Let D₁, D₂ be coprime odd integers with min (D₁, D₂) > 1, and let N (D₁, D₂) denote the number of positive integer solutions (x, m, n) of the equation D₁x²+D^m₂ = 2ⁿ⁺². In this paper, we prove that N (D₁, D₂) ≤ 2 except for N (3, 5) = N (5, 3) = 4 and N (13, 3) = N (31, 97) = 3.

Image inpainting has attracted considerable attention in computer vision and image processing due to its wide range of applications. While deep learning-based methods have shown promising potential, accurately recovering pixel-level details remains a significant challenge, particularly in the presence of large and irregular missing regions. Furthermore, existing methods are limited by unidirectional semantic guidance and a localized understanding of global structural context. In this study, we propose a mask-guided dual-branch Transformer-based framework, named MDT-FI, which effectively balances local detail restoration and global contextual reasoning by explicitly modeling long-range dependencies. MDT-FI consists of three key components: the Interactive Attention Module (IAM), the Spectral Harmonization Module (SHM), and the Lateral Adaptation Network (LAN). The model integrates multi-scale feature interaction, frequency-domain information fusion, and a mask-guided attention mechanism to progressively build cross-level feature associations. This design facilitates multi-level representation learning and optimization, thereby enhancing local texture synthesis while preserving global structural consistency. To further improve perceptual quality, a feature augmenter is employed to assess the fidelity of both texture and structure in the generated results. Extensive experiments on CelebA-HQ, Places2, and Paris Street View demonstrate that MDT-FI significantly outperforms state-of-the-art methods.

The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of character sums analogous to high dimensional Kloosterman sums, and give some interesting identities for it.

Jeffrey Hoffstein et al. (Discrete Appl. Math. 130:37–49, 2003) introduced the Low Hamming Weight products (LHWP) $X=x_{1}x_{2}x_{3}$X=x1x2x3 as random exponent of elements in a group or a ring to improve the operational efficiency, where each $x_{i}$xi has Hamming Weight $\operatorname{Ham}(x_{i})$Ham(xi) in its binary representation. The random power or multiple be used in many cryptographic constructions, such as Diffie–Hellman key exchange, elliptic curve ElGamal variants, and NTRU public-key cryptosystem. But their randomness is just a conjecture, which lacks of the security proof. The main purpose of this paper is using the analytic method and the properties of the character sums to prove the distribution of the Hamming weight products, which is related to their pseudorandomness and unpredictability. It is important to research the application of LHWP in cryptographic constructions. Our theory shows that the LHWP are exponentially close to the uniform distribution, namely, an attack on algorithm (Hoffstein et al. in Discrete Appl. Math. 130:37–49, 2003) needs polynomial time to reach exponentially close probabilities of success.