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ScienceDirect

Indagationes Mathematicae 33 (2022) 992–1000
www.elsevier.com/locate/indag

A modular approach to the generalized
Ramanujan–Nagell equation

Elif Kızıldere Mutlua, Maohua Leb, Gökhan Soydana,∗

a Department of Mathematics, Bursa Uludağ University, 16059 Bursa, Turkey
b Institute of Mathematics, Lingnan Normal College, Zhangjiang, Guangdong, 524048, China

Received 11 November 2021; received in revised form 8 April 2022; accepted 25 April 2022

Communicated by C. Salgado

Dedicated to the memory of Bas Edixhoven.

Abstract

Let k be a positive integer. In this paper, using the modular approach, we prove that if k ≡ 0 (mod 4),
0 < k < 724 and 2k −1 is an odd prime power, then under the GRH, the equation x2

+ (2k −1)y
= kz

as only one positive integer solution (x, y, z) = (k − 1, 1, 2). The above results solve some difficult
ases of Terai’s conjecture concerning this equation.
2022 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

eywords: Polynomial-exponential Diophantine equation; Elliptic curve; S-integral point; Modular approach

1. Introduction

Let Z, N be sets of all integers and positive integers respectively. Let d, k be fixed coprime
positive integers with min{d, k} > 1. A class of polynomial-exponential Diophantine equations
of the form

x2
+ d y

= kz, x, y, z ∈ N (1.1)

is usually called the generalized Ramanujan–Nagell equation. It has a long history and rich
content (see [15]). In 2014, N. Terai [19] discussed the solution of (1.1) in the case d = 2k −1.

e proposed the following conjecture:

∗ Corresponding author.
E-mail addresses: elfkzldre@gmail.com (E.K. Mutlu), lemaohua2008@163.com (M. Le),

gsoydan@uludag.edu.tr (G. Soydan).
https://doi.org/10.1016/j.indag.2022.04.005
0019-3577/© 2022 Royal Dutch Mathematical Society (KWG). Published by Elsevier B.V. All rights reserved.

http://www.elsevier.com/locate/indag
https://doi.org/10.1016/j.indag.2022.04.005
http://www.elsevier.com/locate/indag
http://crossmark.crossref.org/dialog/?doi=10.1016/j.indag.2022.04.005&domain=pdf
mailto:elfkzldre@gmail.com
mailto:lemaohua2008@163.com
mailto:gsoydan@uludag.edu.tr
https://doi.org/10.1016/j.indag.2022.04.005


E.K. Mutlu, M. Le and G. Soydan Indagationes Mathematicae 33 (2022) 992–1000

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Conjecture 1.1. For any k with k > 1, the equation

x2
+ (2k − 1)y

= kz, x, y, z ∈ N (1.2)

as only one solution (x, y, z) = (k − 1, 1, 2).

The above conjecture has been verified in some special cases (see [1, Theorem 3.1],
10, Corollary 1.4], [11, Theorem 1.2] and [19, Proposition 3.3]). Most of solved cases of
onjecture 1.1 focus on the case 4 ∤ k, and very little is known in the case 4 | k. In [10],
.-J. Deng, J. Guo and A.-J. Xu verified Conjecture 1.1 when the case k ≡ 3 (mod 4) with
≤ k ≤ 499. For the case 4 | k, N. Terai [19] used some classical methods to discuss (1.2)

or k ≤ 30. However, his method does not apply for k ∈ {12, 24}. Until 2017, M. A. Bennett
nd N. Billerey [1] used the modular approach to solve the case k ∈ {12, 24}. It follows that

the case 4 | k and 2k − 1 is an odd prime power is a very difficult case to Conjecture 1.1. In
this paper, using the modular approach we prove the following result:

Theorem 1.1. If 4 | k, 30 < k < 724 and 2k − 1 is an odd prime power, then under the
GRH, Conjecture 1.1 is true.

2. Preliminaries

This section introduces some well known notions and results that will be used to prove the
main result.

2.1. The modular approach

The most important progress in the field of the Diophantine equations has been with Wiles’
proof of Fermat’s Last Theorem [18,20]. His proof is based on deep results about Galois
representations associated to elliptic curves and modular forms. The method of using such
results to deal with Diophantine problems, is called the modular approach. After Wiles’ proof,
the original strategy was strengthened and many mathematicians achieved great success in
solving other equations that previously seemed hard. As a result of these efforts, the generalized
Fermat equation

Ax p
+ Byq

= Czr , with 1/p + 1/q + 1/r < 1, (2.1)

where p, q, r ∈ Z≥2, A, B, C are non-zero integers and x, y, z are unknown integers became
a new area of interest. Call an integer solution (x, y, z) to such an equation proper if
gcd(x, y, z) = 1. It was proved that Eq. (2.1) has finitely many proper solutions by H. Darmon
and A. Granville [9, Theorem 2]. We call the triple of exponents (p, q, r ) as in (2.1) the
signature of equation. In the cases 1/p + 1/q + 1/r = 1, the proper solutions give rise to
rational points on certain curves of genus one. It is easily demonstrated that for each p, q, r ,
there exist values of A, B, C such that the equation has infinitely many proper solutions (see
[9, Section 6]). There also exist values of A, B, C such that the equation has no proper
solutions; though, for any A, B, C , there are number fields which contain infinitely many proper
solutions (see [9, Subsection 5.4]). In the cases where 1/p + 1/q + 1/r > 1, the proper
olutions correspond to rational points on certain curves of genus zero. For this case, it is easy
o show that there are infinitely many proper solutions of every equation x p

+ yq
= zr (see

9, Section 7]).
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In recent 30 years, several authors considered many cases of the above equations. Two survey
apers which were written by M.A. Bennett, I. Chen, S. Dahmen and S. Yazdani [2] and M.
. Bennett, P. Mihǎilescu and S. Siksek [3] are good references for the case ABC = 1.
One can find the details concerning modular approach in [7, Chapter 15] and [17].

.2. Signature (n, n, 2)

Here we give recipes for signature (n, n, 2) which was firstly described by M.A. Bennett
nd C. Skinner [4] (see also [13]). We denote by rad(m) the radical of |m|, i.e. the product
f distinct primes dividing m, and by ordp(m) the largest nonnegative integer k such that pk

ivides m.
We always assume that n ≥ 7 is prime, and a, b, c, A, B and C are nonzero integers with

Aa, Bb and Cc pairwise coprime, A and B are nth-power free, C squarefree satisfying

Aan
+ Bbn

= Cc2. (2.2)

e further assume that we are in one of the following situations:
(i) abABC ≡ 1 (mod 2) and b ≡ −BC (mod 4).
(i i) ab ≡ 1 (mod 2) and either ord2(B) = 1 or ord2(C) = 1.
(i i i) ab ≡ 1 (mod 2), ord2(B) = 2 and C ≡ −bB/4 (mod 4).
(iv) ab ≡ 1 (mod 2), ord2(B) ∈ {3, 4, 5} and c ≡ C (mod 4).
(v) ord2(Bbn) ≥ 6 and c ≡ C (mod 4).
In case (v), we will consider the curve

E3(a, b, c) : Y 2
+ XY = X3

+
cC − 1

4
X2

+
BCbn

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X, (2.3)

hich is defined over Q. By [4, Lemma 2.1 ], the conductor of the curve E = E3(a, b, c) is
iven by

N (E) = 2α
· C2

· rad(abAB) (2.4)

here

α =

{
−1 if i = 3, case (v) and ord2(Bbn) = 6,

0 if i = 3, case (v) and ord2(Bbn) ≥ 7.

. Proof of Theorem 1.1

Here and below, we assume that (x, y, z) is a solution of (1.2) with (x, y, z) ̸= (k − 1, 1, 2).
hen, by (1.2), we can get z > y immediately. Obviously, if we can prove that the solution

x, y, z) does not exist, then Conjecture 1.1 is true. The following two lemmas are basic
roperties on the solution (x, y, z).

emma 3.1. Suppose 4 | k. Then 2 ∤ y.

roof. If 2 | y then (1.2) implies x2
+ 1 ≡ 0 (mod 4), impossible. □

emma 3.2. If 2k − 1 is an odd prime power, then 2 ∤ z.

roof. See Lemma 2.6 of [10]. □
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Lemma 3.3. Let F(t) = t + a/t be a function of the real variable t , where a is a constant
with a > 1. Then F(t) is a strictly decreasing function for 1 ≤ t <

√
a.

roof. Since F ′(t) = 1 − a/t2 < 0 for 1 ≤ t <
√

a, where F ′(t) is the derivative of F(t), we
btain the lemma immediately. □

emma 3.4. If k is a power of 2 and 4 | k, then Conjecture 1.1 is true.

roof. Let (x, y, z) be a solution different from (k − 1, 1, 2) and let k = 2r , where r ≥ 2 is a
ositive integer. Then by (1.2), we have

x2
+ (2r+1

− 1)y
= 2r z . (3.1)

ecall that x is odd, hence 2r z/2
+ x and 2r z/2

− x are relatively prime.
If 2 | r z, then from (3.1) we get

2r z/2
+ x = f y, 2r z/2

− x = gy, 2r+1
− 1 = f g, f, g ∈ N, 2 ∤ f g, gcd( f, g) = 1,

hence we obtain

2r z/2+1
= f y

+ gy
= ( f + g)

(
f y

+ gy

f + g

)
. (3.2)

By Lemma 3.1, we have 2 ∤ y. Therefore, since 2 ∤ f g, ( f y
+ gy)/( f + g) is an odd positive

integer. It follows from (3.2) that ( f y
+ gy)/( f + g) = 1, y = 1. Take a = 2r+1

−1 and t = g.
By Lemma 3.3, we have

2r z/2+1
= f + g =

2r+1
− 1

g
+ g ≤ (2r+1

− 1) + 1 = 2r+1. (3.3)

ecall that z > y = 11. We find from (3.3) that f = 2r+1
− 1, g = 1, z = 2 and

x, y, z) = (2r
− 1, 1, 2) = (k − 1, 1, 2), a contradiction.

If 2 ∤ r z, then 2 ∤ r . Since 2 ∤ y, we see from (3.1) that the equation

X2
+ (2r+1

− 1)Y 2
= 4 · 2Z , X, Y, Z ∈ Z, gcd(X, Y ) = 1, Z > 0 (3.4)

as a solution

(X, Y, Z ) = (x, (2r+1
− 1)(y−1)/2, r z − 2). (3.5)

ince 12
+ (2r+1

−1) ·12
= 4 ·2(r−1), (1, 1, r −1) is a solution and clearly the one with smaller

alue of z, hence is what is called a minimal solution in [14]. Then Theorem 2 of [14] implies
hat

r z − 2 = (r − 1)t, t ∈ N. (3.6)

owever, since 2 | r − 1, we get from (3.6) that 2 | r z, a contradiction. Thus, the lemma is
roved. □

emma 3.5. If 4 | k, 2k − 1 is an odd prime power and k is a square, then Conjecture 1.1
s true.

roof. Since k is a square, k = ℓ2, where ℓ is a positive integer with 2 | ℓ. by (1.2), we
ave (ℓz)2

− x2
= (ℓz

+ x)(ℓz
− x) = (2k − 1)y . Since 2k − 1 is an odd prime power and

cd(ℓz
+ x, ℓz

− x) = 1, we have ℓz
+ x = (2k − 1)y and ℓz

− x = 1, whence we get

2ℓz
= (2k − 1)y

+ 1. (3.7)
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Further, since (2k − 1) + 1 = 2ℓ2 and 2 ∤ y by Lemma 3.1, we obtain from (3.7) that

2ℓz
= 2ℓ2

(
(2k − 1)y

+ 1
(2k − 1) + 1

)
,

hence we get

ℓz−2
=

(2k − 1)y
+ 1

(2k − 1) + 1
, (3.8)

here ((2k − 1)y
+ 1)/((2k − 1) + 1) is a positive integer. Notice that 2 | ℓ and 2 ∤

(2k − 1)y
+ 1)/((2k − 1) + 1). We see from (3.8) that z = 2 and (x, y, z) = (k − 1, 1, 2). The

emma is proved. □

For any fixed positive integers m and n with n > 1, there exist unique positive integers f
nd g such that

m = f gn, f is nth-power free. (3.9)

he positive integer f is called the nth-power free part of m, and denoted by f (m). Similarly,
g is denoted by g(m). Obviously, by (3.9), if 2 | m, then we have

ord2(m) = ord2( f (m)) + n ord2(g(m)), 0 ≤ ord2( f (m)) < n. (3.10)

Here, we consider Eq. (1.2) where y ≥ 7 is prime and y = 3 or y = 5, respectively.

.1. The case y ≥ 7 prime

Let k be a positive integer with 4 | k. Suppose that y ≥ 7 is prime. Then Eq. (1.2) becomes

(−1) · (2k − 1)y
+ kz

= x2. (3.11)

hen, the ternary equation (2.1) can be obtained from (3.11) by the substitution

A = −1, a = 2k − 1, B = f (kz), b = g(kz), C = 1, c = (−1)(x−1)/2x, (3.12)

here f (kz) and g(kz) are defined as in (3.9).

emma 3.6. If 4 | k, then b, B and C in (3.12) satisfy the case (v) with ord2(Bbn) > 6.

roof. Since 2 | k, we see from (1.2) that 2 ∤ x . So we have C ≡ (−1)(x−1)/2x ≡ 1
mod 4). In addition, by (3.9) and (3.12), we have Bbn

= f (kz)(g(kz))n
= kz , whence we

et ord2(Bbn) = ord2(kz) = z ord2(k) > 6 (recall that z > y). Thus, the lemma is proved. □

By (2.3) and (3.12), we are interested in the following elliptic curve (called a Frey curve)

E3 : Y 2
+ XY = X3

+
(−1)(x−1)/2x − 1

4
X2

+
kz

64
X. (3.13)

y Lemma 3.6 and (2.4), the conductor of this elliptic curve is

N (E3) = rad(2k − 1) · rad(k). (3.14)

ote that when k = 720, one gets that N (E3) = 43170 and 2k − 1 = 1439 is prime. But
when k = 724, one obtains N (E ) = 523814, outside the range of the Cremona elliptic curve
3

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database [8] where the upper bound for conductors is 500000. We therefore restrict attention
to 30 < k ≤ 720.

Using Lemmas 3.4 and 3.5, we can exclude the cases k = 2r0 with r0 = 6 and k = ℓ2 with
∈ {6, 8, 10, 18, 22, 24}. This leaves 50 values of k to consider. We proceed as follows.
For a given k we compute by (3.14) the conductor of the Frey curve at (3.13). Using

remona’s elliptic curve database [8] we obtain a list of isomorphism classes of elliptic
urves for that conductor. In each class, we must determine whether there exists a model
onsistent with the model at (3.13). For example, when k = 192 and the conductor is 2298,
he isomorphism class of the curve labeled 2298.h4, [1, 0, 0, −184, 1088], contains the curve

[1, −48, 0, 576, 0] (note that this fails to provide a solution to our problem, because the
corresponding values of y, z, equal 1,2, not allowed). Since the curve (3.13) has point (0, 0)
of order 2, it is only necessary to consider isomorphism classes determined by curves with
nontrivial 2-torsion. Suppose a Cremona class representative has nontrivial 2-torsion point T0.
To obtain an isomorphic curve of the form (3.13) we must take the transformation mapping
T0 to (0, 0), and then test the resulting curve to see whether the X− coefficient is of the form
kz

64
. This was programmed into MAGMA. Resulting curves with corresponding (y, z) = (1, 2)

are not allowed, and only one other curve resulted, namely [1, 733/4, 0, 33/16, 0] when
k = 132 with z = 2. But this does not provide a solution to our problem because there is
no corresponding value of x (or y). Finally, thus, we deduce (1.2) has no solutions where
y ≥ 7.

3.2. The case y = 3 or y = 5

Here we solve the Diophantine equations

x2
+ (2k − 1)3

= kz, z > 3 odd, (3.15)

and

x2
+ (2k − 1)5

= kz, z > 5 odd, (3.16)

here 4 | k, 30 < k < 724 and 2k − 1 is an odd prime power.
Write in (1.2) y = 6A + i , z = 3B + j where i = 3 or 5 and 0 ≤ j ≤ 2, A, B ≥ 0. Since

k − 1 is an odd prime power, we have 2k − 1 = pr , where p is an odd prime and r is a
ositive integer. Then we see that(

k B+ j

(2k − 1)2A
,

xk j

(2k − 1)3A

)
s an S-integral point (U, V ) on the elliptic curve

Ei jk : V 2
= U 3

− (2k − 1)i k2 j ,

here S = {p}, 4 | k, 30 < k < 724 and 2k − 1 is a power of p, in view of the restriction
cd(k, x) = 1.

A practical method for the explicit computation of all S-integral points on a Weierstrass
lliptic curve has been developed by A. Pethő, H.G. Zimmer, J. Gebel and E. Herrmann in [16]
nd has been implemented in MAGMA [5]. The relevant routine SIntegralPoints worked
ithout problems for all triples (i, j, k) except for

i, j, k) ∈ {(5, 2, 96), (5, 1, 120), (5, 2, 156), (5, 2, 180), (5, 2, 192), (5, 2, 220),
3, 1, 232), (5, 0, 232), (5, 2, 232), (5, 0, 240), (5, 2, 240), (5, 2, 244), (5, 0, 304),
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(5, 1, 304), (5, 2, 304), (3, 2, 316), (5, 0, 316), (5, 2, 316), (5, 2, 324), (5, 0, 360),
5, 1, 364), (5, 2, 364), (3, 2, 372), (5, 1, 372), (5, 2, 372), (5, 2, 376), (3, 1, 412),
3, 2, 412), (5, 0, 412), (5, 0, 420), (5, 0, 432), (5, 1, 432), (3, 2, 444), (5, 1, 444),
5, 2, 444), (5, 0, 456), (5, 1, 456), (5, 2, 460), (5, 1, 492), (5, 1, 516), (5, 2, 516),

(3, 1, 520), (5, 0, 520), (5, 2, 520), (5, 2, 532), (5, 1, 544), (5, 2, 552), (3, 2, 612),
(5, 0, 612), (5, 1, 612), (5, 2, 612), (5, 1, 616), (5, 0, 640), (5, 2, 640), (3, 2, 652),
(5, 2, 652), (5, 2, 660), (3, 2, 664)(5, 0, 664), (5, 2, 664), (5, 1, 684), (5, 0, 700),
(5, 1, 700), (5, 2, 700), (5, 0, 712), (5, 0, 720), (5, 1, 720)}. The non-exceptional triples (i, j, k)
do not give any positive integer solution to Eq. (3.15) or (3.16).

3.2.1. The elementary approach to some exceptional triples
Thirty-eight of the above exceptional triples have been solved using an elementary approach,

as follows.

Lemma 3.7. If k ≡ 3 or 4 (mod 5), then (3.16) has no solutions (x, z).

Proof. We now assume that (x, z) is a solution of (3.16). If k ≡ 3 (mod 5), then we have
2k − 1 ≡ 0 (mod 5), and by (3.16), x2

≡ kz
− (2k − 1)5

≡ 3z (mod 5). Further, since 2 ∤ z, we
get 1 = (3z/5) = (3/5) = −1, a contradiction, where (∗/∗) is the Legendre symbol. If k ≡ 4
(mod 5), then we have x2

≡ kz
− (2k − 1)5

≡ (−1)z
− 25

≡ −1 − 2 ≡ 2 (mod 5). Hence, we
can get a similar contradiction that 1 = (2/5) = −1. Thus, the lemma is proved. □

Lemma 3.8. If 2 | k and k +1 has an odd prime divisor p with p ≡ ±3 (mod 8), then (3.16)
has no solutions (x, z).

Proof. By (3.16), since 2 ∤ z, we have x2
≡ kz

− (2k − 1)5
≡ (−1)z

− (−3)5
≡ −1 + 243 ≡

242 ≡ 2 · 112 (mod k + 1). It implies that, for every odd prime divisor p of k + 1, we have
1 = (2 · 112/p) = (2/p) = (−1)(p2

−1)/8 and p ≡ ±1 (mod 8). Therefore, if p ≡ ±3 (mod 8),
then (3.16) has no solutions (x, z). □

Notice that if 2 | k and every odd prime divisor p of k + 1 satisfies p ≡ ±1 (mod 8), then
either k + 1 ≡ 1 (mod 8) or k + 1 ≡ −1 (mod 8). Hence, by Lemma 3.8, we can obtain the
following lemma immediately.

Lemma 3.9. If k ≡ 2 or 4 (mod 8), then (3.16) has no solutions (x, z).

Lemma 3.10. For k ∈ {120, 156, 180, 220, 244, 304, 316, 324, 360, 364, 372, 376, 412, 420,

444, 460, 492, 516, 532, 544, 612, 652, 660, 664, 684, 700}, (3.16) has no solutions (x, z).

Proof. By Lemmas 3.7–3.9, (3.16) has no solutions (x, z) for k ∈ {244, 304, 324, 364, 444,

544, 664, 684}, k ∈ {120, 360, 376} and k ∈ {156, 180, 220, 316, 372, 412, 420, 460, 492, 516,

532, 612, 652, 660, 700}, respectively. □

Denote the rank of the elliptic curve Ei jk by r . Here, we separate the above remaining
twentynine exceptional triples (i, j, k) depending on whether r = 0, r = 1 and r = 2,
respectively.
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3.2.2. The case r = 0
For the triples (i, j, k) ∈ {(5, 1, 456), (5, 2, 552), (5, 1, 616), (3, 2, 652), (5, 1, 720)}, there

are no rational points (so no S-integral points) on Ei jk under the assumption that r = 0 which
is proved by descent algorithms of MAGMA.

3.2.3. The case r = 1
For each remaining triple (i, j, k) (in total twentyfour triples), the rank of Ei jk is 1, i.e. r = 1

except for (i, j, k) = (3, 2, 664). We performed two-, four- and eight-descent algorithms or two-
, four-, three- and twelve-descent algorithms for these triples. Since MAGMA found a generator
for each of them, it was successful to show non-existence of S-integral points on Ei jk for the
exceptional twentythree triples.

The rank 1 curves frequently have generators of large height. We can estimate the height of
a generator in advance using the G-Z formula [12], as for example used by A. Bremner [6] in
treating the family of curves y2

= x(x2
+ p). Having an estimate of the height in advance tells

us whether it is likely that standard descent arguments, as programmed into MAGMA, will be
uccessful in finding the generator. For twentythree curves we are considering here, MAGMA
as able to compute generators for all cases, using a combination of three-, four-, eight-, and

welve-descent algorithms.

.2.4. The case r = 2
The single instance of rank 2 was at (i, j, k) = (3, 2, 664). MAGMA found only S-integral

oint (6435758912 : 516297057335360 : 1) on the corresponding curve under the assumption
hat 1 ≤ r ≤ 2 and its generator (402234932 : 8067141520865 : 1). By eight-descent algorithm,

we found two independent points which are generators. So, it is confirmed that r = 2 and this
urve has only S-integral point (6435758912 : 516297057335360 : 1). But it does not give any
ositive integer solution to Eq. (3.15).

To sum up, Theorem 1.1 is proved.
For the computations, we used iMAC computer with the following characteristics: Processor

ntel i5, 2.7 GHz, 16 GB RAM, 1600 MHz DDR3. All computations are done by Magma
2.24-5.
The MAGMA [5] files used to carry out the computations in this paper are available at: htt

://gsoydan.home.uludag.edu.tr/images/programs.zip.

cknowledgments

We are grateful to Professor Andrew Bremner for his generous helps about MAGMA
computations and his useful ideas and would like to thank Professor Benjamin Matschke
for his useful idea that shows how to reduce to S-integral points on Mordell equations
(3.15)–(3.16) and sharing results of computations of his SAGE codes with us and to Professor
Jennifer Balakrishnan for enabling my connection with him. We also cordially thank an
anonymous referee for carefully reading our paper and for giving such constructive comments
which substantially helped improving the quality of the paper. G. S. was supported by the
Research Fund of Bursa Uludag University under Project No: F-2020/8.

References
[1] M. Bennett, N. Billerey, Sums of two S-units via Frey-Hellegouarch curves, Math. Comp. 305 (2017)

1375–1401.
999

http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://gsoydan.home.uludag.edu.tr/images/programs.zip
http://refhub.elsevier.com/S0019-3577(22)00028-3/sb1
http://refhub.elsevier.com/S0019-3577(22)00028-3/sb1
http://refhub.elsevier.com/S0019-3577(22)00028-3/sb1


E.K. Mutlu, M. Le and G. Soydan Indagationes Mathematicae 33 (2022) 992–1000
[2] M.A. Bennett, I. Chen, S.R. Dahmen, S. Yazdani, Generalized fermat equations: a miscellany, Int. J. Number
Theory 11 (2015) 1–28.

[3] M.A. Bennett, P. Mihǎilescu, S. Siksek, The generalized fermat equation, in: J.F. Nash Jr., M. Th Rassias
(Eds.), Open Problems in Mathematics, Springer, New York, 2016, pp. 173–205.

[4] M.A. Bennett, C. Skinner, Ternary diophantine equations via galois representations and modular forms, Canad.
J. Math. 56 (2004) 23–54.

[5] W. Bosma, J. Cannon, C. Playoust, The magma algebra system I, the user language, J. Symbolic Comput.
24 (1997) 235–265.

[6] A. Bremner, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 265, Kluwer Acad. Publ. Dordrecht, 1989.
[7] H. Cohen, Number Theory II: Analytic and Modern Tools, Springer, 2007.
[8] J. Cremona, Elliptic Curve Data, http://johncremona.github.io/ecdata/ and LMFDB - The L-functions and

Modular Forms Database, http://www.lmfdb.org/ or Elliptic curves over Q, http://www.lmfdb.org/EllipticCur
ve/Q/.

[9] H. Darmon, A. Granville, On the equations zm
= F(x, y) and Ax p

+ Byq
= Czr , Bull. London Math. Soc.

27 (1995) 513–543.
[10] M.-J. Deng, J. Guo, A.-J. Xu, A note on the Diophantine equation x2

+ (2c − 1)m
= cn , Bull. Aust. Math.

Soc. 98 (2018) 188–195.
[11] Y. Fujita, N. Terai, On the generalized Ramanujan-Nagell equation x2

+ (2c − 1)m
= cn , Acta Math. Hungar.

162 (2020) 518–526.
[12] B.H. Gross, D.B. Zagier, Heegner points and derivatives of L-series, Invent. Math. 84 (1986) 225–320.
[13] W. Ivorra, A. Kraus, Quelques résultats sur les équations ax p

+ by p
= cz2, Canad. J. Math. 58 (2006)

115–153.
[14] M.-H. Le, Some exponential diophantine equations I: The equation D1x2

− D2 y2
= λkz , J. Number Theory

55 (1995) 209–221.
[15] M.H. Le, G. Soydan, A brief survey on the generalized lebesgue-Ramanujan-Nagell equation, Surv. Math.

Appl. 15 (2020) 473–523.
[16] A. Pethő, H.G. Zimmer, J. Gebel, E. Herrmann, Computing all S-integral points on elliptic curves, Math.

Proc. Cambridge Philos. Soc. 127 (1999) 383–402.
[17] S. Siksek, The modular approach to Diophantine equations, Panor. Synthèses 36 (2012) 151–179.
[18] R. Taylor, A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. Math. 141 (3) (1995) 553–572.
[19] N. Terai, A note on the diophantine equation x2

+ qm
= cn , Bull. Aust. Math. Soc. 90 (2014) 20–27.

[20] A. Wiles, Modular elliptic curves and Fermat’s Last Theorem, Ann. Math. 141 (1995) 443–551.
1000

http://refhub.elsevier.com/S0019-3577(22)00028-3/sb2
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http://refhub.elsevier.com/S0019-3577(22)00028-3/sb6
http://refhub.elsevier.com/S0019-3577(22)00028-3/sb7
http://johncremona.github.io/ecdata/
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http://johncremona.github.io/ecdata/
http://www.lmfdb.org/
http://www.lmfdb.org/
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http://www.lmfdb.org/
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http://www.lmfdb.org/
http://www.lmfdb.org/
http://www.lmfdb.org/
http://www.lmfdb.org/EllipticCurve/Q/
http://www.lmfdb.org/EllipticCurve/Q/
http://www.lmfdb.org/EllipticCurve/Q/
http://www.lmfdb.org/EllipticCurve/Q/
http://www.lmfdb.org/EllipticCurve/Q/
http://www.lmfdb.org/EllipticCurve/Q/
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http://refhub.elsevier.com/S0019-3577(22)00028-3/sb9
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	A modular approach to the generalized Ramanujan–Nagell equation
	Introduction
	Preliminaries
	The modular approach
	Signature (n,n,2)

	Proof of maintheo
	The case y7 prime
	The case y=3 or y=5
	The elementary approach to some exceptional triples 
	The case r=0 
	The case r=1 
	The case r=2 


	Acknowledgments
	References