Integral points on elliptic curves associated with generalized twin primes

Abstract

This article is a continuation of our previous paper [9] concerning elliptic curves Ep,m : y 2 = x(x − 2 m)(x + p), where p and p + 2 m are primes. There we proved inter alia that E p,1 has at most two non-torsion integral points, and E p,2 has no such points. Now by using completely dierent methods, namely an analysis of local height functions, we try to get upper bounds for the number of integral points and for the number of multiples of such points on our curves for any m. In particular, we show that no even multiples of an integral point on Ep,m are also integral, and if Ep,m has rank 1 and p ≡ 3 (mod 4) then there are at most twelve non-torsion integral points in the union of the non-identity component and the certain subset of the identity component.

Full text

INTEGRAL POINTS ON ELLIPTIC CURVES ASSOCIATED
WITH GENERALIZED TWIN PRIMES
TOMASZ JDRZEJAK
Abstract.
This article is a continuation of our previous paper [9] concern-
ing elliptic curves
Ep,m :y2=x(x−2m)(x+p)
, where
p
and
p+ 2m
are
primes. There we proved inter alia that
Ep,1
has at most two non-torsion in-
tegral points, and
Ep,2
has no such points. Now by using completely dierent
methods, namely an analysis of local height functions, we try to get upper
bounds for the number of integral points and for the number of multiples of
such points on our curves for any
m
. In particular, we show that no even mul-
tiples of an integral point on
Ep,m
are also integral, and if
Ep,m
has rank 1
and
p≡3 (mod 4)
then there are at most twelve non-torsion integral points in
the union of the non-identity component and the certain subset of the identity
component.
1.
Introduction
It is believed that there exist innitely many twin primes (this is a famous Twin
Prime Conjecture). More generally, one expects that (for a xed positive even
integer
k
) there exist innitely many primes
p
such that
p+k
is also a prime (c.f.
rst part of Conjecture B in [7]). These conjectures are still open and hard to prove.
One well known result is Chen's theorem [2] stating that there are innitely many
primes
p
such that
p+ 2
is a prime or a product of two primes. However, in the
related problem (the so called Bounded Gap Conjecture) Zhang [19] showed that
there are bounded gaps between consecutive primes innitely often. More precisely,
he proved that
lim inf
n→∞ (pn+1 −pn)≤7×107
where
pn
denotes the
n
th prime (note
that the Twin Prime Conjecture says that
lim inf
n→∞ (pn+1 −pn) = 2
). Nowadays, this
bound has been reduced to
246
unconditionally, and to
6
under the assumption of
the generalized Elliott-Halberstam conjecture (see [12]). Consequently, there is at
least one positive even integer (less than
247
) which can be written innitely often
as the dierence of two consecutive primes. It is worth mentioning the classical
result due to Romano [13] that the set of integers of the form
p+ 2m
, where
p
is
a prime and
m
is a positive integer, has positive lower density in
N
. On the other
hand, there are innitely many primes
p
such that
p+ 2m
is a composite number
for any
m
[18]; one of them is
p= 47867742232066880047611079
[3].
By the well known Siegel Theorem [14], any elliptic curve
E
over
Q
has only
nitely many integral points. However, searching for the number of integral points
in families of elliptic curves is not easy in general. Other question related to this
subject is to ask which multiples of a non-torsion integral point
P
may be integral.
2010
Mathematics Subject Classication.
11G05, 11G50.
Key words and phrases.
canonical height, elliptic curves, local heights, multiples of integral
points, number of integral points.
1

2 TOMASZ JDRZEJAK
It is widely expected (for instance, because it is implied by the ABC conjecture)
that there is a uniform bound on the number of integers
n
such that
nP
is integral.
Many authors have dealt with these problems. For example, Ingram [8] proved
that there is a uniform constant
C
and a quantity
M(P)
bounded above by the
global Tamagawa number of
E
such that
nP
is integral for at most one value of
n > CM (P)16
. Moreover, he showed that if
N
is square-free and
E
is the congruent
number curve
y2=x3−N2x
then there is at most one value
n > 1
such that
nP
is integral. Subsequently, Fujita and Terai [6] proved that if the congruent number
curve has rank one then it contains at most 17 integral points. Next, Fujita and
Nara [5] showed that if the Fermat elliptic curve
x3+y3=m
(
m
is cube-free
integer) has rank one then it has at most two integral points, and if has rank two
then it contains at most six such points. They also proved that the Fermat elliptic
curve of rank
r
has
≤3r−1
integral points.
In this paper, we consider elliptic curves associated to the generalized twin
primes, i.e., the family of elliptic curves over
Q
given by
(1.1)
Ep,m :y2=x(x−2m)(x+p)
,
where
p
,
q
are odd primes such that
q−p= 2m
,
m≥1
. Such curves were considered
for the rst time by D¡browski and Wieczorek in [4]. Here we give some informa-
tion concerning this family. Note that
Ep,m (Q)tors =Ep,m [2] = {∞,(0,0) ,(2m,0),
(−p, 0)}
(see [11, Main Theorem 1]) and
rp,m := rank Ep,m (Q)≤2
(see [10, Propo-
sition 4.19]). Moreover, this bound can only be obtained for
m= 3
or
m > 4
and
certain special primes
q≡1 (mod 8)
. Clearly,
Ep,m (R)
has two connected compo-
nents: the non-identity component consisting of the points with
x
-coordinates in
the interval
[−p, 0]
, and the identity component consisting of the ane points with
x
-coordinates
≥2m
and the point at innity. Note also that under the assump-
tion of the Parity Conjecture
rp,m = 1
if (
m= 3
and
p≡5 (mod 8)
) or
m= 4
or (
m≥5
and
p≡3,5,7 (mod 8)
) (see [4, Corollary 2]). Consequently, we have
Ep,m (Q)'(Z/2Z)2×Zr
where
r∈ {0,1,2}
. We also know that the reduction at
p
and
q
is multiplicative.
Suppose that we have a solution of
y2=x(x−2m)(x+p)
in nonzero integers
x
and
y
. Then the point
(x, y)
is non-torsion and we call it a
non-torsion integral
point on
Ep,m
(however note that the equation (1.1) is not minimal in general). If
Ep,m
has a non-torsion integral point then, by the consideration above, we have
rank Ep,m (Q)∈ {1,2}
. Clearly, two non-torsion integral points on
Ep,m
may dier
by the torsion point. For example,
E5,3
has rank one and its non-torsion integral
points
(−1,6)
and
(40,240)
dier by the torsion point
(0,0)
(notice that
E5,3
has
only four non-torsion integral points).
2.
Results
In this section, we state the main results of this paper. In [9] we considered
the same family, and proved among others that
Ep,1
has at most two non-torsion
integral points, and
Ep,2
has no such points. We have also listed all possible integral
points as solutions of certain systems of Pell-like equations. The purpose of this
paper is to describe the upper bounds for the number of integral points (at least
in certain subsets) in this family for
m > 3
. We are also interested in multiples of
non-torsion integral points on
Ep,m
. For instance, we show that no even multiples
of an integral point on
Ep,m
are also integral, and if
Ep,m
has rank 1 and
p≡

INTEGRAL POINTS ON ELLIPTIC CURVES 3
3 (mod 4)
then there is at most twelve non-torsion integral points with
x
-coordinates
≤2m152p
. If moreover conjecture 1 is true, then this number reduces to eight. Our
main method are an analysis of the local height functions and estimations of the
Nèron-Tate height on elliptic curves.
Theorem 1.
Let
p≡3 (mod 4)
or (
p≡1 (mod 4)
and
m= 3
) or
m≤2
. Assume
that the subgroup
Γ
of
Ep,m (Q)
which contains
Ep,m [2]
is generated (modulo tor-
sion) by the single non-torsion point
P∈Ep,m (Q)
and (without loss of generality)
x(P)>0
. Let
Q
be an integral point in
Γ
.
1) If
Q
belongs to the non-identity component of
Ep,m (R)
then
Q=nP +T
,
where
|n| ≤ 1
and
T∈ {(0,0) ,(−p, 0)}
.
2) If
Q
has
x
-coordinate in the interval
[2m,2m152p]
then
Q=nP +T
, where
|n| ≤ 3
and
T∈ {∞,(2m,0)}
.
Corollary 1.
Let
p≡3 (mod 4)
or (
p≡1 (mod 4)
and
m= 3
) or
m≤2
. If
Ep,m (Q)
has rank one then there are at most twelve non-torsion integral points
with
x
-coordinate
≤2m152p
.
Theorem 2.
Assume that
P
is a non-torsion point in
Ep,m (Q)
and
n
is an integer.
If
nP
is an integral point then
n
is odd. Assume moreover that
p≡3 (mod 4)
or
(
p≡1 (mod 4)
and
m= 3
) or
m≤2
.
1) If
nP
belongs to the non-identity component then
n=±1
.
2) If
x(nP )∈[2m,2m152p]
then
n∈ {±1,±3}
.
3.
Proofs
In this section, we prove our main results. The proofs split in a natural way
into several lemmata. We start with two lemmata concerning multiples of integral
points which are of independent interest.
Lemma 1.
If
E
is an elliptic curve over
Q
and a point
P∈E(Q)
is not integral
(with respect to a given Weierstrass equation of
E
) then for any nonzero integer
n
a point
nP
is not integral too.
Proof.
See [15, Exercise 9.12].

Lemma 2.
If
P∈Ep,m (Q)
is a non-torsion point then
x(2P)/∈Z
. In conse-
quence,
nP
is not integral for any even
n
.
Proof.
By Lemma 1 we may assume that
P= (x, y)
, where
x, y
are nonzero integers.
By the duplication formula, we obtain
x(2P) = x2+ 2mp2
4x(x−2m) (x+p)
.
Clearly, if
x
is odd then
x(2P)
is not an integer, so assume that
x= 2αx1
where
x1
is odd, and
α
is a positive integer. Substituting, we get
(3.1)
x(2P) = 22α−1x2
1+ 2m−1p2
2αx1(2αx1−2m) (2αx1+p)
,
so if
α≥m
then
ord2(22α−1x2
1+ 2m−1p2) = 2m−2< α+m= ord2(2αx1(2αx1−
2m)(2αx1+p))
, and consequently
x(2P)/∈Z
, hence we assume that
α < m
, in par-
ticular
m > 1
. Now suppose that
x(2P)∈Z
. Then
x1
divides
22α−1x2
1+ 2m−1p2
,
hence any prime factor of
x1
divides
2m−1p
. Therefore
x1=±pβ
where
β≥0
. If

4 TOMASZ JDRZEJAK
β > 1
then
ordp(22α−1x2
1+ 2m−1p2= 2
but
ordp(2αx1(2αx1−2m) (2αx1+p) =
1 + β > 2
, so
β≤1
. The cases
x1= 1
or
−p
are impossible since they im-
ply
0< x = 2α<2m
or
x=−2αp < −p
. First consider the case
x1=−1
,
i.e.,
x=−2α
. By [9, Theorem 2.4] we obtain,
α=m−3
(so in particu-
lar
m≥4
) and
p−2m−3
is a square, say
c2
(
c
is a positive integer). Hence
the denominator of (3.1) equals
22m−69p−2m−3= 22m−6(3c)2
, and numerator
of (3.1) equals
22µ−222m−6−µ+ 2m−µp2= 22µ−222m−6−µ9+2m−µc22
where
µ= min (2m−6, m)
. Since
x(2P)∈Z
, we have
c= 3
or
c= 9
. In any case
c= 3c0
but then the prime
q=p+ 2m= 9c02+ 2m−3+ 2m= 9 c02+ 2m−3
which is a con-
tradiction. Now let
x1=p
, i.e.,
x= 2αp
(
0< α < m
). Again by [9, Theorem 2.4]
we get
m≥4
,
3≤α≤m−1
,
2-α
, and
p=b2+ 2m−α, q =pc2−2αb2, c2= 2α+ 1
where
b, c
are positive relatively prime odd integers. In this case the denominator
and the numerator of (3.1) are equal to
22αp2(p−2m−α) (2α+ 1) = 22αp2(bc)2
and
22µ−2p222α−µp+ 2m−µ2= 22µ−2p222α−µb2+ 2m−µc22
respectively (
µ=
min (2α, m)
). Since
bc
divides
22α−µb2+ 2m−µc2
, we get
b=c= 1
, and conse-
quently
p+ 2m=q=p−2α
which is an absurd. Therefore always
x(2P)/∈Z
, and
nP = 2 (kP )
is not an integral point for
n= 2k
, which nish the proof.

Our next proofs involve estimations of the canonical height and an analysis of
local height function. Now we introduce some notation and facts about height
functions. Further details may be found for example in Silverman's books [15, 16].
Let
E
be an elliptic curve over
Q
. For
P∈E(Q)
with
x(P) = a/b
where
a
and
b
are relatively prime integers, the
naive height
h:E(Q)→R
is dened
by
h(P) = log max (|a|,|b|)
(we put also
h(∞) = 0
). The
canonical height
(or
Néron-Tate height
)
b
h:E(Q)→R
is dened by
b
h(P) = lim
n→∞
h(2nP)
4n
.
The canonical height is a quadratic form on
E(Q)
modulo torsion. In particular,
b
h(nP ) = n2b
h(P)
, and
b
h(P) = 0
if and only if
P
is a torsion point. Nèron and Tate
(see e.g. [16]) showed that
b
h
decomposes into the sum of the local height functions
b
hp:E(Qp)→R
where
p
is a place in
Q
, i.e.,
p
is a prime or innity. Hence for any
P∈E(Q)
we have
b
h(P) = X
p≤∞ b
hp(P)
.
In fact this sum is nite since
b
hp(P) = 0
for any point
P
and for almost all
p
. For ex-
ample, if
p
is a prime of the good reduction of
E
then
b
hp(P) = 1/2 max (0,−vp(x(P)))
where
vp(x) = ordp(x) log p
and
P6=∞
. There are also the (more complicated)
formulae for computations of non-archimedean local heights
b
hp
in the other cases
(when the reduction at
p
is not good and a point
P
is singular after the reduc-
tion, see [16, pp. 478-479]). To estimate the archimedean contribution
b
h∞
to the
canonical height we use Tate's series:
(3.2)
b
h∞(P) = 1
2log |x(P)|+1
8
∞
X
k=0
log 
z2kP

4k−1
12 log |∆E|
,
where
z
is a certain function depends on the curve
E
(we omit details but
E
must
be given by the minimal equation) and
∆E
is the discriminant of
E
.

INTEGRAL POINTS ON ELLIPTIC CURVES 5
Lemma 3.
If
p≡3 (mod 4)
or (
p≡1 (mod 4)
and
m≤3
) then for any
P∈
Ep,m (Q)\ {∞}
we have
b
h(P)≤1
4log a2+ 2mpb2+1
12 log 1 + p
2m
,
where
x(P) = a/b
and
gcd (a, b) = 1
.
Proof.
Under our assumptions a global minimal model of
Ep,m
is given by (1.1). We
shall consider two cases. In archimedean case we use Tate's series (3.2) where now
z(P) = 1+2mpt22
with
t= 1/x (P)
. Clearly,
Ep,m (R)
has two components,
and for any point
Q
in the identity component
E0
p,m (R)
we have
1≤z(Q)≤
(1 + p2−m)2
. Since
2kP∈E0
p,m (R)
(for
k≥1
), we get
b
h∞(P)≤1
2log |x(P)|+1
8log |z(P)|+1
8
∞
X
k=1
2 log (1 + p2−m)
4k−1
12 log |∆E|=
1
4log x(P)2+ 2mp+1
12 log 1 + p
2m−1
12 log |∆E|
.
Now consider non-archimedean case. Let
l
be a prime. By [16, p. 478], we obtain
b
hl(P)≤1
2max (0,−vl(x(P))) + 1
12vl(∆E)
.
Note that
x(P) = a/b
, and so
max (0,−vl(x(P))) = vl(b)
. Since
X
l
vl(∆E) = X
l
log lordl(∆E)= log Y
l
lordl(∆E)= log |∆E|
,
where the above sum is over all (nite) primes, combining the inequalities for local
heights, we nally get
b
h(P)≤1
4log a2+ 2mpb2+1
12 log 1 + p
2m
,
which completes the proof.

Lemma 4.
If
p≡3 (mod 4)
or (
p≡1 (mod 4)
and
m= 1,3
) or (
p≡1 (mod 8)
and
m= 2
) then for any non-torsion point
P∈Ep,m (Q)
we have
b
h(P)≥1
16 log (2m(p+ 2m)) .
Proof.
See [4, Proposition 7].

Proof of Theorem 1.
We can assume that
m≥3
because by [9, Theorems 2.1, 2,2,
2.3],
Ep,2
has no non-torsion integral points at all,
Ep,1
has no non-torsion integral
points in the non-identity component, and at most two such points in the identity
component. By assumption,
Q=nP +T
and
x(Q) = a
where
T
is a torsion point,
and
n, a
are integers. Let us rst assume that
Q /∈E0
p,m (R)
. Then
a∈[−p, 0]
and
T∈ {(0,0) ,(−p, 0)}
(since
P
,
(2m,0)
and
∞
belong to the identity component
E0
p,m (R)
). Therefore by Lemma 3, we get
b
h(P)≤1
4log a2+ 2mp+1
12 log 1 + p
2m≤1
4log p2+ 2mp+1
12 log 1 + p
2m
.
On the other hand, by Lemma 4 and properties of the canonical height, we obtain
b
h(P) = b
h(nQ +T) = b
h(nQ) = n2b
h(Q)≥n2
16 log (2m(p+ 2m))
.

6 TOMASZ JDRZEJAK
Consequently, after some calculations, we have
n2≤4 log p+ 16/3 log (p+ 2m)−4m/3 log 2
log (p+ 2m) + mlog 2
.
It is not dicult to check (we used Mathematica's [17] command Maximize) that
the function on the right side reaches its maximum for
m= 3
and
p= 5
, hence
n2<3.74
. So
|n| ≤ 1
, which proves the rst part.
Now suppose that
a∈[2m,2m152p]
. Consequently,
T∈ {∞,(2m,0)}
. As before
by Lemmata 3 and 4, we obtain
n2
16 log (2m(p+ 2m)) ≤b
h(P)≤1
4log 22mp2154+ 2mp+1
12 log 1 + p
2m
,
which implies
n2≤4 log 22mp2154+ 2mp+ 4/3 log (1 + p/2m)
log (p+ 2m) + mlog 2
.
We checked (by using Mathematica) that the function on the right side reaches its
maximum for
m= 3
and
p= 7
(and it is circa
15.9519
). Therefore
|n| ≤ 3
, which
completes the proof.

Proof of Corollary 1.
By remark 2 we know that there are at most two non-torsion
integral points in the non-identity component. If
Ep,m (Q)
has rank 1 then
Γ =
Ep,m (Q)
, and by Theorem 1 and Lemma 2, any non-torsion integral point with
x
-coordinate in
[2m,2m152p]
is contained in the set
{±P, ±P+ (2m,0) ,±2P+
(2m,0) ,±3P, ±3P+ (2m,0)}
, and we are done.

Proof of Theorem 2.
It follows immediately from Lemma 2 and the proof of Theo-
rem 1.

4.
Remarks and Conjectures
In this section we make ve remarks about our theorems and methods, and state
two conjectures. We also discuss numerical computations.
Remark 1.
Note that Theorem 1 and Corollary 1 are not 'empty' since for e.g.,
m= 4
or (
m≥5
and
p≡3,5,7 (mod 8)
) the root number of
Ep,m (Q)
is
−1
so
conjecturally
rank Ep,m (Q)=1
. In fact, we have many examples of curves
Ep,m
with a positive rank.
Remark 2.
Note that by
[9, Theorems 2.1-2.4]
, if
p≡3 (mod 4)
or
m≤3
, we
have at most two non-torsion integral points in the non-identity component. If they
exist, they have
x
-coordinate
−1
(for
m= 3
), and
−22m−3−12
(for
m≥4
).
Remark 3.
The bound
2m152p
looks arbitrary, but that's not the whole truth.
Indeed, as we showed in
[9, Theorem 2.4]
, any non-torsion integral point in the
identity component (with an exception in one case) has
x
-coordinate of the form
2αpβa2
where
0< α ≤m
,
0≤β≤1
, and
a
is a positive integer. Moreover, almost
all integral points (with one exception) that we found numerically have
x
-coordinates
≤2m132p
(see below for details).
Remark 4.
In
[4, Proposition 7]
there are also lower bounds for
b
h(P)
for other
p
and
m
than in Lemma 4. We are able to compute upper bounds for the canon-
ical height for
p≡1 (mod 4)
and
m≥4
(c.f. Lemma 3) but in this case the

INTEGRAL POINTS ON ELLIPTIC CURVES 7
equation (1.1) is not minimal. We must use the minimal model i.e.,
y2+xy =
x3+p−2m−1
4x2−2m−4px
but we are interested in the integral solutions of (1.1).
Remark 5.
Note that the family of the congruent number elliptic curves (considered
in
[6]
) is the family of quadratic twists of one curve
y2=x3−x
, and the family of
Fermat elliptic curves (considered in
[5]
) is the family of cubic twists of one curve
x3+y3= 1
. Our two-parameter family
Ep,m
is not a family of quadratic twists nor
cubic twists, and perhaps it is harder to estimate the number of integral points in
this case.
We have made some numerical calculations in Magma [1]. We were looking
for integral points on
Ep,m
for
m≤10
and
p≤p1000 = 7919
(by using the
command IntegralPoints) but for some
m
and
p
Magma was not able to answer.
We also checked certain bigger
m
or
p
, and e.g., we found the integral point
(2019550184120871208,2870000639061829344674420880)
on
E178566897581,3
and the
integral point
(16331640832,2087108321525760)
on
E23593,12
. Among all tested
curves
Ep,m
we found only one example of curve with eight non-torsion integral
points (all other our examples have at most six integral points):
m= 9, p = 89, q =
601,(x, y) = (−64,±960),(712,±10680),(2312,±99960),(481312,±333771360)
. Note
that
E89,9(Q)
has rank two, and the points
P1= (−16/169,−144240/2197)
,
P2=
(−200/9,24040/27)
generate
E89,9(Q)
modulo torsion. We have also the following
relations:
(−64,−960) = P2+(2m,0)
,
(712,−10680) = P2+(−p, 0)
,
(2312,99960) =
P1+P2
,
(481312,−333771360) = P1+ (0,0)
. In addition, for all found non-torsion
integral points
P
on the tested
Ep,m
we also checked that
3P
is not integral. There-
fore numerical computations performed in Magma, Theorems 1, 2, Corollary 1, and
other reasons mentioned below suggest the following.
Conjecture 1.
If
P
is a non-torsion point in
Ep,m (Q)
then
3P
is not integral
point.
Conjecture 2.
The curve
Ep,m
has at most 8 non-torsion integral points.
Notice that if Conjecture 1 is true then from the proof of Corollary 1 we get that
if
Ep,m (Q)
has rank one then it has at most 8 non-torsion integral points with
x
-
coordinates
≤2m152p
(indeed,
±3P= 3 (±P)
and
±3P+(2m,0) = 3 (±P+ (2m,0))
are not integral). Similarly, in this case by Theorem 2, we immediately obtain that
if
nP
is a non-torsion integral point and
x(nP )≤2m152p
then
n=±1
.
Here we explain why we believe in Conjecture 1. We found the 'triplication'
formula, i.e., for a point
P= (x, y)
on
Ep,m
we have
x(3P) = xF (x)2
G(x)2
where
F(x) = x4+2m+13px2+ 2m+2p(p−2m)x−22m3p2
,
G(x) = −3x4−4 (p−2m)x3+
2m+13px2+22mp2
. Assume that
P
is integral (otherwise by Lemma 1,
3P
is not inte-
gral too). Then
x(3P)∈Z⇔G(x)2|xF (x)2⇔gcd G(x)2, xF (x)2=G(x)2
,
but we can prove that for any integer
x
, the only prime divisors of
gcd G(x)2, xF (x)2
are
2, p
, and
q=p+2m
. Since it is reasonable to suppose that for any
x=x(P)∈Z
the integer
G(x)
has (at least one) another prime factor, we think that
x(3P)/∈Z
.
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8 TOMASZ JDRZEJAK
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University of Szczecin, Institute of Mathematics, Wielkopolska 15, 70-451 Szczecin,
Poland
E-mail address
:
tjedrzejak@gmail.com