You shall know a number by the company it keeps: Neighborhood effects in number processing.

Little is known about whether number-theoretic properties shape the mental representation and processing of integers. Guided by neighborhood effects in word processing, we investigate whether numbers reside in mental neighborhoods defined by their factors, with prime numbers living alone, and composite numbers inhabiting neighborhoods of varying sizes. We propose that processing a number requires 'searching' through its task-irrelevant factors to find the task-relevant factor; we therefore predict that in a factoring task, more highly composite numbers will be processed more slowly. We find evidence for this prediction in both a multiplication production task (e.g., x*9 = 72; what is x?) and a factor verification task (e.g., "Is 9 a factor 72?"). This neighborhood effect is primarily seen in 'table facts', i.e., numbers that are answers in the well-practiced 12 × 12 times table. These findings set the stage for future studies of how people understand number-theoretic notions such as primality.