There are widespread fears that conversational artificial intelligence (AI) could soon exert unprecedented influence over human beliefs. In this work, in three large-scale experiments ( N = 76,977 participants), we deployed 19 large language models (LLMs)—including some post-trained explicitly for persuasion—to evaluate their persuasiveness on 707 political issues. We then checked the factual accuracy of 466,769 resulting LLM claims. We show that the persuasive power of current and near-future AI is likely to stem more from post-training and prompting methods—which boosted persuasiveness by as much as 51 and 27%, respectively—than from personalization or increasing model scale, which had smaller effects. We further show that these methods increased persuasion by exploiting LLMs’ ability to rapidly access and strategically deploy information and that, notably, where they increased AI persuasiveness, they also systematically decreased factual accuracy.