Endpoints of
US Route 57

Note the part (in parentheses) about "US 57 suggested": normally it would have been AASHO's responsibility to determine the number of a new US route, but in this case TXDoT suggested the specific designation they wanted: US 57.  And "suggested" is not the best word for it: the number "57" was really the whole point. Following is an excerpt from TXDoT's Sep. 1970 application to AASHO:

However, there was one big obstacle: for the past few decades AASHO had taken the position that they would not approve any more single-state US routes. And, as a matter of fact, just a few months earlier, AASHO had brought up the issue again by strongly encouraging the state DOTs to eliminate their single-state US routes by re-designating them as state routes.  A few years prior to this, TXDoT had renumbered this particular route as state hwy. 57, so AASHO could have responded by pointing out that state routes were just as good as US routes, and that it would not be possible to bring this particular highway into the US route system.  But TXDoT had obviously anticipated that line of reasoning; following was their argument for why a state route designation was not good enough:

Apparently those arguments were enough to convince AASHO to grant an exception (and they have not approved any more single-state US routes since then).  The fact that US 57 was conceptualized as a unified international route explains why the highway is signposted north/south, despite having a much more east/west bearing in Texas.  And actually, the number "57" does make a little sense when viewed in context with nearby US 59; the map linked above shows both routes.

Research and/or image credits: Karin and Martin Karner; Alex Nitzman; Dale Sanderson; Michael Summa; Stephen Taylor; Charles Turner