"On no account" means that something should never happen or under no circumstances in any situation. Synonyms for this phrase could include "under no condition," "in no way," "under no circumstances," "by no means," "never," or "in no case." These phrases all convey a sense of absolute prohibition and extreme caution. They are often used to communicate the severity of a situation or to emphasize the importance of avoiding a particular action. When used in writing, it is important to choose the synonym that best fits the context and tone of the work.