lemma 2.8. Sequence (a(n))n∈N is k-automatic if and only if it is km-
automatic for all m ∈ N≥2.
Proof (Sketch). If we have a k-automaton generating a sequence, then we
can easily manipulate it to create a km-automaton generating the same
sequence. The main idea is to take the transition function to be the m-th
composition of the original transition function with itself (a digit in base
km can be seen as m digits in base k).
On the other hand, let Q be the set of states of the km-automaton
generating a sequence, and ρ be its transition function. We take Q′ =
Q× {0, 1, . . . , km−1 − 1} × {0, 1, . . . ,m− 1} and
ρ′((q, r, s), σ) =
{
(q, kr + σ, s+ 1) if s < m− 1
(ρ(q, kr + σ), 0, 0) if s = m− 1;
this way we accumulate base k digits until we collect m of them and then
use the original transition function. 
3. New results
Let’s start with some facts that we will be using in our proof:
Proposition 3.1 (Legendre’s formula [6]). For any prime p and positive
integers a, n, we have
vpa(n!) =
⌊
n− sp(n)
a(p− 1)
⌋
.
Proposition 3.2 (Result from [7]). For any positive integers b, c such that
ln(b)
ln(c) 6∈ Q there exists a constant d such that for each integer n > 25 we
have
sb(n) + sc(n) >
log logn
log log logn+ d − 1.
Automaticity of the sequence of the last nonzero digits of n! in a fixed base 287
The next proposition is a known fact, but we haven’t found it clearly
stated anywhere; it can be easily proven using Dirichlet’s approximation
theorem or Equidistribution theorem.
Proposition 3.3. For any positive integers a, b, c such that ln(b)ln(c) 6∈ Q there
exist infinitely many triples of non-negative integers d, e, f with 1 ≤ f < be
such that
cd = a · be + f.
In other words, there are infinitely many powers of c with base b notation
starting with given string of digits.
After such an introduction we can finally state our results. The following
lemma and theorem are the main steps in proving when (`b(n!))n∈N is not
automatic.
Lemma 3.4. Let P be a non-empty finite set of prime numbers and p be its
greatest element. Let a > 0, k > 1 be integers. Then there exists an integer
a′ such that maxi∈P {si(a′)} = sp(a′) and [a]k is prefix of [a′]k.
Proof. If k is not a power of p, then by Proposition 3.3 there exist infinitely
many triples (d, e, f) of non-negative integers with 1 ≤ f+1 < ke such that
pd = a · ke + (f + 1).
Furthermore we have
sp(pd − 1) = d(p− 1) > (p− 1)
ln(pd − 1)
ln(p) ,
and from the definition of sq, for any prime q the following holds
sq(pd − 1) < (q − 1)
(
ln(pd − 1)
ln(q) + 1
)
.
Because p is the greatest number in P , then for any q ∈ P, q 6= p, we have
sq(pd − 1)− sp(pd − 1) < ln(pd − 1)
(
q − 1
ln(q) −
p− 1
ln(p)
)
+ q − 1.
The right side of this inequality is negative for d big enough, so because
0 ≤ f < ke we can take a′ = pd − 1. When k = pt we can notice that for
any integer d
sp(a · ptd + ptd − 1) = sp (a) + td (p− 1)
> (p− 1)
(
ln(a · ptd + ptd − 1)
ln(p) −
ln (a)
ln(p) − 1
)
,
and hence it is enough to take a′ = a·ptd+ptd−1 for d sufficiently large. 
288 Eryk Lipka
Theorem 3.5. Let P be a finite set of prime numbers with at least two
elements and p be its greatest element, also let c > 0 be a real number. Let
us define sets
A− =
{
n ∈ Z+ : max
i∈P
{si(n)} = sp(n)
}
,
A+ =
{
n ∈ Z+ : max
i∈P
{si(n)} − sp(n) ≥ c
}
.
Then there does not exist a deterministic finite automaton with output that
assigns one value to integers in A− and another value to those in A+.
Proof. Let’s suppose that we have such an automaton (Q,Σk, ρ, q0,∆, τ) for
some k. Because Q is finite, there exists an internal state S ∈ Q such that
for infinitely many positive integers c1 < c2 < . . . we have ρ (q0, [pci ]k) = S.
Now, by Lemma 3.4, there exists an integer a′ ∈ A− which can be obtained
from pc1 by appending some suffix. Hence we can fix positive integers e, f <
ke such that a′ = pc1 ·ke + f . Let the sequence of digits (f1, f2, . . . , fe) be a
representation of f in base k, possibly with added leading zeros. By T ∈ Q
we denote an internal state such that
T = ρ
(
q0,
[
a′
]
k
)
= ρ (S, f1f2 . . . fe) .
This means that for every i ∈ N+ we have
ρ (q0, [pci · ke + f ]k) = ρ (q0, [p
ci ]k f1f2 . . . fe) = ρ (S, f1f2 . . . fe) = T ,
and this implies that τ (ρ (q0, [pci · ke + f ]k)) = τ (T ) does not depend on
the value of i.
On the other hand, when ci > dlogp(f)e we have sp (pci · ke + f) =
sp (ke) + sp (f) which is a constant. However, due to Proposition 3.2 we
know that for any q ∈ P, q 6= p, the value of sq (pci · ke + f) is increasing
with ci. Hence for ci big enough we have pci · ke + f ∈ A+.
All but finitely many integers of the form pci · ke + f are elements of A+
but at least one (namely pc1 · ke + f) is an element of A−. This proves that
such an automaton cannot assign different values to members of those two
sets. 
Now we will show that lb(n!) can be automatic for some b.
Lemma 3.6. If b = pa, p ∈ P, a ∈ N then the sequence (`b(n!))n∈N is
b-automatic.
Proof. First, we notice that `b (xy) = `b (`b (x) `b (y)), so
`b ((bn)!) = `b
`b(n!) bn∏
i=n+1
`b (i)
 .
Automaticity of the sequence of the last nonzero digits of n! in a fixed base 289
Because `b (bx) = `b (x) we can rewrite the product in the following way:
`b ((bn)!) = `b
`b(n!) bn∏
i=1
b-i
`b (i)
 = `b
`b(n!) n∏
i=1
`b
j=b−1∏
j=1
j
 .
We denote mi = `b (i!) and obtain `b ((bn)!) = `b
(
`b(n!)mnb−1
)
. Now we
take the string of digits n1n2 . . . nl = [n]b and obtain the following formula
`b(n!) = `b (〈n1n2 . . . nl〉b!)
= `b ((b · 〈n1n2 . . . nl−1〉b + nl)!) =
= `b
(
(b · 〈n1n2 . . . nl−1〉b)!
nl∏
i=1
(b · 〈n1n2 . . . nl−1〉b + i)
)
= `b (`b ((b · 〈n1n2 . . . nl−1〉b)!) `b (nl!))
= `b
(
mnl`b (〈n1n2 . . . nl−1〉b!)m
〈n1n2...nl−1〉b
b−1
)
,
which by iteration leads to
(3.1) `b (〈n1n2 . . . nl〉b!) = `b
(
mn1mn2 . . .mnl`b
(
mrb−1
))
,
where r = 〈n1n2 . . . nl−1〉b + . . .+ 〈n1n2〉b + 〈n1〉b. Now, by Euler’s Theorem
m
ϕ(b)
b−1 ≡ m
pa−pa−1
b−1 ≡ 1 (mod b) so we only need to know the value of r
(mod pa − pa−1).
r =
l−1∑
i=1
bi−1 l−i∑
j=1
nj

=
l−1∑
j=1
nj +
l−1∑
i=2
pa(i−1) l−i∑
j=1
nj

≡
l−1∑
i=1
ni + pa−1
l−1∑
i=2
 l−i∑
j=1
nj
 (mod (pa − pa−1)).
Hence
(3.2) r ≡
l−1∑
i=1
ni + pa−1
l−2∑
i=1
(l − 1− i)ni (mod (pa − pa−1)).
Finally, we can define an automaton (Q,Σb, ρ, q0,∆, τ) generating the se-
quence (`b(n!))n∈N in the following way:
• the input alphabet Σb = {0, 1, 2, . . . , b− 1};
• the output alphabet ∆ = {1, 2, . . . , b− 1};
• the set of states Q = ∆× Σpa−pa−1 × Σp−1;
• the initial state q0 = (1, 0, 0);
290 Eryk Lipka
• the output function τ(u, v, w) = `b(u ·mv+p
a−1w
b−1 );
• the transition function ρ ((u, v, w) , s)
=
(
`b (u ·ms) , (v + s) (mod(pa − pa−1)), (w + v) (mod(p− 1))
)
.
With this definition we have ρ (q0, [n]b) = (u, v, w) where
• u = `b (mn1mn2 . . .mnl);
• v =
∑l−1
i=1 ni (mod
(
pa − pa−1
)
);
• w =
∑l−2
i=1 (l − 1− i)ni (mod (p− 1)).
Hence using equations (3.1) and (3.2) we see, that `b(n!) = τ (u, v, w). 
Now we are ready to prove the main result.
Theorem 3.7. Let b = pa11 p
a2
2 . . . with a1 (p1 − 1) ≥ a2 (p2 − 1) ≥ . . .. The
sequence (`b(n!))n∈N is p1-automatic if a1 (p1 − 1) > a2 (p2 − 1) or b = p
a1
1
and not automatic otherwise.
Proof. Let n 0. For b = p1a1 the sequence is b-automatic from Lemma 3.6;
by Lemma 2.8 it is also p1-automatic. If b has more than one prime factor
and a1 (p1 − 1) > a2 (p2 − 1) we take b′ = bpa11
so p1 - b′. From Proposi-
tion 3.1 and the definition of vb′ we have
vb′(n!) = min
i>1
vpaii
(n!)
= min
i>1
⌊
n− spi(n)
ai (pi − 1)
⌋
> min
i>1
⌊
n− (logpi(n) + 1) (pi − 1)
ai (pi − 1)
⌋
≥ min
i>1
⌊
n
ai (pi − 1)
− logpi(n)− 1
⌋
>
⌊
n
a1 (p1 − 1)
⌋
>
⌊
n− sp1(n)
a1 (p1 − 1)
⌋
= vpa11 (n!),
which leads to b′|`b(n!). Thus `b(n!) ∈ {b′, 2b′, 3b′, . . . (pa11 − 1) b′}, so the
value of `b(n!) can be computed from the value of `b(n!) (mod pa11 ). Now,
we can be sure that there exist integers c, d satisfying the equations
n! = b(vb(n!))`b(n!) + b(vb(n!)+1)c
n! = p
a1
(
v
p
a1
1
(n!)
)
1 `pa11
(n!) + p
a1
(
v
p
a1
1
(n!)+1
)
1 d.
We notice that vb(n!) = mini≥1 vpaii (n! = vpa11 (n!), which leads to
b
(
v
p
a1
1
(n!)
)
`b(n!) + b
(
v
p
a1
1
(n!)+1
)
c = p
a1
(
v
p
a1
1
(n!)
)
1 `pa11
(n!) + p
a1
(
v
p
a1
1
(n!)+1
)
1 d.
Automaticity of the sequence of the last nonzero digits of n! in a fixed base 291
After division of the above equality by p
a1
(
v
p
a1
1
(n!)
)
1 we obtain the following(
b′
)(v
p
a1
1
(n!)
)
`b(n!) + pa11
(
b′
)(v
p
a1
1
(n!)+1
)
c = `pa11 (n!) + p
a1
1 d.
From the above equation, we can notice that `b(n!)(b′)
v
p
a1
1
(n!)
≡ `pa11 (n!)
(mod pa11 ), hence to finish this part of the proof we just need to con-
struct a p1-automaton that returns the value of vpa11 (n!) (mod ϕ(p
a1
1 )). By
Proposition 3.1 this value can be computed from (n− sp1(n)) (mod ϕ(p
a1
1 )·
a1(p1 − 1)), and this expression is p1-automatic as we already mentioned
in Example 2.5.
Now, in the last case, when a1 (p1 − 1) = a2 (p2 − 1), let I = {i :
ai (pi − 1) = a1 (p1 − 1)}. Without loss of generality we can assume p1 =
maxi∈I pi. By Legendre’s formula (Proposition 3.1) we have
max
i∈I
spi(n) = sp1(n)⇒ vpa11 (n!) = mini∈I vp
ai
i
(n!) = vb(n!)
⇒ pa11 - `b(n!),
max
i∈I
spi(n) > a1(p1 − 1) + sp1(n)⇒ vpa11 (n!) > mini∈I vp
ai
i
(n!) = vb(n!)
⇒ pa11 |`b(n!).
Hence, by Theorem 3.5, there is no finite automaton that can, for given
n, tell whether pa11 divides `b(n!) or not. This completes the proof, as any
finite automaton generating the sequence (`b(n!))n∈N should distinguish
those two sets. 
References
[1] J.-P. Allouche & J. Shallit, Automatic Sequences. Theory, Applications, Generalizations,
Cambridge University Press, 2003.
[2] J. Byszewski & J. Konieczny, “A density version of Cobham’s theorem”, https://arxiv.
org/abs/1710.07261, 2017.
[3] J.-M. Deshouillers, “A footnote to The least non zero digit of n! in base 12”, Unif. Distrib.
Theory 7 (2012), no. 1, p. 71-73.
[4] ———, “Yet another footnote to The least non zero digit of n! in base 12”, Unif. Distrib.
Theory 11 (2016), no. 2, p. 163-167.
[5] J.-M. Deshouillers & I. Ruzsa, “The least non zero digit of n! in base 12”, Publ. Math. 79
(2011), no. 3-4, p. 395-400.
[6] A.-M. Legendre, Théorie des nombres, Firmin Didot frères, 1830.
[7] C. L. Stewart, “On the representation of an integer in two different bases”, J. Reine Angew.
Math. 319 (1980), p. 63-72.
Eryk Lipka
Potok 448
38-404 Krosno, Poland
Current address Institute of Mathematics
Pedagogical University of Cracow; Poland
E-mail: eryklipka0@gmail.com


Automaticity of the sequence of the last nonzero digits of $n!$ in a fixed base

Taylor Dupuy

David Zureick-Brown

Deligne–Illusie Classes as Arithmetic Kodaira–Spencer Classes

Yuval Z. Flicker

Conjugacy classes of finite subgroups of $\protect \mathrm{SL}(2,F)$, $\protect \mathrm{SL}(3,\protect \bar{F})$

Christophe Reutenauer

On the Stern–Brocot expansion of real numbers

Gérald Tenenbaum

Gérald TENENBAUM
Normal largest gap between prime factors
Tome 31, no 3 (2019), p. 747-749.
<http://jtnb.centre-mersenne.org/item?id=JTNB_2019__31_3_747_0>
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Centre de diffusion des revues académiques de mathématiques
http://www.centre-mersenne.org/
Journal de Théorie des Nombres
de Bordeaux 31 (2019), 747–749
Normal largest gap between prime factors
par Gérald TENENBAUM
Résumé. Désignons par {pj(n)}ω(n)j=1 la suite croissante des facteurs premiers
distincts d’un entier n. Nous explicitions les détails de la preuve d’un énoncé
d’Erdős impliquant que, pour toute fonction ξ(n) tendant vers l’infini avec n,
nous avons
f(n) := max
16j<ω(n)
log
(
log pj+1(n)
log pj(n)
)
= log3 n+O(ξ(n))
pour presque tout entier n.
Abstract. Let {pj(n)}ω(n)j=1 denote the increasing sequence of distinct prime
factors of an integer n. We provide details for the proof of a statement of
Erdős implying that, for any function ξ(n) tending to infinity with n, we have
f(n) := max
16j<ω(n)
log
(
log pj+1(n)
log pj(n)
)
= log3 n+O(ξ(n))
for almost all integers n.
In private correspondence [4], E. Sofos asked for normal upper and lower
bounds for the arithmetic function
f(n) := max
16j<ω(n)
log
(
log pj+1(n)
log pj(n)
)
,
where {pj(n)}ω(n)j=1 denotes the increasing sequence of distinct prime fac-
tors of an integer n. The answer to this question is actually provided by a
statement of Erdős in [1], where it is asserted without proof that, for any
c > 0, the density of the set of those integers n such that ef(n) > c log2 n
is 1 − e−1/c. (Here and in the sequel, logk denotes the k-fold iterated log-
arithm.) Indeed, given any function ξ(n) tending to infinity and letting c
tend sufficiently slowly to zero, we infer that, for almost all n, we have
f(n) > log3 n − ξ(n), while by letting c tend slowly to infinity we obtain
that f(n) 6 log3 n + ξ(n) also holds for a set of asymptotic density 1.
Interesting related results appear in Erdős’ articles [2] and [3].
In this short note, we provide the not so obvious details of the proof, in
the spirit of the paper [1].
Manuscrit reçu le 22 février 2019, accepté le 15 juin 2019.
2010 Mathematics Subject Classification. 11N56, 14G42.
Mots-clefs. Distribution of prime factors, normal order, largest gap.
748 Gérald Tenenbaum
Theorem 1 (Erdős [1]). Let c > 0. Then the inequality
f(n) > log3 n− log(1/c)
holds on a set of integers n of asymptotic density 1− e−1/c. In particular,
given any function ξ(n)→∞, we have
log3 n− ξ(n) 6 f(n) 6 log3 n+ ξ(n)
for almost all integers n.
Proof. Let x be a large parameter, write Z := c log2 x, and let χp(n) denote
the indicator function of the set of those integers n 6 x that are divisible
by the prime p but by no prime q such that p < q 6 pZ . For squarefree m,
put χm(n) :=
∏
p|m χp(n). Taking into account the slow growth of log2 x, it
is sufficient to show that, as x tends to infinity,
(1) N(x) := |{n 6 x : p | n, p 6 x1/Z ⇒ χp(n) = 0}| ∼ e−1/cx.
Indeed, n 6 x is counted by N(x) if, and only if, pj+1(n) 6 pj(n)Z when-
ever 1 6 j < ω(n). Let P+(n) (resp. P−(n)) denote the largest (resp. the
smallest) prime factor of an integer n, with the convention that P+(1) = 1
(resp. P−(1) = ∞), and let µ designate the Möbius function. By the
inclusion–exclusion principle (see, e.g., [6, p. 39]) or the Möbius inversion
formula, we thus have, for x > 1,
(2) N(x) =
∑
m6x
m∈M
µ(m)
∑
n6x
χm(n),
where M denotes the set of all squarefree integers m 6 x such that
P+(m) 6 x1/Z , and pj+1(m) > pj(m)Z (1 6 j < ω(m)).
Moreover, as is familiar in Brun’s sieve method, we obtain a lower or an
upper bound for the left-hand side by restricting the outer sum to integersm
whose number of prime factors is at most equal to an odd or an even bound.
Let k > 1 be fixed. For each m ∈M, define Pm :=
∏
p|m
∏
p<q6pZ q. Then
(3) Nk(x) :=
∑
m∈M
ω(m)=k
∑
n6x
χm(n) =
∑
m∈M
ω(m)=k
∑
ν6x/m
(ν,Pm)=1
1.
From a standard sieve result (see, e.g., the lemma in [5]), we infer that the
inner sum is
(4) ∼ xϕ(Pm)
mPm
= x
mZk
{
1 +O
( 1
logP−(m)
)
+ o(1)
}
provided P+(m) 6 xo(1/Z). The contribution of the remaining integers m
will be treated as an error term. Indeed, in this latter case we may plainly
Normal largest gap between prime factors 749
assume that xεx < P+(m)Z 6 x, with εx tending to 0 arbitrarily slowly.
Then the inner sum is classically (see, e.g., [6, Exercise 85])
 xϕ(Pm)
mPm
 x
mZk
and the corresponding contribution to Nk(x) is
x
Zk
∑
xεx/Z<p6x1/Z
1
p
∑
P+(h)6x
ω(h)=k−1
1
h
 x log(1/εx)log2 x
= o(x).
Carrying (4) back into (3) and summing over m, we obtain, for k > 0,
Nk(x) =
x
Zk
∑
pZk 6x
pZj 6pj+1 (16j<k)
1 +O(1/ log p1) + o(1)
p1 · · · pk
∼ x(log2 x)
k
k!Zk ∼
x
ckk! ·
Thus, for arbirary ` > 1, we have
x
∑
06k62`+1
(−1)k
ckk! + o(x) 6 N(x) 6 x
∑
06k62`
(−1)k
ckk! + o(x),
and the required result follows by selecting ` arbitrarily large. 
Acknowledgements. The author wishes to express warm thanks to Régis
de la Bretèche for valuables discussions on this matter.
References
[1] P. Erdős, “Some remarks on prime factors of integers”, Can. J. Math. 11 (1959), p. 161-167.
[2] ———, “On some properties of prime factors of integers”, Nagoya Math. J. 27 (1966),
p. 617-623.
[3] ———, “On the distribution of prime divisors”, Aequationes Math. 2 (1969), p. 177-183.
[4] E. Sofos, private e-mail message, August 30, 2018.
[5] G. Tenenbaum, “Cribler les entiers sans grand facteur premier”, Philos. Trans. R. Soc.
Lond., Ser. A 345 (1993), p. 377-384.
[6] ———, Introduction to analytic and probabilistic number theory, 3rd ed., Graduate Studies
in Mathematics, vol. 163, American Mathematical Society, 2015.
Gérald Tenenbaum
Institut Élie Cartan
Université de Lorraine
BP 70239
54506 Vandœuvre-lès-Nancy Cedex, France
E-mail: gerald.tenenbaum@univ-lorraine.fr
URL: http://www.iecl.univ-lorraine.fr/~Gerald.Tenenbaum/


Normal largest gap between prime factors

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Automaticity of the sequence of the last nonzero digits of $n!$ in a fixed base

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Normal largest gap between prime factors

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