1 Introduction
Diophantine analysis is a significant area within number theory. Diophantine equations appear naturally when we look for integer or rational solutions of algebraic or geometric problems. Such equations arise in many areas, such as the study of integer points on curves, approximation of real numbers by rational numbers, dynamical systems and cryptography. In cryptography, elliptic curves defined by such equations are used in modern public-key encryption systems. As another example, the geometric problem of finding right-angled triangles whose side lengths are integers leads, by the Pythagorean theorem, to the equation
$x^2+y^2=z^2$
.
In 1637, Fermat proposed that for any integer
$n \geq 3$
, the equation
$x^n + y^n = z^n$
admits no positive integer solutions. In 1990, Ribet [Reference Ribet10] demonstrated that the equation
$x^n + y^n = 2z^n$
lacks nontrivial solutions for
$n> 2$
(the trivial solutions include
$x = y = z$
). In 1995, Elkies [Reference Elkies, Coates and Yau4] examined the equation
$x^3 + y^3 = nz^3.$
In contrast to these scalar equations, Mallick and Mishra [Reference Mallick and Mishra8] showed that the Diophantine equation
${X^n+Y^n=2Z^n}$
admits infinitely many solutions within various structured matrix classes when
$n=3$
. A question that remains unanswered is whether there is a relation between the set of solutions of a matrix Diophantine equation and the solutions of the corresponding scalar equation over certain number fields. Answering this will provide a key motivation for studying matrix Diophantine equations.
In 1937, Erdős and Obláh [Reference Erdős and Obláth5] established that the equation
$x^p \pm y^p = n!$
possesses no solutions for prime exponents
$p> 2$
. In 2025, Bhoi et al. [Reference Bhoi, Rout and Panda9] showed that Diophantine equations with binary recurrence sequences possess only finitely many solutions. Guy [Reference Guy6, page 219] asked for a proof that the equation
has no integer solutions when n is a positive integer greater than 2. Contrary to this conjecture, Jena [Reference Jena7] demonstrated in 2018 that the Diophantine equation
$x^n + y^n = n!z^n$
admits infinitely many primitive integer solutions in the specific case
$n = 3$
.
Matrix equations such as
$AX = B$
,
$AX + XB = C$
(Sylvester equation),
$A^{T}X + XA - XBR^{-1}B^{T}X + Q = 0$
(Riccati equation),
$\dot {X}(t) = AX(t) + X(t)B$
(matrix differential equation),
$AX = \lambda BX$
(generalised eigenvalue problem),
$AX - XA = C$
(commutator equation),
$X^{2} + AX + B = 0$
(matrix polynomial equation) and
$X^n+Y^n=Z^n$
(matrix Fermat equation) arise naturally in a wide range of mathematical and engineering contexts, including control theory, signal processing, systems theory, stability analysis, partial differential equations, data fitting, machine learning, quantum mechanics and operator theory. These equations often contain very important information about a system, such as its structure and whether it is stable. Strong theories and fast, reliable methods to solve matrix equations are major topics of research.
Let
$d\neq 0$
be a fixed integer. Białek and Grytczuk [Reference Białek and Grytczuk1] introduced the matrix class
and studied the solutions of the Fermat equation
$X^n + Y^n = Z^n$
in
$G_2(d)$
.
Clearly,
$G_2(d)$
is closed under the usual multiplication. Basic results dealing with
$G_2(d)$
, which are required to prove our three main results stated below, are collected in Section 2. In 1998, Cao and Grytczuk [Reference Cao and Grytczuk2] obtained partial results on Fermat-type equations in the set
$G_2(d)$
under certain conditions.
We establish relationships between solutions of the matrix Diophantine equation
for
$2 \times 2$
integer matrices in
$G_2(d)$
and solutions of the corresponding scalar equation (1.1) in the number field
$\mathbb {Q}(\sqrt {d})$
in the following two theorems.
Theorem 1.1. Let
$n> 1$
be an integer. The matrix Diophantine equation
admits a solution in
$G_2(d)$
if and only if there exist numbers
$ \theta , \eta , \omega \in \mathbb {Q}(\sqrt {d}) $
satisfying
Theorem 1.2. Let K be a number field and let
$ n \in \mathbb {N} $
. For any
$ p, q, r \in K $
satisfying the equation
the matrices
satisfy the related matrix Diophantine equation
We next introduce a determinant-fixed family of matrices associated with
$G_2(d)$
. For a fixed
$l \in \mathbb {Z}$
and a nonzero square-free integer d, define
The final result of this article, stated below, concerns a solution of a matrix Diophantine equation in
$G_2(d,l)$
.
Theorem 1.3. Let
$X, Y, Z \in G_2(d,l).$
Then the Diophantine equation
admits only trivial solutions for
$n>2$
.
The equations
$x^n+y^n=n!z^n$
and
$x^n+y^n=2z^n$
arise in supercongruences, norm equations, elliptic curves and symmetry problems. The connections developed in Theorems 1.1 and 1.2 linking matrix Diophantine equations and scalar Diophantine equations may serve as a useful foundation for further research on the arithmetic and structural properties of such equations.
2 Background results
In proving the theorems, we make use of the following lemmas.
Lemma 2.1 [Reference Białek and Grytczuk1, Lemma 1]
If
$A^n= \big (\begin {smallmatrix} \alpha & \beta \\ d\beta & \alpha \end {smallmatrix}\big )$
with
$A=\big (\begin {smallmatrix} a_1 & a_2 \\ da_2 & a_1 \end {smallmatrix}\big )\in G_2(d)$
and
$n \geq 2$
, then
and
Lemma 2.2 [Reference Białek and Grytczuk1, Lemma 3]
Let
$d\neq 0$
be a fixed integer and let
$X = \big (\begin {smallmatrix} x_1 & x_2 \\ x_3 & x_4 \end {smallmatrix}\big ) \in M_2(\mathbb {Z}).$
If, for some integer
$n\geq 2$
, the matrix X satisfies
$X^n = \big (\begin {smallmatrix} \alpha & \beta \\ d\beta & \alpha \end {smallmatrix}\big ),$
where
$\alpha , \beta \in \mathbb {Z}\setminus \{0\}$
, then
$X\in G_2(d).$
3 Proof of the main results
The proof techniques for Theorems 1.1 and 1.2 are adapted from those employed in Theorems 1 and 2 of Białek and Grytczuk [Reference Białek and Grytczuk1]. For completeness, we give a concise proof of these results below.
Proof of Theorem 1.1
Let
$d \neq 0$
be a fixed integer, and suppose that
$X, Y, Z \in G_{2}(d)$
are given by
Assume that the matrices
$X, Y, Z$
satisfy the matrix Diophantine equation
for some integer
$n \ge 2$
. By Lemma 2.1,
where
for
$i=1,2,3$
. Thus, by (3.1),
Now, substituting
$\alpha _i,~ \beta _i$
for
$i=1, 2, 3$
in (3.2) and adding the resulting equations, we obtain
Setting
$x_1 + y_1 \sqrt {d}=\theta , \ x_2 + y_2 \sqrt {d}= \eta $
and
$x_3 + y_3 \sqrt {d}=\omega $
, we have
Conversely, suppose that
$\theta , \eta , \omega \in \mathbb {Q}(\sqrt {d})$
satisfy
Then we may write
and their conjugates
for some integers
$x_i, y_i, 1\leq i \leq 3$
. Taking conjugates on both sides of (3.3) yields
From equations (3.3) and (3.4),
By defining
equations (3.5) and (3.6) become
Now consider the matrices
where
$\beta _i \neq 0$
for
$ 1\leq i \leq 3$
. By (3.7),
From Lemmas 2.1 and 2.2, it follows that there exist matrices
$ X, Y, Z $
satisfying
$X_1 = X^n, \ Y_1 = Y^n$
and
$Z_1 = Z^n$
. Substituting these into (3.8) gives
$X^n + Y^n = n!Z^n$
and the matrices
Remark 3.1. Ribet [Reference Ribet10] showed that the Diophantine equation
$x^n + y^n = 2z^n$
lacks nontrivial solutions for
$n> 2$
, except for the case
$x = y = z$
. Similar to Theorem 1.1, we can establish a relationship between the solutions of the matrix analogue of this equation in
$G_2(d)$
and its solutions in the number field
$\mathbb {Q}(\sqrt {d})$
.
More generally, let
$n> 1$
and
$a_n, b_n, c_n$
be integers. Then, the matrix Diophantine equation
admits a solution in
$G_2(d)$
if and only if there exist numbers
$\theta, \eta, \omega \in \mathbb{Q}(\sqrt{d}) $
satisfying
Proof of Theorem 1.2
For any matrix of the form
$P = (\begin{smallmatrix} 0 & 1 \\ p & 0 \end{smallmatrix}),$
we have
$P^2=pI$
. Consequently, with
$P, Q, R$
as in the statement of the theorem,
Thus,
$P^{4n} + Q^{4n} = n! R^{4n}$
, as required.
As in Remark 3.1, there is an extension of Theorem 1.2 to the matrix Diophantine equation
$a_nP^{4n} + b_nQ^{4n} = c_nR^{4n}$
for any integers
$a_n, b_n, c_n$
.
If we set
$n = 1$
and
$K = \mathbb {Q}$
in Theorem 1.2, we obtain the matrix form of Fermat’s equation of exponent
$4$
with rational entries, which admits infinitely many solutions. This gives the following corollary. (For more details see [Reference Domiaty3].)
Corollary 3.2. Let
$ K = \mathbb {Q}, p, q, r \in \mathbb {Z}\setminus \{0\}$
and
$n=1.$
Then the matrix Diophantine equation
$ P^4 + Q^4 = R^4 $
admits infinitely many solutions of the form
where
$p = ( u^2 - v^2) w, ~ q = 2uvw$
and
$r = ( u^2 + v^2 ) w$
, for
$u> v, w\geq 1$
.
Remark 3.3. Let K be a number field and let
$ n \in \mathbb {N} $
. For any
$ p, q, r \in K $
satisfying the equation
the matrices
satisfy the related matrix Diophantine equation
$P^{4n} + Q^{4n} = 2R^{4n}.$
Proof of Theorem 1.3
Suppose that
$X = \big (\begin {smallmatrix} x_{1} & y_{1} \\ d y_{1} & x_{1} \end {smallmatrix}\big ),~ Y = \big (\begin {smallmatrix} x_{2} & y_{2} \\ d y_{2} & x_{2} \end {smallmatrix}\big ),~ Z = \big (\begin {smallmatrix} x_{3} & y_{3} \\ d y_{3} & x_{3} \end {smallmatrix}\big )$
is a solution of (1.2) with
$\det X= \det Y = \det Z = l.$
By Lemma 2.1 and in a similar manner to the proof of Theorem 1.1, we obtain
and
where
$\sigma _i = x_i + y_i \sqrt {d}$
and
$\rho _i = x_i - y_i \sqrt {d}$
for
$i=1,2,3.$
The assumption
$X^n+Y^n=2Z^n$
yields
and consequently,
Furthermore,
$\det X= \det Y= \det Z = l = x_i^2-dy_i^2={\sigma _i}{\rho _i}$
for
$i= 1, 2, 3.$
Multiplying equations (3.10) now gives
and, since
$\sigma _i\rho _i=l$
for each i, this gives
If
$l={\sigma _i}{\rho _i}=0,$
then
${\sigma _i}=0$
or
${\rho _i}=0$
. In this case,
$\alpha _i = \tfrac 12( {\sigma _i}^n + {\rho _i}^n)= 2^{n-1}x_i^n$
for
$i=1, 2, 3.$
Thus, (3.9) implies that
But, from the result given by Ribet [Reference Ribet10], (3.12) has only a trivial solution
$x_1=x_2=x_3$
for
$n>2$
. Similarly, we can obtain
$ y_1=y_2=y_3$
so that
$X=Y=Z$
.
We can now assume that
$l\neq 0$
. Since
$l={\sigma _1}{\rho _1}={\sigma _2}{\rho _2}$
, dividing both sides of (3.11) by
${\sigma _1}{\rho _1}$
gives
Taking
$({\rho _2}/{\rho _1})^n=t$
, we can rewrite (3.13) as
$-2+t+{1}/{t}=0$
so that
$t=1$
and consequently
$({\rho _2}/{\rho _1})^n=1.$
Here we have two cases.
Case I. If n is odd, then
${\rho _2}/{\rho _1}=1$
so that
$\rho _2= \rho _1.$
From (3.10),
$ {\rho _1}^n + {\rho _2}^n= 2 {\rho _3}^n$
so
${\rho _1}^n = {\rho _3}^n$
. Therefore, we have
$ {\rho _1} = {\rho _2}= {\rho _3},$
which implies that
$x_1=x_2=x_3$
and
$y_1=y_2=y_3$
and so
$X=Y=Z$
. Hence, (1.2) has only a trivial solution.
Case II. If n is even, then
${\rho _2}/{\rho _1}=\pm 1$
, that is,
$\rho _2=\pm \rho _1.$
From (3.10),
${\rho _1}^n + {\rho _2}^n= 2 {\rho _3}^n$
so that
${\rho _3} = \pm {\rho _1}$
. Therefore, we have
$X=Y=Z$
,
$X=Y=-Z$
,
$X=-Y=Z$
or
$-X=Y=Z$
. Hence, (1.2) has only trivial solutions.
