The report, innocuously titled A Study of Lunar Research Flights, focuses almost exclusively on the scientific benefits from such a mission.

[In 1958, United States Air Force Physicist Leonard] Reiffel envisioned soft landing three identical scientific instrument packages carrying seismometers and radiation detectors at random on the visible face of the moon. These in situ stations would complement optical and spectroscopic observations from ground-based observatories and high altitude telescopes hoisted by balloons. Studying the nuclear reaction would teach scientists about our satellite’s environment and history, knowledge that would ultimately help us understand the Earth.

But it's the military aspect of this proposal that's really interesting.

The Nazis had explored piloted space planes that could skip off the atmosphere to drop bombs on America; the United States Air Force considered adopting this technology with the Dyna-Soar. Throughout the 1950s, unmanned missiles overtook piloted bombers as they became increasingly sophisticated and their range improved.

But space-based military platforms became the sought after military high ground after Sputnik. With a satellite passing over the United States every 90 minutes, the worry was that the enemy could very precisely drop a bomb from orbit. Militarizing space thus became a feature in most long-range space plans in America. The idea was that whoever controlled the skies controlled the world.

Detonating a nuke on the moon would give American scientists a first hand look at the realities of nuclear weapons in space -- what to expect from those detonated by either the United States or by the Soviet Union. This experiment would teach scientists how to detect nuclear material in space. It would give them a clearer understanding of how capable and effective nuclear weapons would be in space, going a long way to determine whether nuclear warfare would even be feasible in space.