There are two separate contests on the table. One calls for big-picture, strategic ideas to re-envision the code, with prizes up to $20,000. The other calls for tactical changes that might increase efficiency, with up to $35,000 in cash prizes on offer.

Daniel Merlino, 30, is going for both, and he’s willing to put in some long hours whether or not he wins. He likes a challenge, he said.

Last week, Mr. Merlino, a computer science student at the University of North Texas, came across a BBC News story about the NASA request. He said he doesn’t know Fortran yet but has taught himself other programming languages in the past.

“It’s our responsibility as students to continuously search for these types of situations,” he said. “And stagnation in the technology industry doesn’t really work for you, so you always have to push forward.”

NASA has used crowdsourcing before. It asked members of the general public to come up with ways for astronauts to exercise in space in 2015 and do laundry in microgravity in 2010. Last year, it even held the Space Poop Challenge, which sought a mechanism for astronauts to take care of important business without shedding their spacesuits.

But this new challenge stands out, Mr. Hetle said, because it requires in-depth knowledge of programming codes and algorithms. The winning entry could revolutionize the way scientists use the Pleiades.

The supercomputer in question doesn’t look that glamorous. Found inside the Ames Research Center in Mountain View, Calif., it consists of rows and rows of units that look a lot like high-tech vending machines.