Authorship verification (AV) is a fundamental task in natural language processing (NLP) and computational linguistics, with applications in forensic analysis, plagiarism detection, and identification of deceptive content. Existing AV techniques, including traditional stylometric and deep learning approaches, face limitations in terms of data requirements and lack of explainability. To address these limitations, this paper proposes PromptAV, a novel technique that leverages Large-Language Models (LLMs) for AV by providing step-by-step stylometric explanation prompts. PromptAV outperforms state-of-the-art baselines, operates effectively with limited training data, and enhances interpretability through intuitive explanations, showcasing its potential as an effective and interpretable solution for the AV task.