Hacettepe Journal of Mathematics and Statistics
Volume 38 (2) (2009), 161 – 171
ELLIPTIC DIVISIBILITY SEQUENCES IN
CERTAIN RANKS OVER FINITE FIELDS
∗†
B. Gezer and O. Bizim∗
Received 09 : 02 : 2009 : Accepted 04 : 05 : 2009
Abstract
We develop techniques first studied by Morgan Ward to characterize
sequences which arise from elliptic curves and which contain a zero
term. We first define elliptic divisibility sequences over finite fields by
noting that they are not the sequences which arise by reduction from
integer sequences. After that, we give general terms of these sequences
over the finite fields Fp (p > 3 is a prime) and then we determine elliptic
curves and singular curves associated with them.
Keywords: Elliptic divisibility sequences, Singular sequences, Elliptic curves, Singular
curves.
2000 AMS Classification: 11B50, 11A 07, 11G05.
1. Introduction
A divisibility sequence is a sequence (hn) (n ∈ N) of positive integers with the prop-
erty that hm|hn if m|n. The oldest example of a divisibility sequence is the Fibonacci
sequence. There are also divisibility sequences satisfying a nonlinear recurrence relation.
These are the elliptic divisibility sequences and this recurrence relation comes from the
recursion formula for elliptic division polynomials associated with an elliptic curve.
An elliptic divisibility sequence (or EDS) is a sequence of integers (hn) satisfying a
non-linear recurrence relation
(1.1) hm+nhm−n = hm+1hm−1h
2
n − h 2n+1hn−1hm
and with the divisibility property that hm divides hn whenever m divides n for all
m ≥ n ≥ 1.
∗Uludag University, Faculty of Science, Department of Mathematics, Görükle, 16059, Bursa,
Turkey.
E-mail: (B. Gezer) betulgezer@uludag.edu.tr (O. Bizim obizim@uludag.edu.tr
†Corresponding author
†This work was supported by The Scientific and Technological Research Council of Turkey,
Project No: 107T311.
162 B. Gezer, O. Bizim
EDSs are a generalization of a class of integer divisibility sequences called Lucas
sequences [10]. EDSs were of interest because they were the first non-linear divisibility
sequences to be studied. Morgan Ward wrote several papers detailing the arithmetic
theory of EDSs, [11, 12]. For the arithmetic properties of EDSs, see also [2, 3, 4, 5, 9].
Shipsey and Swart, [5, 9], were interested in the properties of EDSs reduced modulo
primes. Shipsey [5] used EDSs to study some applications to cryptography and the elliptic
curve discrete logarithm problem (ECDLP). The Chudnovsky brothers considered prime
values of EDSs in [1]. EDSs are connected to the heights of rational points on elliptic
curves and the elliptic Lehmer problem.
2. Some preliminaries on elliptic divisibility sequences and ellip-
tic curves
There are two useful formulas (known as duplication formulas) used to calculate the
terms of an EDS. The duplication formulas are obtained by setting first m = r+1, n = r
and then m = r + 1, n = r − 1 in (1.1):
(2.1) h 3 32r+1 = hr+2hr − hr−1hr+1,
(2.2) h2rh2 = h
2 2
r(hr+2hr−1 − hr−2hr+1)
for all r ∈ N.
A solution of (1.1) is proper if h0 = 0, h1 = 1, and h2h3 6= 0. Such a proper solution
will be an EDS if and only if h2, h3 and h4 are integers with h2|h4. The sequence (hn)
with initial values h1 = 1, h2, h3 and h4, is denoted by [1 h2 h3 h4]. The discriminant of
an elliptic divisibility sequence (hn) is defined by the formula:
∆(h , h , h ) = h h15−h3h12+3h2h10−20h h3h7+3h3h5+16h6h4+8h2h3 2 42 3 4 4 2 3 2 4 2 4 3 2 4 2 3 2 4 3h2+h4.
An elliptic divisibility sequence (hn) is said to be singular if and only if its discriminant
∆(h2, h3, h4) vanishes.
In this work we discuss the behavior of some special EDSs over a finite field Fp, where
p > 3 is a prime, and also the elliptic curves associated with (hn). To classify EDSs
modulo p we need to know the rank of an EDS.
An integer m is said to be a divisor of the sequence (hn) if it divides some term with
positive suffix. Let m be a divisor of (hn). If ρ is an integer such that m |hρ and there
is no integer j such that j is a divisor of ρ with m |hj , then ρ is said to be the rank of
apparition of m in (hn). In the following theorem Ward said that the multiples of p are
regularly spaced in (hn).
2.1. Theorem. [12] Let p be a prime divisor of an elliptic divisibility sequence (hn), and
let ρ be its smallest rank of apparition. Let hρ+1 ≇ 0 (p), then
hn ≡ 0 (p) if and only if n ≡ 0 (ρ).
A sequence (sn) of rational integers is said to be numerically periodic modulo m if
there exists a positive integer π such that
(2.3) sn+π ≡ sn (m)
for all sufficiently large n. If (2.3) holds for all n, then (sn) is said to be purely periodic
modulo m. The smallest such integer π for which (2.3) is true is called the period of (sn)
modulo m. All other periods are multiples of it.
The following theorem of Ward shows us how the period and rank are connected.
Elliptic Divisibility Sequences 163
2.2. Theorem. [12] Let (hn) be an EDS and p an odd prime whose rank of apparition ρ
is greater than 3. Let a1 be an integral solution of the congruence a ≡ h21 (p) and lethρ−2
e and k be the exponents of a1 and a2 ≡ hρ−1 (p). Then (hn) is purely periodic modulo
p, and its period π is given by the formula π(hn) = τρ, where τ = 2
α[e, k]. Here [e, k]
is the least common multiple of e and k, and the exponent α is determined as follows:

+1 if e and k are both odd

 if e and k are both even and both divisible
α = −1
by exactly the same power of 2


 0 otherwise.
We will now give a short account of material about elliptic curves. More details of the
theory of elliptic curves can be found in [6, 8]. Consider an elliptic curve defined over
the rational numbers determined by a short Weierstrass equation y2 = x3 + ax + b with
coefficients a, b ∈ Q and discriminant ∆ = − 16(4a3 + 27b2).
Ward proved that EDSs arise as values of the division polynomials of an elliptic
curve. We will write ψn(P ) for ψn evaluated at the point P = (x1, y1). The following
theorem shows us the relations between EDSs and the elliptic curves (for further details
see [5, 7, 9, 12]).
2.3. Theorem. [7] Let (hn) be an elliptic divisibility sequence [1 h2 h3 ch2]. Then there
exists an elliptic curve E : y2 = x3 + ax+ b where a, b ∈ Q, and a non singular rational
point P = (x1, y1) on E such that ψn(x1, y1) = hn for all n ∈ Z where ψn is the n-th
polynomial of E. These quantities are given by
( )
3 (−h162 − 4ch122 + (16h33 − 6c2)h82 + (8ch3 3 4(2.4) a = 3 3 − 4c )h2− ,(16h63 + 8c2h3 43 + c )
 
h242 + 6ch
20
2 − (24h33 − 15c2)h162 − (60ch33 − 20c3)h122
(2.5) b = 2.33  +(120h63 − 36c2h33 + 15c4)h8 + (−48ch6 + 12c3h3)h4 2 3 3 2 ,
+(64h93 + 48c
2h63 + 12c
4h33 + c
6)
( )
(2.6) P = (x 8 4 3 2 3 41, y1) = 3(h2 + 2ch2 + 4h3 + c ),−108h3h2 ,
and
∆ = 28312h9h8(ch123 2 2 +(−h33 +3c2)h82 +(−20ch3 +3c3)h4 +(16h63 2 3 +8c2h33 + c4)).
By Theorem 2.3, we can say that the EDS [1 h2 h3 ch2] is associated with the elliptic
curve E : y2 = x3 + ax + b and the rational point P ∈ E. Note that if E is a singular
curve, then possibly P is a singular point. In this case we move P to any non singular
point P ′ on E.
In the following theorem, Ward showed that the discriminant of an elliptic divisibility
sequence is equal to the discriminant of the elliptic curve associated with this sequence.
2.4. Theorem. [12] Let (hn) be an elliptic divisibility sequence in which h2h3 6= 0, and
let E be an elliptic curve associated with (hn). Then the discriminant of (hn) is equal to
discriminant of the elliptic curve E.
3. Elliptic divisibility sequences in certain ranks
In this section we work with elliptic divisibility sequences in certain ranks over Fp,
where p > 3 is a prime, and we discuss some properties of these sequences. Firstly, we
define elliptic sequences and then elliptic divisibility sequences over Fp.
164 B. Gezer, O. Bizim
3.1. Definition. An elliptic sequence over Fp is a sequence of elements of Fp satisfying
the formula
h 2m+nhm−n = hm+1hm−1hn − hn+1h 2n−1hm .
If (hn) is an elliptic sequence over Fp then (hn) is an elliptic divisibility sequence over
Fp since any non-zero element of Fp divides any other. Therefore, in this paper, the term
elliptic sequence over Fp will mean an elliptic divisibility sequence over Fp. Of course,
the concept of the rank of an elliptic divisibility sequence over Fp is the same as that for
an elliptic divisibility sequence defined above.
Note that, as for integral sequences, elliptic divisibility sequences satisfy the further
conditions that h0 = 0, h1 = 1, that two consecutive terms of (hn) cannot vanish over
Fp and if some term is zero, then multiples of this term are zero too, that is; if h2 = 0
then h4 = 0 and so h2n = 0 for all n ∈ N. This relation is shown below:
3.2. Lemma. Let (hn) be an elliptic divisibility sequence with rank ρ over Fp. Then
hρn ≡ 0 (p).
Proof. Let (hn) be an elliptic divisibility sequence over Fp. If (hn) has rank ρ then
hρn ≡ 0 (p) since hρ divides hρn as ρ divides ρn. 
Now we consider the EDSs with rank two. We know that if h2 = 0 then we must have
h2n = 0 for all integers n =6 0. Thus every term of the sequence with even subscript is
zero. Ward proved that such a sequence is given by the following formula for all odd n:
n2n −1
(hn) = (−1)⌊ 4 ⌋h 83 ,
where ⌊x⌋ denotes the greatest integer in x.
3.1. Sequences with rank three. Now consider the EDSs with rank three. We know
that if h3 = 0 then we must have h3n = 0 for all integers n 6= 0.
3.3. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 0 h4], (h2, h
∗
4 ∈ Fp).
Then (hn) is given by the following formula:
k(k+1) (k+2a−2)(k+2a−3)
(3.1) hn = h 2 23k+a = ε h4 h2 ,
{
+1 if n ≡ 1, 2, 4, 5 (12)
where ε =
−1 if n ≡ 7, 8, 10, 11 (12).
Proof. It is clear that the result is true for n = 4. Hence we assume that n > 4. If (hn) is
an EDS, then we know that
h 2 2n+2hn−2 = hn+1hn−1h2 − h3h1hn .
It suffices to prove our main result by induction based on equation (3.1). Now first
suppose that n + 1 ≡ 4 (12) and let the equation (3.1) be true for n + 1. Then since
n+1 ≡ 4 (12), we have n+1 = 3(4r+1)+1, (r ∈ N) and so n+2 = 3(4r+1)+2. Thus
2
we find that h 8r +6r+1 8r
2+10r+3
n+2 = h4 h2 . On the other hand we see that
2 2
h = h8r +6r+1h8r +2rn+1 4 2
hn = 0
8r2+2r 8r2h = h h +6r+1n−1 4 2
h = h8r
2+2r 2
n−2 4 h
8r −2r
2 .
Elliptic Divisibility Sequences 165
2 2
Substituting these expressions into (3.1) gives h = h8r +6r+1h8r +10r+3n+2 4 2 . Thus we
proved this theorem for n + 1 ≡ 4 (12). Other cases can be proved by induction in the
same way. 
We know that if (hn) is a proper EDS, then h2 | h4, so we may write h4 = ch2 where
c ∈ F∗p. Thus we can give a new formula for the general terms of EDSs with rank three
and parameter c.
3.4. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 0 h4], (h4 = ch2 and
c ∈ F∗p). Then (hn) is given by the following formula:
k(k+1) (k+a−1)2
hn = h3k+ 2a = ε c h2
{
+1 if n ≡ 1, 2, 4, 5 (12)
where ε =
−1 if n ≡ 7, 8, 10, 11 (12).
Proof. The theorem can be proved by induction in the same way as Theorem 3.1. 
3.2. Sequences with rank four. Now let (hn) be an elliptic divisibility sequence with
rank four, namely consider the sequences whose fourth term is zero. We know that if
h4 = 0, then h4n = 0 for all integers n =6 0. Firstly we give the general term of (hn) with
rank four in the following theorem:
3.5. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 0] and (h2, h ∈ F∗3 p).
Then (hn) is given by the following formula:
2
(3.2) hn = h4k+a = ε h
β
2h
2k +ak+α
3 ,
{ {
+1 if n ≡ 1, 2, 3 (8) 1 2 − 3 1 if 2 | nwhere ε = , α = a a+ 1 and β = .
−1 if n ≡ 5, 6, 7 (8) 2 2 0 if 2 ∤ n.
Proof. If (hn) is an EDS, we know that
hn+2hn−2 = hn+1hn−1h
2
2 − h3h 21hn .
Then it suffices to prove our main result by induction based on equation (3.2). It is clear
that the result is true for n = 5. Hence we assume that n > 5.
Now first suppose that n+1 ≡ 2 (8) and let the equation (3.2) be true for n+1. We wish
2
to show that this equation is also true for n+ 2. We want to see that hn+2 = h
8r +6r+1
3
is true, where n+2 = 4 ·2r+3, r ∈ N. On the other hand we know from the assumption
that h = −h8r2−2r 2n−2 3 and similarly h = h8r + 2rn 3 . Substituting these relations into
equation (3.2) gives
8r2h (−h − 2r
2
) = −h16r + 4r+1n+2 3 3
2
and so we obtain that h = h8r +6r+1n+2 3 . Thus we proved this theorem for n+1 ≡ 2 (8).
Other cases of the theorem can be proved by induction in the same way. 
Now we give the period of (hn) with rank four in the following theorem:
3.6. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 0], (h2, h3 ∈ F∗p)
and q the order of h3. Then the period of (hn) is
{
4(p− 1) if h3 is a primitive root in Fp
π(hn) =
8r otherwise
{
q if q is odd
where r =
q if q is even.
2
166 B. Gezer, O. Bizim
Proof. It is clear that the rank of (hn) is 4 since h
h2
4 = 0, that is ρ = 4. Since a1 = =hρ−2
h2 = 1 and a2 = hρ −1 = h3, by Theorem 2.2 we see that the orders of a1 and a2 areh2
e = 1 and k = p− 1 if h3 is a primitive root in Fp, and k = q otherwise. Thus [e, k] = k.
If h3 is a primitive root in Fp, then α = 0 and in this case τ = 2
α[e, k] = p − 1. Then
π(hn) = 4(p− 1), since ρ = 4.
If h3 is not a primitive root in Fp then the order of h3 is q. So in this case α = 0 or
1, then τ = q or 2q. Hence π(hn) = 4q or 8q since ρ = 4. 
3.3. Sequences with rank five. Now let (hn) be an elliptic divisibility sequence with
rank five. We know that if h5 = 0, then we must have h5n = 0 for all integers n 6= 0.
The general term of (hn) is determined in the following theorem:
3.7. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 h4], (h2, h , h ∈ F∗3 4 p),
and having rank five. Then (hn) is given by the following formula:
2 −(5k2+2ak+β)
(3.3) hn = h = ε h
5k +2ak+α
5k+a 3 h2 ,
{
+1 if n ≡ 1, 2, 3, 4 (10)
where ε = , α = 1a2 − 3a+ 1 and β = a2 − 4a+ 3.
−1 if n ≡ 6, 7, 8, 9 (10) 2 2
( )3
h3
Proof. Since (hn) is an EDS with rank five and h5 = h
3 3
4h2 − h3, we have h4 = .h2
It is clear that the result is true for n = 6. Hence we assume that n > 6. If (hn) is an
EDS, we know that
h h = h h h2 2n+2 n−2 n+1 n−1 2 − h3h1hn .
It suffices to prove our main result by induction based on equation (3.3). Now first
suppose that n + 1 ≡ 2 (10), and let the equation (3.3) be true for n + 1. We want
2
to see that h = h20r
2+12r+1 −(20r +12r)
n+2 3 h2 is true, where n + 2 = 5 · 2r + 3, r ∈ N.
2 −(20r2+4r)
On the other hand we know from the assumption that h = h20r +4rn 3 h2 and
2 −(20r2−4r)
hn−2 = −h20r −4r3 h2 . Substituting these expressions into the equation (3.3),
we have
( ) ( )
2 − 2− (20r
2−4r) 2
h h20r −4r h = −h h20r2+4r −(20r +4r)n+2 3 2 3 3 h2
2 −(20r2+12r)
and so h 20r +12r+1n+2 = h3 h2 . Thus we have proved this theorem for n + 1 ≡
2(10). Other cases can be proved by induction in the same way. 
Now we give the period of (hn) with rank five in the following theorem:
3.8. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 h4], (h2, h3 ∈ F∗p)
with rank five and q the order of h2 . Then the period of (h ) is
h n3
{
5 (p− 1) if h2 is a primitive root in F
π(h ) = 2 h
p
3
n
10r otherwise
{
q if q is odd
where r =
q if q is even.
4
h2 h2
Proof. We know that the rank of (hn) is ρ = 5. Since a1 = = and a2 = hρ−1 =
hρ−2 h3
( )3
h3
h4 = , by Theorem 2.2, let e and k be the orders of a1 and a2 respectively. If
h2
h h2 3
Elliptic Divisibility Sequences 167
is a primitive root in Fp, then e = p − 1, k = p−1 when 3 divides p − 1, and e = p − 1,3
h2
k = p− 1 when 3 does not divide p− 1. If is not a primitive root in Fp, then e = q,
h3
k = q
h2
when 3 divides q, and e = q, k = q when 3 does not divide q. If is a primitive
3 h3
root in Fp, then α = −1, since p− 1 and p−1 are divisible by the same power of two, and3
p− 1
in this case τ = 2α[e, k] = . Then π(h ) = 5n (p− 1).
2 2
If h2 is not a primitive root in Fp, then α = 1 when q is odd, and α = −1 when qh3
is even; and τ = 2q and q , respectively. Then π(hn) = 10q if q is odd and
5q if q is
4 2
even. 
3.4. Sequences with rank six. Now let (hn) be an elliptic divisibility sequence with
rank six. We know that if h6 = 0 then we must have h6n = 0 for all integers n 6= 0. We
determine the general term of (hn) in the following theorem:
3.9. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 h4], (h2, h3, h4 =
ch2 ∈ F∗p), and with rank six. Then (hn) is given by the following formula:
(3.4) h = h = ε hα hβ c3k
2+ak+γ
n 6k+a 2 3 ,
{
+1 if n ≡ 1, 2, 3, 4, 5 (12)
where ε = and
−1 if n ≡ 7, 8, 9, 10, 11 (12)
{ { {
1 if 2 | n 1 if 3 | n 0 if a ≤ 3
α = β = γ =
0 if 2 ∤ n, 0 if 3 ∤ n, a− 3 if a > 3.
( )2
h3 2− 2 h4Proof. Since (hn) is an EDS with rank six and h6 = (h5h2 h4) we have h5 = .h2 h2
It is clear that the result is true for n = 7. Hence we assume that n > 7. If (hn) is an
EDS we know that
h 2 2n+2hn−2 = hn+1hn−1h2 − h3h1hn
Then we prove our main result by induction based on equation (3.4). Now first suppose
that n + 1 ≡ 2 (12) and let the equation (3.4) be true for n + 1. We want to see that
2
hn+2 = h3 c
12r +6r is true, where n + 2 = 6 · 2r + 3, r ∈ N. On the other hand we
2
know from the assumption that h = c12r +2r and h = −c12r2−2rn n−2 . Substituting these
expressions into (3.4) we have
( ) ( )
− 12r2− 2
2
h c 2r = −h 12r +2rn+2 3 c ,
2
and so h = h c12r +6rn+2 3 . Thus we have proved this theorem for n+ 1 ≡ 2(10). Other
cases can be proved by induction in the same way. 
Now we give the period of (hn) in the following theorem:
3.10. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 h4], (h , h ∈ F∗2 3 p)
with rank six and q the order of h2 . Then the period of (hn) ish4
{
6(p− 1) if h2 is a primitive root in F
h pπ(hn) = 4
12r otherwise
{
q if q is odd
where r =
q if q is even.
2
168 B. Gezer, O. Bizim
h2
Proof. We know that the rank of (hn) is ρ = 6. Let e and k be the orders of a1 = =
hρ−2
( )2
h2 h4
and a2 = hρ−1 = h5 = , respectively, where a1 and a2 are as in Theorem 2.2.
h4 h2
If h2 is a primitive root in Fp, then e = p − 1 and k = p−1 . In this case α = 0 andh4 2
τ = p− 1, so that π(hn) = 6(p− 1).
h2
If is not a primitive root in Fp, then there are two cases. In the first case, let q
h4
be even. Then e = q and k = q, so that α = 1 and τ = q. Hence π(hn) = 6q. In the
second case, let q be odd. Then e = q and k = q , so that α = 0 and τ = 2q. Hence
2
π(hn) = 12q. 
4. Elliptic divisibility sequences in certain ranks and the
associated curves
In this section we determine the curves associated with (hn) for ranks two, three, four,
five and six.
First we find the associated curves for a (hn) with rank two. Note that all elliptic
divisibility sequences with rank two are singular since their discriminant is zero and so
they are associated with a singular curve.
4.1. Theorem. Let (hn) be a singular elliptic divisibility sequence [1 0 h3 ch2 = 0],
(c ∈ Fp and h3 ∈ F∗p). Then (hn) is associated with a singular curve given by the
equation
(4.1) E : y2 = x3 − 27(4h3 + c2)2x+ 54(4h33 3 + c2)3,
and if P = (x1, y1) is a non-singular point on E then P = (3(h
3 + c23 ), 0).
Proof. Putting h2 = 0 in the equations (2.4), (2.5), (2.6), we have
a = −27(16h6 + 8c2h33 3 + c4) = −27(4h3 2 23 + c ) ,
b = 54(64h9 + 48c2h6 + 12c4h3 + c63 3 3 ) = 54(4h
3 2
3 + c )
3
and P = (3(4h33 + c
2), 0). 
These singular curves have singular point as a cusp or a node. Now we determine
when these curves have cusps, namely when they have the form y2 = x3.
4.2. Theorem. Let (hn) be an elliptic divisibility sequence [1 0 h 0], (h ∈ F∗3 3 p). Then
{
h3 ∈ Qp if p ≡ 1 (4)
(hn) is associated with a singular curve with a cusp ⇐⇒
h3 ∈/ Qp if p ≡ 3 (4),
{
⇐⇒ h3 ∈/ Qp if p ≡ 1 (4)(hn) is associated with a singular curve with a node
h3 ∈ Qp if p ≡ 3 (4),
where Qp denotes the set of quadratic residues in Fp.
2
Proof. The theorem can be proved by putting h33 = − c in (4.1). In this case the point4
P is a singular point on E. 
The elliptic divisibility sequence [1 0 h3 h4 = ch2 = 0], (c ∈ Fp and h ∗3 ∈ Fp), is an
improper EDS. So, when we determine the fourth term h4 = ch2, we choose all elements
of Fp for the number c. Therefore, such sequences can be associated with more than one
curve. For example, in F5, the sequence [1 0 1 0] is associated with the singular curves
y2 = x3 + 3x+ 1 for c = 0; y2 = x3 for c = 1, 4; and y2 = x3 + 2x+ 3 for c = 2, 3.
Elliptic Divisibility Sequences 169
Now we find the curves associated with (hn) having rank three. Note that all elliptic
divisibility sequences with rank three are singular since their discriminant is zero and so
they are associated with a singular curve.
4.3. Theorem. Let (hn) be a singular elliptic divisibility sequence [1 h2 0 ch2] and
(c, h2 ∈ F∗p). Then (hn) is associated with the singular curve E given by the equation
(4.2) E : y2 = x3 − 27(h42 + c)4x+ 54(h42 + c)6,
and if P = (x1, y1) is a non-singular point on E then P = (3(h
4
2 + c)
2, 0).
Proof. The theorem can be proved in the same way as Theorem 4.1. 
Now we see that when these singular EDSs are associated with the curve y2 = x3.
4.4. Theorem. Let (hn) be a singular elliptic divisibility sequence [1 h2 0 ch2], (c, h2 ∈
F∗p). Then (hn) is associated with the singular curve E : y
2 = x3 if and only if h4 = −h52.
Proof. The theorem can be proved by putting h42 = −c in (4.2). In this case the point P
is a singular point on E. 
Now we will find elliptic curves associated with (hn) having rank four.
4.5. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 0], (h2, h3 ∈ F∗p).
Then (hn) is associated with an elliptic curve E given by the equation:
E : y2 = x3+27(−h162 +16ch33h82−16h63)x+54(h242 −24h3h16 +120h6h83 2 3 2+64h93),
and if P = (x1, y1) is a point on E then P = (3(h
8
2 + 4h
3 3 4
3),−108h3h2).
Proof. Since h4 = 0 and since h2h3 6= 0 we obtain c = 0. Putting c = 0 in (2.4), (2.5),
(2.6), we find that
(4.3) a = 27(−h16 + 16ch3h82 3 2 − 16h63),
(4.4) b = 54(h242 − 24h3 163h2 + 120h6 83h2 + 64h93),
(4.5) P = (3(h8 3 3 42 + 4h3),−108h3h2).

Now we determine which of these curves are singular curves.
4.6. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 0], (h ∈ F∗2 p) and
h8
h3 = 23 . Then (hn) is associated with the singular curve E given by the equation E :16
y2 = x3 − 27h16x − 54h242 2 and if P = (x1, y1) is a non singular point on E then P =16 64
( )
15h82 − 27h
12
, 2 .
4 4
h8
Proof. Since E is a singular curve if and only if (hn) is a singular sequence, and h
3 2
3 = ,16
putting this equation in (4.3), (4.4) and (4.5) we have desired result. 
Note that in this case we have no singular curve of the form y2 = x3 since h2 6= 0.
Now we find elliptic curves associated with (hn) having rank five.
4.7. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 h4], (h2, h h ∈ F∗3, 4 p)
and let h5 = 0. Then (hn) is associated with the elliptic curve E given by the equation:
E : y2 = x3 + 27(−h162 + 12h122 c− 14h8 2 4 3 42c − 12h2c − c )x
(4.6)
+ 54(h24 20 16 2 8 42 − 18h2 c+ 75h2 c + 75h2c + 18h4 52c + c6),
and if P = (x1, y1) is a point on E then P = (3(h
8
2 + 4h
3
3),−108h3h43 2).
170 B. Gezer, O. Bizim
Proof. Since h = 0 we obtain h3 = h4c. Putting h3 45 3 2 3 = h2c in (2.4), (2.5) and (2.6) we
find the desired results. 
We will see when singular curves arise:
4.8. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 ch2], (c, h2, h ∈ F∗3 p
where p > 5), and having rank five. Then there exists a singular curve E associated to
(hn) if and only if p ≡ 1, 9 (10).
Proof. Since (hn) is an EDS of rank five and h5 = h h
3 − h34 2 3 = 0 we have h4 = ch2 =
( )3
h3
. If the elliptic divisibility sequence [1 h2 h3 ch2] is associated with a singular
h2
curve E then we know that this sequence is singular. That is,
h9 h12
∆ = −h4 6 3 3 16 12 8 22h3 + 11 + = −h + 11h c+ h c = 0.
h4 h12
2 2 2
2 2
So, we have −h82 + 11h4 22c+ c = 0. If we substitute h42 = t in (4.6), then we have
√
4 11± 5 5t1,2 = h2 = c.2
Thus (hn) is associated with a singular curve if and only if 5 is a quadratic residue in Fp.
But, 5 is a quadratic residue in Fp if and only if p ≡ 1, 9 (10). 
Now we will find singular curves associated with (hn).
4.9. Theorem. Let (h ) be an elliptic divisibility sequence [1 h h ch ], (c, h , h ∈ F∗n √ 2 3 2 2 3 p
where p > 5), having rank five and satisfying h4 = 11±5 52 c, where 5 is a quadratic residue2
in Fp. Then (hn) is associated with the singular curve E given by the equation
√
( )
16605 ± 7425 5 √( )
(4.7) E : y2 = x3 − c4x− 411750 ± 184140 5 c6,
2
and if P = (x1, y1) is a non-singular point on E then
√
(( ) )
573± 255 5 √( )
P = (x1, y1) = c
2, − 6642± 2970 5 c3 .
2
√
11± 5 5
Proof. The theorem can be proved by substituting h42 = c in (4.6). 2
Now we will see that all EDSs are associated with the singular curve y2 = x3 when
p = 5.
4.10. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 ch2], (c, h2, h3 ∈
F∗5), and having rank five in F5. Then (hn) is associated to the singular curve E : y
2 = x3.
Proof. Considering the equation (4.7) in F5 gives the desired result. 
Now we find elliptic curves associated with (hn) having rank six:
4.11. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 h4], (h2, h3, h4 ∈
F∗p), and let h6 = 0. Then (hn) is associated with an elliptic curve E given by the
equation:
E : y2 = x3 + 27(−h162 + 12h12c− 30h8c22 2 + 12h4 3 42c − 9c )x
(4.8)
+ 54(h242 − 18h202 c+ 99h162 c2 − 180h12 32 c + 135h8 4 4 5 62c + 54h2c − 27c )
and if P = (x 8 4 2 8 2 41, y1) is a point on E then P = (3(h2 + 6ch2 − 3c ), −108(ch2 − c h2)).
Elliptic Divisibility Sequences 171
Proof. Since h6 = 0 we obtain h
3
3 = ch
4 2
2 − c . Putting h3 = ch4 − c23 2 in (2.4), (2.5) and
(2.6) we find the desired results. 
Now we see when associated singular curves arise:
4.12. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 h
∗
4], (h2, h3 ∈ Fp)
having rank six. Then there exists a singular curve E associated to (hn) if and only if
h5
h4 = ch 22 = .9
Proof. Since (hn) is an EDS with rank six and h6 =
h3 (h 25h2 − h24) = 0 we have h5 =h2
( )2
h4
and h33 = ch
4 2
2 − c . If the elliptic divisibility sequence [1 h2 h3 ch2] is associatedh2
with the singular curve E then we know that this sequence is singular. That is,
∆ = h162 c− h12 32 h3 + 3h12 22 c − 20h82c3 + 16h4h62 3 + 8h42c2h33 + h42c4 = 0.
If we substitute h33 = ch
4 − c22 in this equation we have 9c = h42. 
Now we find the singular curves associated with (hn).
4.13. Theorem. Let (hn) be an elliptic divisibility sequence [1 h2 h3 ch2], (c, h2, h3 ∈
∗ h5Fp), having rank six and let h4 = ch = 22 . Then (hn) is associated with the singular9
curve E given by the equation E : y2 = x3 − 3888c4x− 93312c6, and if P = (x1, y1) is a
non-singular point on E then P = (396c2,−7776c3).
Proof. The theorem can be proved by substituting h42 = 9c in (4.8). 
References
[1] Chudnovsky, D.V. and Chudnovsky, G.V. Sequences of numbers generated by addition in
formal groups and new primality factorization tests, Adv. in Appl. Math. 7, 385–434, 1986.
[2] Einsiedler, M., Everest, G. and Ward, T. Primes in elliptic divisibility sequences, LMS J.
Comput. Math. 4, 1–13, 2001 (electronic).
[3] Everest, G., van der Poorten, A., Shparlinski, I. and Ward, T. Recurrence Sequences (Math-
ematical Surveys and Monographs 104, AMS, Providence, RI, 2003).
[4] Everest, G. and Ward, T. Primes in divisibility sequences, Cubo Mat. Educ. 3, 245–259,
2001.
[5] Shipsey, R. Elliptic divisibility sequences (Ph.D. Thesis, Goldsmith’s (University of London),
2000).
[6] Silverman, J.H. The Arithmetic of Elliptic Curves (Springer-Verlag, 1986).
[7] Silverman, J.H. and Stephens, N. The sign of an elliptic divisibility sequence, Journal of
Ramanujan Math. Soc. 21, 1–17, 2006.
[8] Silverman, J. H. and Tate, J. Rational Points on Elliptic Curves (Undergraduate Texts in
Mathematics, Springer, 1992).
[9] Swart, C. S. Elliptic curves and related sequences (PhD. Thesis, Royal Holloway (University
of London), 2003).
[10] Tekcan, A., Gezer, B. and Bizim, O. Some relations on Lucas numbers and their sums,
Advanced Studies in Comtemporary Mathematics 15 (2), 195–211, 2007.
[11] Ward, M. The law of repetition of primes in an elliptic divisibility sequences, Duke Math.
J. 15, 941–946, 1948.
[12] Ward, M. Memoir on elliptic divisibility sequences, Amer. J. Math. 70, 31–74, 1948.