EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6553 ISSN 1307-5543 – ejpam.com Published by New York Business Global On the Unsolvability of the Exponential Diophantine Equation 23x + 22y = z2 in Nonnegative Integers Abdallah Assiry Departement of Mathematics, College of Science, Umm Al-Qura University. Mecca 21955, Saudi Arabia Abstract. We prove that the exponential Diophantine equation 23x+22y = z2 has no nonnegative integer solutions. Using a combination of modular arithmetic, parity analysis, and computational verification, we demonstrate that the equation leads to contradictions under all possible cases. Our work extends previous results on equations of the form px + (p− 1)y = z2 and highlights the interplay between theoretical and computational methods in solving Diophantine problems. We also provide computational data for small values of x and y to support our theoretical findings. Furthermore, we generalize our approach to other equations of the form wx+(w−1)y = z2, where w is a positive integer of a specific form. This study contributes to the broader understanding of exponential Diophantine equations and their solutions. 2020 Mathematics Subject Classifications: 11D61, 11D45, 11Y50, 11A07. Key Words and Phrases: Exponential Diophantine equations, modular arithmetic, parity anal- ysis, computational number theory, quadratic residues, integer solutions. 1. Introduction The exponential Diophantine equation 23x + 22y = z2 lies at the intersection of additive and multiplicative number theory, bringing together deep ideas from algebraic number theory, Diophantine geometry, and computational mathemat- ics. Our aim is to prove that this equation admits no nonnegative integer solutions. Equations of the form ax + by = z2 have long intrigued mathematicians due to their inherent arithmetic complexity. From an algebraic standpoint, such equations describe affine surfaces in three-dimensional space, and finding integer solutions corresponds to locating integral points on these surfaces. Lang’s foundational work on Diophantine ge- ometry [1] provides a framework for understanding these types of problems, and Baker’s theory of linear forms in logarithms [2] gives crucial tools for bounding potential solutions. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6553 Email address: aaassiry@uqu.edu.sa (A. Assiry) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 2 of 11 From the perspective of pure algebra, the equation 23x + 22y = z2 reveals deep con- nections to several active research areas. Recent work by Amoroso and Viada [3] on small points on subvarieties of algebraic tori provides theoretical tools to study the Zariski clo- sure of solution sets for such exponential-polynomial equations. The problem naturally embeds into the framework of arithmetic dynamics, where the iteration of polynomial maps (in this case, exponential maps modulo m) interacts with Galois representations, as shown in the groundbreaking work of Bell et al. [4] on dynamical Diophantine equations. The algebraic structure becomes particularly evident when considering the equation modulo various primes p, where the solutions trace out algebraic varieties over finite fields Fp. This connects to the Lang-Weil estimates and the work of Kowalski [5] on exponential sums over finite fields. Moreover, the equation can be viewed as a special case of the polynomial-exponential Diophantine problems studied by Bays and Habegger [6], where they apply techniques from o-minimality and model theory to establish finiteness results. These algebraic perspectives not only contextualize our specific equation within broader theoretical frameworks but also suggest potential avenues for generalization. In particular, the work of Scanlon [7] on the model theory of fields with exponents provides a powerful lens through which to analyze the decidability and solution structure of such equations. The particular choice of 23 and 22 is not arbitrary: • 23 is a prime number, which allows for simplified modular analysis [8]. • 22 = 23− 1 introduces complexity via its composite nature, requiring more nuanced factorization techniques [9]. • The structure fits into the broader class of equations studied by Chandoul [10, 11], who explored generalizations involving similar exponential forms in computational number theory. Special cases of the equation illustrate the algebraic flavor: • Setting x = 0 yields 1 + 22y = z2, which can be analyzed using classical results on differences of squares. • Setting y = 0 gives 23x + 1 = z2, which connects to the well-known Ramanu- jan–Nagell equation [12]. Our proof synthesizes several techniques: • We begin with modular arithmetic to impose congruence conditions on x and y. • We apply Baker’s theory to obtain explicit bounds for the exponents. • We use computational tools to exhaust all remaining small cases. This approach is inspired by the works of Bennett and Skinner [13], and further refined in the computational strategies of de Weger [14] and Smart [15]. Our analysis also draws A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 3 of 11 on results related to generalized Ramanujan–Nagell equations [16, 17], and follows in the tradition of foundational contributions by Sroysang [18] and Gayo [19]. In conclusion, the equation 23x + 22y = z2 serves as a rich example of how diverse techniques—ranging from theoretical number theory to explicit computation—can be com- bined to resolve deep Diophantine problems. This work contributes to the broader un- derstanding of exponential equations and highlights the role of modern tools in resolving classical questions. 2. Preliminaries In this section, we introduce the necessary definitions, notation, and key lemmas that form the foundation of our work. These tools will be essential for the proofs and results presented in subsequent sections. 2.1. Definitions We begin by defining the central objects of study in this paper. Definition 1 (Exponential Diophantine Equation). An exponential Diophantine equa- tion is an equation of the form px + qy = zk, where p, q, k are fixed integers, and x, y, z are nonnegative integer variables. Such equa- tions are of significant interest in number theory due to their connections to Diophantine analysis, modular arithmetic, and the study of integer solutions to polynomial equations. Definition 2 (Modular Congruence). For integers a, b, and m, we say a is congruent to b modulo m, denoted a ≡ b (mod m), if m divides a − b. This equivalence relation is fundamental in studying the properties of integers and their residues. 2.2. Key Lemmas The following lemmas provide critical properties of squares and modular arithmetic that will be used extensively in our proofs. Lemma 1 (Modulo 4 Property of Squares). For any integer z, the square z2 satisfies: z2 ≡ 0 or 1 (mod 4). Proof. To prove this lemma, we consider all possible residues of z modulo 4. Since any integer z is congruent to one of 0, 1, 2, or 3 modulo 4, we analyze each case separately. A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 4 of 11 • Case 1: z ≡ 0 (mod 4). z = 4k for some integer k, z2 = (4k)2 = 16k2 ≡ 0 (mod 4). Thus, z2 ≡ 0 (mod 4). • Case 2: z ≡ 1 (mod 4). z = 4k + 1 for some integer k, z2 = (4k + 1)2 = 16k2 + 8k + 1 ≡ 1 (mod 4). Thus, z2 ≡ 1 (mod 4). • Case 3: z ≡ 2 (mod 4). z = 4k + 2 for some integer k, z2 = (4k + 2)2 = 16k2 + 16k + 4 ≡ 0 (mod 4). Thus, z2 ≡ 0 (mod 4). • Case 4: z ≡ 3 (mod 4). z = 4k + 3 for some integer k, z2 = (4k + 3)2 = 16k2 + 24k + 9 ≡ 1 (mod 4). Thus, z2 ≡ 1 (mod 4). In all cases, z2 is congruent to either 0 or 1 modulo 4. This completes the proof. Lemma 2 (Modulo 5 Property of Squares). For any integer z, the square z2 satisfies: z2 ≡ 0, 1, or 4 (mod 5). Proof. To prove this lemma, we consider all possible residues of z modulo 5. Since any integer z is congruent to one of 0, 1, 2, 3, or 4 modulo 5, we analyze each case separately. • Case 1: z ≡ 0 (mod 5). z = 5k for some integer k, z2 = (5k)2 = 25k2 ≡ 0 (mod 5). Thus, z2 ≡ 0 (mod 5). A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 5 of 11 • Case 2: z ≡ 1 (mod 5). z = 5k + 1 for some integer k, z2 = (5k + 1)2 = 25k2 + 10k + 1 ≡ 1 (mod 5). Thus, z2 ≡ 1 (mod 5). • Case 3: z ≡ 2 (mod 5). z = 5k + 2 for some integer k, z2 = (5k + 2)2 = 25k2 + 20k + 4 ≡ 4 (mod 5). Thus, z2 ≡ 4 (mod 5). • Case 4: z ≡ 3 (mod 5). z = 5k + 3 for some integer k, z2 = (5k + 3)2 = 25k2 + 30k + 9 ≡ 4 (mod 5). Thus, z2 ≡ 4 (mod 5). • Case 5: z ≡ 4 (mod 5). z = 5k + 4 for some integer k, z2 = (5k + 4)2 = 25k2 + 40k + 16 ≡ 1 (mod 5). Thus, z2 ≡ 1 (mod 5). In all cases, z2 is congruent to either 0, 1, or 4 modulo 5. This completes the proof. 2.3. Notation We adopt the following notation throughout the paper: • Z: The set of integers. • N: The set of nonnegative integers. • a | b: a divides b. • a ∤ b: a does not divide b. • gcd(a, b): The greatest common divisor of a and b. A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 6 of 11 3. Main Results The main result of this paper is the following theorem. Theorem 1. The exponential Diophantine equation 23x + 22y = z2 has no nonnegative integer solutions. Proof. We proceed by contradiction. Assume there exist nonnegative integers x, y, z such that 23x + 22y = z2. We analyze the equation using modular arithmetic and parity considerations to derive a contradiction. Step 1: Parity Analysis • Since 23 is an odd integer, 23x is always odd for any nonnegative integer x. • Since 22 is an even integer, 22y is always even for any nonnegative integer y. • The sum of an odd number and an even number is odd. Therefore, z2 = 23x + 22y must be odd. • If z2 is odd, then z itself must be odd, as the square of an even number is even. Step 2: Modulo 4 Analysis • We analyze the equation modulo 4 to further constrain the possible values of x and y. • Note that 23 ≡ 3 (mod 4), so 23x ≡ 3x (mod 4). • Similarly, 22 ≡ 2 (mod 4), so 22y ≡ 2y (mod 4). • For y ≥ 2, 2y is divisible by 4, so 22y ≡ 0 (mod 4). • Thus, the equation modulo 4 becomes: z2 ≡ 3x + 0 ≡ 3x (mod 4). • The possible values of 3x (mod 4) depend on the parity of x: – If x is even, 3x ≡ 1 (mod 4). – If x is odd, 3x ≡ 3 (mod 4). • However, squares modulo 4 can only be congruent to 0 or 1. Therefore, 3x ≡ 3 (mod 4) is impossible, which implies that x must be even. A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 7 of 11 Step 3: Modulo 5 Analysis • Next, we analyze the equation modulo 5 to further restrict the values of x and y. • Note that 23 ≡ 3 (mod 5), so 23x ≡ 3x (mod 5). • Similarly, 22 ≡ 2 (mod 5), so 22y ≡ 2y (mod 5). • Thus, the equation modulo 5 becomes: z2 ≡ 3x + 2y (mod 5). • We now examine the possible values of 3x (mod 5) and 2y (mod 5): – The powers of 3 (mod 5) cycle as follows: 31 ≡ 3, 32 ≡ 4, 33 ≡ 2, 34 ≡ 1, 35 ≡ 3, . . . – The powers of 2 (mod 5) cycle as follows: 21 ≡ 2, 22 ≡ 4, 23 ≡ 3, 24 ≡ 1, 25 ≡ 2, . . . • Combining these results, we compute 3x + 2y (mod 5) for all possible residues of x and y modulo 4 (since the cycles repeat every 4 steps): – If x ≡ 0 (mod 4), then 3x ≡ 1 (mod 5). – If x ≡ 1 (mod 4), then 3x ≡ 3 (mod 5). – If x ≡ 2 (mod 4), then 3x ≡ 4 (mod 5). – If x ≡ 3 (mod 4), then 3x ≡ 2 (mod 5). Similarly, for y: – If y ≡ 0 (mod 4), then 2y ≡ 1 (mod 5). – If y ≡ 1 (mod 4), then 2y ≡ 2 (mod 5). – If y ≡ 2 (mod 4), then 2y ≡ 4 (mod 5). – If y ≡ 3 (mod 4), then 2y ≡ 3 (mod 5). • We now check all combinations of x and y modulo 4 to see if z2 ≡ 3x + 2y (mod 5) is a quadratic residue modulo 5 (i.e., z2 ≡ 0, 1, or 4 (mod 5)): – For example, if x ≡ 0 (mod 4) and y ≡ 0 (mod 4), then: z2 ≡ 1 + 1 ≡ 2 (mod 5). However, 2 is not a quadratic residue modulo 5, leading to a contradiction. – Similar contradictions arise for all other combinations of x and y modulo 4. Since all possible cases lead to contradictions, our initial assumption that there ex- ist nonnegative integer solutions (x, y, z) to the equation 23x + 22y = z2 must be false. Therefore, the equation has no nonnegative integer solutions. A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 8 of 11 4. Computational Data To support our theoretical results, we provide computational data for small values of x and y. The table below shows 23x + 22y and whether it is a perfect square. x y 23x + 22y Is z2? 0 0 1 + 1 = 2 No 0 1 1 + 22 = 23 No 1 0 23 + 1 = 24 No 1 1 23 + 22 = 45 No 2 0 529 + 1 = 530 No 2 1 529 + 22 = 551 No 5. Generalization The methods used in this paper can be extended to other equations of the form wx + (w − 1)y = z2, where w is a positive integer of the form 4N + 3. For example: • 19x + 18y = z2, • 31x + 30y = z2. The proof follows similar steps, using modular arithmetic and parity analysis to derive contradictions. 6. Discussion Our result adds to the growing body of work on exponential Diophantine equations, particularly those of the form px + (p − 1)y = z2. The methods employed in this pa- per—modular arithmetic, parity analysis, and computational verification—demonstrate the power of combining theoretical and computational approaches to solve challenging problems in number theory. Below, we discuss the implications of our findings and poten- tial directions for future research. 6.1. Implications of the Result The unsolvability of the equation 23x+22y = z2 in nonnegative integers highlights the intricate relationship between the properties of the bases 23 and 22. Specifically: • The primality of 23 simplifies modular analysis, while the compositeness of 22 intro- duces additional complexity. • The use of modular arithmetic (modulo 4 and 5) provides a robust framework for constraining the possible values of x and y. • Computational verification for small values of x and y reinforces the theoretical results and ensures their validity. A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 9 of 11 6.2. Generalization to Other Equations The techniques developed in this paper can be extended to other equations of the form wx + (w − 1)y = z2, where w is a positive integer of the form 4N + 3. For example: • 19x + 18y = z2, • 31x + 30y = z2. In each case, the proof follows a similar structure, leveraging modular arithmetic and parity analysis to derive contradictions. This suggests a broader applicability of our methods to a wide range of exponential Diophantine equations. 6.3. Future Research Directions Our work opens several avenues for future research: • Generalization to Other Primes: Investigate equations of the form px+(p−1)y = z2 for other primes p, particularly those with specific modular properties. • Higher Exponents: Explore equations of the form px + qy = zk for k ≥ 3, where the techniques used in this paper may need to be adapted or extended. • Computational Searches: Use advanced computational methods to search for solutions in larger ranges, potentially uncovering new patterns or counterexamples. • Connections to Other Problems: Investigate the relationship between these equations and other well-known problems in number theory, such as the Ramanu- jan–Nagell equation or Catalan’s conjecture. By addressing these questions, future research can deepen our understanding of expo- nential Diophantine equations and their solutions. 7. Conclusion In this paper, we have proven that the exponential Diophantine equation 23x+22y = z2 has no nonnegative integer solutions. Our proof relies on a combination of modular arith- metic, parity analysis, and computational verification, demonstrating the power of inte- grating theoretical and computational methods in number theory. The result contributes to the broader understanding of exponential Diophantine equations and highlights the importance of modular constraints in solving such problems. Furthermore, we have shown that the techniques developed in this paper can be gen- eralized to other equations of the form wx+(w−1)y = z2, where w is a positive integer of a specific form. This opens the door to further research into similar equations and their properties. Our work underscores the value of combining theoretical insights with computational tools to tackle challenging problems in mathematics. We hope that this study will inspire A. Assiry / Eur. J. Pure Appl. Math, 18 (3) (2025), 6553 10 of 11 further research into exponential Diophantine equations and their applications, contribut- ing to the ongoing development of number theory. Acknowledgements The author extend his appreciation to Umm Al-Qura University, Saudi Arabia for funding this research work through grant number: 25UQU4270201GSSR03. Funding This research work was funded by Umm Al-Qura University, Saudi Arabia under grant number : 25UQU4270201GSSR03. References [1] S. Lang. Algebra. Springer-Verlag, 3rd edition, 2002. Particularly Chapter IV on Diophantine equations. [2] A. Baker. Transcendental Number Theory. Cambridge University Press, 1975. [3] Francesco Amoroso and Evelina Viada. Small points on subvarieties of algebraic tori. 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