ON THE NUMBER OF DISTINCT PRIME FACTORS
OF A SUM OF SUPERPOWERS
arXiv:1511.08784v1 [math.NT] 27 Nov 2015
PAOLO LEONETTI AND SALVATORE TRINGALI
Abstract. Given k, ℓ ∈ N+ , let xi,j be, for 1 ≤ i ≤ k and 0 ≤ j ≤ ℓ, some fixed integers.
P
Q
j
+
Then, define sn := ki=1 ℓj=0 xn
i,j for every n ∈ N .
We prove that there exist infinitely many n for which the number of distinct prime factors
of sn is greater than the super-logarithm of n to base C, for some real constant C > 1, if and
Q
(2n)i
only if there do not exist nonzero integers a0 , b0 , . . . , aℓ , bℓ such that s2n = ℓi=0 ai
and
Qℓ
(2n−1)i
s2n−1 = i=0 bi
for all n. (In fact, we prove a slightly more general result in the same
spirit, where the numbers xi,j are rational.)
In particular, for c1 , x1 , . . . , ck , xk ∈ N+ the number of distinct prime factors of the sum
+
n
c 1 xn
1 + · · · + ck xk is bounded, as n ranges over N , if and only if x1 = · · · = xk .
1. Introduction
Given k, ℓ ∈ N+ , let xi,j be, for 1 ≤ i ≤ k and 0 ≤ j ≤ ℓ, some fixed rationals. Then, consider
the Q-valued sequence (sn )n≥1 obtained by taking
sn :=
k Y
ℓ
X
j
xni,j
(1)
i=1 j=0
for every n ∈ N+ (see Section 2 for notation used, but not defined, in this introduction); we
refer to sn a sum of superpowers of degree ℓ, for it is more general, and has typically (i.e., except
for trivial cases) a much faster growth rate, than a sum of powers. Note that (sn )n≥1 includes
as a special case any integer sequence (tn )n≥1 of general term
ℓi
k Y
X
f
yi,ji,j
(n)
,
(2)
i=1 j=1
where, for each i = 1, . . . , k, we let ℓi ∈ N+ and yi,1 , . . . , y1,ℓi ∈ Q \ {0}, while fi,1 , . . . , fi,ℓi
are polynomials in one variable with integral coefficients. Conversely, sequences of the form (1)
can be viewed as sequences of the form (2), the latter being prototypical of scenarios where
polynomials are replaced with more general functions N+ → Z (see also Section 4).
Now, let ω(x) denote, for each x ∈ Z \ {0}, the number of distinct prime divisors of x, and
define ω(0) := ∞. Then, for x ∈ Z and y ∈ N+ we let ω(xy −1 ) := ω(xδ −1 ) + ω(yδ −1 ), where δ
is the greatest common divisor of x and y.
2010 Mathematics Subject Classification. Primary: 11A05, 11A41, 11A51. Secondary: 11R27.
Key words and phrases. Integer sequences, number of distinct prime factors, sum of powers, super-logarithm
and tetration.
2
P. Leonetti and S. Tringali
In addition, given n ≥ 2 and C > 1, we write slogC (n) for the super-logarithm of n to base C,
that is the largest integer l ≥ 0 for which C ⇈ l ≤ n, where C ⇈ 0 := 1 and C ⇈ l := C C⇈(l−1)
for l ≥ 1, cf. [5] for the notation; note that slogC (n) → ∞ as n → ∞.
The main goal of the present paper is to provide necessary and sufficient conditions for the
boundedness of the sequence (ω(sn ))n≥1 . More precisely, we have:
Main Theorem. There exists a base C > 1 such that ω(sn ) > slogC (n) for infinitely many n,
hence (ω(sn ))n≥1 is an unbounded sequence, if and only if there do not exist nonzero rational
Qℓ
Qℓ
(2n−1)j
(2n)j
for all n.
and s2n−1 = j=0 bj
numbers a0 , b0 , . . . , aℓ , bℓ such that s2n = j=0 aj
The theorem will be proved in Section 3 (working with rationals may look pointless at first,
but will eventually result into a simplification of the arguments), along with the following:
Corollary. Let c1 , . . . , ck ∈ Q+ and x1 , . . . , xk ∈ Q \ {0}. Then, (ω(c1 xn1 + · · · + ck xnk ))n≥1 is
a bounded sequence only if |x1 | = · · · = |xk |, and this condition is also sufficient provided that
Pk
−1
is the sign of xi .
i=1 εi ci 6= 0, where, for each i ∈ J1, kK, εi := xi · |xi |
Results in the spirit of the above theorem have been proved by various authors in the special
case of Z-valued sequences raising from the solution of non-degenerate linear homogeneous
recurrence equations with (constant) integer coefficients of order ≥ 2, namely in relation to a
sequence (un )n≥1 of general term
un = αn1 f1 (n) + · · · + αnh fh (n),
(3)
where α1 , . . . , αh are the nonzero (and pairwise distinct) roots of the characteristic polynomial
of the recurrence under consideration and f1 , . . . , fh nonzero polynomials in one variable with
coefficients from the smallest field extension of the rational field containing α1 , . . . , αh , see [10,
Theorem C.1]. In this respect, we recall that a recurrence is said to be non-degenerate if its
characteristic polynomial has at least two distinct nonzero complex roots (so that h ≥ 2 in the
above) and the ratio of any two distinct characteristic roots is not a root of unity.
More specifically, it was proved by van der Poorten and Schlickewei [14] and, independently,
by Evertse [4, Corollary 3], using Schlickewei’s p-adic analogue of Schmidt’s subspace theorem
[8], that the greatest prime factor of un tends to ∞ as n → ∞. In a similar note, effective lower
bounds on the greatest prime divisor and on the greatest square-free factor of a sequence of type
(3) were obtained under mild assumptions by Shparlinski [11] and Stewart [12, 13], based on
variants of Baker’s theorem on linear forms in the logarithms of algebraic numbers [2].
On the other hand, Luca has shown in [6] that if (vn )n≥1 is a sequence of rational numbers
satisfying a recurrence of the form
g0 (n)vn+2 + g1 (n)vn+1 + g2 (n)vn = 0
for n ∈ N+ ,
where g0 , g1 and g2 are univariate polynomials over the rational field and not all zero, and
(vn )n≥n0 is not binary recurrent (viz., a solution of a linear homogeneous second-order recurrence
equation with integer coefficients) for some n0 ∈ N+ , then there exists a real constant c > 0 such
Distinct prime factors of a sum of superpowers
3
that the product of the numerators and denominators (in the reduced fraction) of the nonzero
rational terms of the finite sequence (vn )1≤i≤n has at least c log(n) prime factors as n → ∞.
With this said, it is perhaps worth mentioning that the present manuscript has been originally
motivated by a (so-far fruitless) attempt by the authors to obtain a “suitable generalization” of
the following classical result due to Zsigmondy [15] to sums of powers:
Zsigmondy’s theorem. Let a, b, n ∈ N+ be such that a > b, n ≥ 2, and neither (a, b, n) =
(2, 1, 6), nor a + b is a power of 2 and n = 2. Then, there exists a prime p such that p | an − bn ,
but p ∤ am − bm for m < n.
Continuing with the notation and the hypotheses as in the above statement, we have, in fact,
that ω(an − bn ) ≥ σ0 (n) − 2 for all n, with σ0 (n) the number of (positive integer) divisors of n.
To see why, let d > 1 be a divisor of n. By Zsigmondy’s theorem, there then exists a prime
p such that p | ad − bd , but p ∤ ai − bi for 1 ≤ i < d, unless we have either (a, b, d) = (2, 1, d), or
a + b is a power of 2 and d = 2. This yields the above inequality, because ad − bd | an − bn and
(a, b) = (2, 1) only if a + b is not a power of two. On the other hand, it is known, e.g., from [9]
P
that n1 ni=1 σ0 (i) is asymptotic to log(n) as n → ∞.
It follows that there exist a constant c ∈ R+ and infinitely many n for which ω(an − bn ) >
c log(n), which can be viewed as an analogue of our main theorem (though much stronger than
the latter in the special case of the sequences to which Zsigmondy’s theorem applies) and served
as a starting point for our investigations.
2. Notation and conventions
Through the paper, R, Q, Z, and N are, respectively, the sets of reals, rationals, integers,
and nonnegative integers. Each of these sets is endowed with its usual addition, multiplication,
and (total) order ≤, and we assume they have been constructed in a way that N ⊆ Z ⊆ Q ⊆ R.
Unless noted otherwise, the letters h, i, j, l and κ, with or without subscripts, will stand for
nonnegative integers, the letters m and n for positive integers, the letters p and q for (positive
rational) primes, and the letters A, B and C for real numbers.
For a, b ∈ R ∪ {∞} we write [a, b] for the closed interval {x ∈ R ∪ {∞} : a ≤ x ≤ b}, ]a, b[ for
the open interval [a, b] \ {a, b}, and Ja, bK for [a, b] ∩ Z (we use ∞ in place of +∞); moreover, for
a set X ⊆ R we let X + := X ∩ ]0, ∞[.
We refer to [1] and [3], respectively, for basic aspects of number theory and real analysis
(including notation and terms not defined here). In particular, the only topology considered on
R will be the usual topology, and we will use without explicit mention some of the most basic
properties of the upper limit of a real sequence, see especially [3, Theorem 18.3].
We denote by | · | the usual absolute value on R and by log the natural logarithm, and assume
the convention that an empty sum (of real numbers) is 0 and an empty product is equal to 1.
3. Proofs
Proof of Main Theorem. The “only if” part is straightforward, and the claim is trivial if at least
one of the sequences (s2n )n≥1 and (s2n−1 )n≥1 is eventually zero.
4
P. Leonetti and S. Tringali
Therefore, we can just focus on the two cases below, in each of which we have to prove that
there exists B > 1 such that ω(sn ) > slogB (n) for infinitely many n.
Q
(2n)j
for all n, so we have
Case (i): There do not exist a0 , . . . , aℓ ∈ Q such that s2n = ℓj=0 aj
in particular that k ≥ 2 and sn 6= 0 for infinitely many n. Notice also that |xi,j | 6= 1 for some
P
(i, j) ∈ J1, kK × J1, ℓK, as otherwise s2n = ki=1 xi,0 .
Without loss of generality, we assume that xi,j 6= 0 for all (i, j) ∈ J1, kK × J0, ℓK, and actually
Qℓ
Qℓ
j
(2n)j
that xi,j > 0 for j 6= 0, since j=0 xi,j = xi,0 · j=1 |xi,j |(2n) .
Accordingly, we may also assume, as we do, that (xi,1 , . . . , xi,ℓ ) ≺ (xj,1 , . . . , xj,ℓ ) for 1 ≤ i <
j ≤ k, where ≺ denotes the binary relation on Rℓ defined by taking (u1 , . . . , uℓ ) ≺ (v1 , . . . , vℓ )
if and only if |ui | < |vi | for some i ∈ J1, ℓK and |uj | = |vj | for i < j ≤ ℓ: This is because the
ℓ-tuples (xi,1 , . . . , xi,ℓ ) cannot be equal to each other for all i ∈ J1, kK, and on the other hand,
if two of these tuples are equal then we can add up some terms in (1) so as to obtain a sum of
superpowers of degree ℓ, but with fewer summands.
Now, for each (i, j) ∈ J1, kK × J0, ℓK let αi,j , βi,j ∈ Z be such that αi,j > 0 and xi,j = α−1
i,j βi,j ,
and consequently set x̃i,j := αj xi,j , where αj := α1,j · · · αk,j ; note that x̃i,j is a nonzero integer,
Pk Qℓ
Qℓ
j
j
and x̃i,j > 0 for j 6= 0. Then, define un := i=1 j=0 x̃ni,j and vn := j=0 αjn .
It is clear that (un )n≥1 and (vn )n≥1 are integer sequences and (x̃i,1 , . . . , x̃i,ℓ ) ≺ (x̃j,1 , . . . , x̃j,ℓ )
for 1 ≤ i < j ≤ k. Moreover, sn = un vn−1 and, hence, ω(sn ) ≥ ω(un ) − ω(vn ) = ω(un ) − ω(v1 )
for all n, which shows it is sufficient to prove the existence of a base B > 1 such that ω(u2n ) >
slogB (2n) for infinitely many n, and ultimately implies, along with the rest, that we can further
assume (again, as we do) that xi,j is an integer for all (i, j) ∈ J1, kK × J0, ℓK.
Putting it all together, it follows that xi,j ≥ 2 for some (i, j) ∈ J1, kK × J1, ℓK, and in addition
there exists N ∈ N+ such that
ℓ
X Y
nj x
(4)
i,j 6= 0 for all n ≥ N and ∅ 6= I ⊆ J1, kK.
i∈I j=0
We claim that it is enough, as well, to assume (once more, as we do) that δ0 = · · · = δℓ = 1,
where for each j ∈ J0, ℓK we let δj be the greatest common divisor of x1,j , . . . , xk,j .
In fact, define, for 1 ≤ i ≤ k and 0 ≤ j ≤ ℓ, ξi,j := δj−1 xi,j , and let (wn )n≥1 and (s̃n )n≥1 be
Pk Qℓ
Qℓ
j
nj
, respectively. Then s2n =
the integer sequences of general term j=0 δjn and i=1 j=0 ξi,j
+
w2n s̃2n , and hence ω(s2n ) ≥ ω(s̃2n ), for every n ∈ N . On the other hand, there cannot exist
Qℓ
Qℓ
j
(2n)j
for all n, as otherwise s2n = j=0 (δj ãj )(2n) (which
ã0 , . . . , ãℓ ∈ Z such that s̃2n = j=0 ãj
is absurd). This leads to the claim.
Q
Q
With the above in mind, let P be the set of all (positive) prime divisors of z := ki=1 ℓj=1 xi,j ,
and note that P is finite and nonempty, as the previous considerations yield |z| ≥ 2.
Next, let υp (x) denote, for a prime p and a nonzero integer x, the exponent of the largest
power of p dividing x. Then, for every n ∈ N+ we can write


k
X
Y (i)
xi,0
s2n =
pep (2n) ,
(5)
i=1
p∈P
5
Distinct prime factors of a sum of superpowers
Pℓ
(i)
where for each i = 1, . . . , k and p ∈ P we let ep be the function N+ → N : n 7→ j=0 nj υp (xi,j ).
Since we assumed δ0 = · · · = δℓ = 1, it is easily seen that for every p ∈ P there are i, j ∈ J1, kK
(i )
(i)
(j)
(i)
for which ep 6= ep , and there exist ip ∈ J1, kK and np ≥ N such that ep p (n) < ep (n) for all
(ip )
(i)
n ≥ np and i ∈ J1, kK for which ep 6= ep . Let nP := maxp∈P np (recall that P is a nonempty
(i )
(i)
(i)
finite set), and for each p ∈ P and i ∈ J1, kK define ∆ep := ep − ep p . Then set


k
Y (ip )
X
Y
(i)
xi,0
πn :=
(6)
pep (n) and σn :=
p∆ep (n) .
i=1
p∈P
p∈P
We observe that |s2n | = π2n · |σ2n |; furthermore, if n ≥ nP then σn is an integer, and actually
a nonzero integer by (4). Thus, ω(s2n ) ≥ ω(σ2n ) for n ≥ nP , and it will suffice to show that for
some B > 1 there exist infinitely many n such that ω(σ2n ) > slogB (2n).
On the other hand, having assumed that (x1,1 , . . . , x1,ℓ ) ≺ · · · ≺ (xk,1 , . . . , xk,ℓ ) and xi,j > 0
for all (i, j) ∈ J1, kK × J1, ℓK yields, together with (5), that
j
lim
n→∞
Y
p
(k)
(i)
ep
(n)−ep
(n)
= lim
n→∞
p∈P
n
ℓ Y
xk,j
xi,j
j=1
= ∞ for each i ∈ J1, k − 1K,
(7)
and consequently
|sn | ∼ |xk,0 | ·
ℓ
Y
j
xnk,j = |xk,0 | ·
j=1
Y
(k)
pep
(n)
as n → ∞
(8)
p∈P
and
|σ2n | =
Y
(k)
|s2n |
∼ |xk,0 | ·
p∆ep (2n)
π2n
as n → ∞.
(9)
p∈P
We want to show that the sequence (σ2n )n≥1 is eventually (strictly) increasing in absolute value.
(k)
Lemma 1. There exists p ∈ P such that ∆ep (n) → ∞ as n → ∞.
(k)
(i )
Proof of Lemma 1. Suppose the contrary is true. Then, for each p ∈ P we must have ep = ep p ,
(k)
since ∆ep (n) can be understood as a homogeneous polynomial with integral coefficients in the
(i )
(k)
(k)
variable n and ∆ep (n) = ep (n) − ep p (n) ≥ 0 for n ≥ nP . Therefore, we get by (7) that
Y (ip )
Y (i)
Y (k)
Y (ip )
pep (n) ≤
pep (n) ≤
pep (n) =
pep (n)
p∈P
p∈P
p∈P
p∈P
(i )
(i)
for all n ≥ nP and i ∈ J1, kK. But this is absurd, as it implies that ep = ep p for all p ∈ P and
(ip )
Q
i ∈ J1, kK, and hence, in view of (5), s2n = (x1,0 + · · · + xk,0 ) · p∈P pep (2n) for all n.
With this in hand, let A > |xk,0 | ·
Qℓ
j=1 xk,j . Then, using that
(k)
∆ep is a polynomial function
(k)
∆ep
is eventually nonde(k)
creasing for every p ∈ P (recall that
and ∆ep (n) ≥ 0 for large
n), we get from (8) and Lemma 1 that there exists n0 ≥ nP such that, for every n ≥ n0 ,
2
2
σ2n
≤ s2n
< A(2n)
ℓ
and |σ2n+2 | > |σ2n |.
(10)
6
P. Leonetti and S. Tringali
Now, denote by Qn the set of all prime divisors of σn and let Q⋆n := Qn \ P; we note that Qn
is finite for n ≥ n0 (recall that σn 6= 0 for n ≥ nP ). Next, let
λ := max υp (σ2n0 ) + max max ∆ep(i) (2n0 ),
p∈P 1≤i≤k
p∈P
and then
α := k · max |xi,0 | ·
1≤i≤k
Y
pλ
and β :=
p∈P
Y
pα−1 (p − 1).
p∈P
Lastly, suppose for a fixed κ ∈ N that we have determined even integers r0 , . . . , rκ ∈ N+ such
Q
that n0 ≤ r0 ≤ · · · ≤ rκ , and define βκ := β · p∈Q⋆r pυp (σrκ ) (p − 1).
κ
By taking r0 := 2n0 and rκ+1 := 2βκ + rκ , the recursive construction we have just described
then yields a (strictly) increasing sequence (rκ )κ≥0 of even integers ≥ n0 with the property that,
however we choose a prime p ∈ P and an index i ∈ J1, kK, it holds
α−1
(i)
∆e(i)
(q − 1) for all q ∈ P
p (rκ+1 ) ≡ ∆ep (rκ ) mod q
(11)
υq (σrκ )
∆ep(i) (rκ+1 ) ≡ ∆e(i)
(q − 1) for all q ∈ Q⋆rκ ,
p (rκ ) mod q
(12)
and
(i)
where we use that rκ+1 ≡ rκ mod m whenever m | βκ and, as was already mentioned, ∆ep is
essentially an integral polynomial. In particular, (11) and a routine induction imply that
(i)
α−1
∆e(i)
(q − 1) for all p, q ∈ P, i ∈ J1, kK and κ ∈ N.
p (rκ ) ≡ ∆ep (2n0 ) mod q
Also, we get from (10) and rκ ≥ 2n0 that there exists B > A such that, for all κ,
Y
ℓ
ℓ
rκ+1 ≤ rκ + 2β ·
pυp (σrκ ) (p − 1) ≤ rκ + 2β σr2κ < rκ + 2βArκ < B rκ .
(13)
(14)
p∈Qrκ
Based on these premises, we now prove a few more lemmas; to ease notation, we will denote by
(i )
(i)
Ip , for each p ∈ P, the set of all indices i ∈ J1, kK such that ep 6= ep p , and let Ip⋆ := J1, kK \ Ip .
Lemma 2. Qrκ ⊆ Qrκ+1 for every κ.
(i)
Proof of Lemma 2. Pick κ ∈ N and q ∈ Qrκ . If i ∈ Ip , then ∆ep (n) = 0 for all n, and hence
(i)
(i)
p∆ep (n) = 1. If, on the other hand, i ∈ Ip⋆ , then ∆ep (n) > 0 for n ≥ nP , with the result that
(i)
(i)
(i)
p∆ep (n) ≡ 0 mod q if q = p, and p∆ep (n) ≡ pm mod q if p 6= q and ∆ep (n) ≡ m mod (q − 1),
in the light of Fermat’s little theorem.
So putting it all together, we get from (11), (12) and rκ+1 > rκ ≥ 2n0 > nP that
(i)
p∆ep
(rκ+1 )
(i)
≡ p∆ep
(rκ )
mod q
for all p ∈ P and i ∈ J1, kK,
which in turn implies that




k
k
Y
X
Y
X
(i)
(i)
xi,0
xi,0
p∆ep (rκ )  ≡ σrκ ≡ 0 mod q.
p∆ep (rκ+1 )  ≡
σrκ+1 ≡
i=1
p∈P
i=1
p∈P
This concludes the proof, by the arbitrariness of κ ∈ N and q ∈ Qrκ .
Lemma 3. Let q ∈ P and κ ∈ N. Then υq (σrκ ) ≤ α − 1.
7
Distinct prime factors of a sum of superpowers
Proof of Lemma 3. The claim is straightforward if κ = 0, since r0 = 2n0 and υq (σ2n0 ) ≤ λ < α.
So assume for the rest of the proof that κ ≥ 1. Then, we have from (6) that




X
X
Y
Y
(i)
(i)
xi,0
xi,0
(15)
σn =
p∆ep (n)  for all n.
p∆ep (n)  +
i∈Iq
i∈Iq⋆
p∈P
q6=p∈P
If i ∈ Iq , n > 2n0 and n ≡ 2n0 mod β then q α divides
(i)
(i)
(i)
Q
p∈P
p∆ep
(n)
(i)
, because n | ∆ep (n) and
∆ep (n) 6= 0, hence α < β < β + 2n0 ≤ n ≤ ∆ep (n).
On the other hand, it is seen by induction that rκ ≡ 2n0 mod β (recall that rκ ≡ rκ−1 mod β).
Thus, we get from the above, equations (15) and (13), [1, Theorem 2.5(a)], and Euler’s totient
theorem that




X
Y
X
Y
(i)
(i)
xi,0
xi,0
σrκ ≡
p∆ep (rκ )  ≡
p∆ep (2n0 )  mod q α .
(16)
i∈Iq⋆
i∈Iq⋆
q6=p∈P
q6=p∈P
But ∅ 6= Iq⋆ ⊆ J1, kK, so it follows from (4) that


ℓ
X
Y
1 XY
j
(i)
(2n
)
0
∆ep (2n0 ) 

xi,j = xi,0
p
0<
π
⋆
2n0
⋆ j=0
≤ max |xi,0 | ·
1≤i≤k
q6=p∈P
i∈Iq
i∈Iq
X Y
p
(i)
∆ep
(2n0 )
≤ k · max |xi,0 | ·
1≤i≤k
i∈Iq⋆ q6=p∈P
Y
pλ = α < q α ,
p∈P
which, together with (16), yields that υq (σrκ ) < α.
Lemma 4. Let κ ∈ N+ and q ∈ Qrκ . Then υq (σrκ ) = υq (σrκ+1 ).
Proof of Lemma 4. If q ∈
/ P, then we have from (6) and (12), [1, Theorem 2.5(a)], and Euler’s
totient theorem that σrκ+1 ≡ σrκ mod q υq (σrκ )+1 , and we are done.
If, on the other hand, q ∈ P, then we get from Lemma 3 that υq (σr1 ) ≤ α − 1, which, along
with (16), entails that σrκ ≡ σr1 mod q υq (σr1 )+1 , and consequently υq (σrκ ) = υq (σr1 ).
At long last, we are almost there. In fact, since the sequence (σrκ )κ≥0 is (strictly) increasing
in modulus (as was mentioned before) and rκ is even and ≥ 2n0 for all κ, it follows from Lemmas
2-4 that ∅ 6= Qrκ ( Qrκ+1 , and hence ω(σrκ ) < ω(σrκ+1 ), for all κ ∈ N+ . By a routine induction,
this in turn implies that ω(σrκ ) ≥ κ for all κ.
On the other hand, if we let C := max(B ℓ ℓ, r1ℓ ), then we get from (14) and another induction
that rκℓ < C ⇈ κ for all κ ∈ N+ , which, together with the above considerations, ultimately leads
to slogC (rκ ) < κ ≤ ω(σrκ ), and hence to the desired conclusion.
Qℓ
(2n−1)j
for all n. Then,
Case (ii): There do not exist b0 , . . . , bℓ ∈ Q such that s2n−1 = j=0 bj
we are reduced to the previous case by taking
yi,j :=
ℓ
Y
h=j
(−1)h−j (h
j)
xi,h
for 1 ≤ i ≤ k and 0 ≤ j ≤ ℓ,
8
P. Leonetti and S. Tringali
and noticing that for every n ∈ N+ we have s2n−1 = t2n , where (tn )n≥1 is the integer sequence
Pk Q ℓ
nj
of general term i=1 j=0 yi,j
(we omit further details).
Proof of Corollary. Suppose to a contradiction that there are c1 , . . . , ck ∈ Q+ and x1 , . . . , xk ∈
Q \ {0} such that |xi | 6= |xj | for some i, j ∈ J1, kK and (ω(un ))n≥1 is bounded, where un :=
Pk
n
i=1 ci xi for all n, and let k the minimal positive integer for which this is pretended to be
true.
Then k ≥ 2, and there is no loss of generality in assuming, as we do, that |x1 | ≤ · · · ≤ |xk | 6=
|x1 |. Furthermore, we get from the main theorem of this paper that there must exist c, x ∈ Q+
such that u2n = cx2n . So now, we have two cases, each of which will lead to a contradiction
(the rest is trivial and we may omit details):
Pk
Case (i): x ≤ |xk |. We have cy 2n = i=1 ci yi2n for all n, where yi := |xi | · |xk |−1 for 1 ≤ i ≤ k
and y := x · |xk |−1 . Let h be the maximal index in J2, kK such that yh−1 < yk , which exists
because y1 < yk . Since 0 < y ≤ 1 and 0 < yi < 1 for 1 ≤ i < h, we find that
c · lim y 2n = ch + · · · + ck ,
n→∞
which is possible only if y = 1, as ch , . . . , ck > 0. But then c = c1 + · · · + ck , and consequently
Ph−1
2n
i=1 ci yi = 0 for all n, which is absurd, because h ≥ 2 and c1 , . . . , ch−1 > 0.
Pk
Case (ii): x > |xk |. Then c = i=1 ci zi2n for all n, where zi := |xi | · x−1 for 1 ≤ i ≤ k. But
Pk
this is still absurd, since z1 , . . . , zk ∈ ]0, 1[, and hence i=1 ci zi2n → 0 as n → ∞.
4. Closing remarks
We list here some questions we hope to pick up in future work: Let τ be a (strictly) increasing
function from N+ into itself. What can be said about the behavior of ω(sτ (n) ) as n → ∞? In
particular, is it true that lim supn→∞ ω(sτ (n) ) = ∞? Is it possible to obtain nontrivial bounds
on the greatest prime divisors of sτ (n) as n → ∞? And what about the asymptotic growth of
the average of the function R+ → N : x 7→ #{n ≤ x : ω(sτ (n) ) ≥ h} for a fixed h ∈ N+ ? (If S
is a set, we write #S for the cardinality of S.)
In this paper, we have considered the case where τ is the identity or, more in general, a
polynomial function (by the considerations made in relation to equation 2 in the introduction).
So it could be interesting to answer the above questions under the assumption that τ is, e.g., a
geometric progression, which however may be hard, as an affirmative answer would then imply
the existence of infinitely many composite Fermat numbers (to the best of our knowledge, still
a longstanding open problem).
On the other hand, the basic question addressed in the present manuscript has the following
algebraic generalization (we refer to [7, Ch. 1] for background on divisibility and related topics
in ring theory): Given a unique factorization domain F = (F, +, ·), let θi,j be, for 1 ≤ i ≤ k and
0 ≤ j ≤ ℓ, some fixed elements in F , and for x ∈ F let ω F (x) denote the number of non-associate
primes dividing x, where two nonzero elements in F are non-associate (in F) if their ratio is not
Pk Qℓ
nj
a unit of F. What can be said about the sequence (ϑn )n≥1 of general term i=1 j=0 θi,j
if
Distinct prime factors of a sum of superpowers
9
the sequence (ω F (ϑn ))n≥1 is bounded? More specifically, does anything in the lines of our main
theorem hold true?
Acknowledgments
During the writing of this paper, P.L. was supported by a PhD scholarship from Università
“Luigi Bocconi”, and S.T. by NPRP grant No. [5-101-1-025] from the Qatar National Research
Fund (a member of Qatar Foundation) and, partially, by the French ANR Project No. ANR12-BS01-0011. The statements made herein are solely the responsibility of the authors.
The authors are thankful to Carlo Sanna (Università di Torino, IT) for useful comments.
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Università L. Bocconi, via Roentgen 1 – 20136 Milano, Italy
E-mail address: [email protected]
Centre de mathématiques Laurent Schwartz, École polytechnique – 91128 Palaiseau, France
E-mail address: [email protected]
URL: http://www.math.polytechnique.fr/~tringali/