Thrust requirements from Saturn 1 rocket's first stage

Question

What was the need / trigger to develop Saturn I rocket, and who defined / what dictated the thrust requirements from Saturn I, so that eight number of H1 engines had to be clustered to achieve that thrust?

Answer 1

According to Wikipedia, the origin of the Saturn I was a US Department of Defense requirement to launch large communication and reconnaissance satellites:

The Saturn project was started as one of a number of proposals to meet a new Department of Defense (DoD) requirement for a heavy-lift vehicle to orbit a new class of communications and "other" satellites. The requirements called for a vehicle capable of putting 20,000 to 40,000 pounds (9,100 to 18,100 kg) into orbit, or accelerating 13,200 to 26,200 pounds (6,000 to 11,900 kg) to trans-lunar injection.

For a given set of propellants, in this case RP-1 kerosene and liquid oxygen for the first stage, and liquid hydrogen and liquid oxygen for upper stages, it's straightforward to use the rocket equation to make rough estimates of the required thrust and fuel tankage for each stage to get a given payload to orbit. Wernher von Braun and his team did the math and worked out that they needed about 6.7 MN thrust on the first stage to build such a launcher. This was a reasonable fit for the F-1 engine currently under development, but the F-1 wouldn't be available in the needed timeframe.

As an alternative, they turned to an engine being developed as a simpler, more powerful version of the S-3D engine family used on the Atlas, Thor, and Jupiter missiles. This was called the X-1 at the time, but evolved into the H-1. Given the target thrust for the first stage, and the expected performance of the H-1, 8 engines was the right number to use.