Trademark is more like reading tea leaves than applying facts to a formula.

The closeness of the marks has to be examined. The more similar they are, the more likely there is infringement.

Target market must be examined. If the two entities are competing or essentially in the same market, the more likely there is infringement.

Similarities in display and marketing must be examined. The more similar these factors are, the more likely there is infringement.

Trademark is a murky area of law involving many many factors that all get weighed in different dimensions. However, in this case I do not feel that Naughty American University is infringing on the National American University trademark.

NAU is definitely not unique to any one university, nor are any of the terms within it unique. If the website was the same name as the university, the university may have a chance to win its claims but since Naughty America University is not the same as National American University AND -- big AND here -- they are not competing in the same market, this makes it much less likely that National American University will be successful. If National American University was selling porn of any kind there would be a real case here, but not when one sells degrees and another sells images of naked people.

I do believe Northern Arizona and National American U's can both have trademarks using NAU because there are other factors involved -- namely the National American and Northern Arizona are not the same, but that is a completely different battle. They ARE competing in the same market but this fight is not between them.