This is set at SETI (the Search for Extra-Terrestrial Intelligence), which also exists in a non-fictional context.

Note that Sagan lists 1 as a prime number. As he died a few years ago, it’s impossible to tell if (i) this was a careless mistake (ii) Sagan for some reason thought 1 should be considered prime (iii) Sagan thought extra-terrestrial intelligences would or could consider 1 to be prime.

[excerpt from p. 78-79]

"...Now let’s take a closer look at those moving pulses. Assuming that this is binary arithmetic, has anybody converted it into base ten? Do we know what the sequence of numbers is? Okay, here, we can do it in our heads...fifty-nine, sixty-one, sixty-seven...seventy-one...Aren’t these all prime numbers?"

A little buzz of excitement circulated through the control room. Ellie’s own face momentarily revealed a flutter of something deeply felt, but this was quickly replaced by a sobriety, a fear of being carried away, an apprehension about appearing foolish, unscientific.

"Okay, let’s see if I can do another quick summary. I’ll do it in the simplest language. Please check if I’ve missed anything. We have an extremely strong, not very monochromatic signal. Immediately outside the bandpass of this signal there are no other frequencies reporting anything besides noise. The signal is linearly polarized, as if it’s being broadcast by a radio telescope. The signal is around nine gigahertz, near the minimum in the galactic radio noise background. It’s the right kind of frequency for anyone who wants to be heard over a big distance. We’ve confirmed sidereal motion of the source, so it’s moving as if it’s up there among the stars and not from some local transmitter. NORAD tells us that they don’t detect satellites - ours or anybody else’s - that match the position of this source. Interferometry excludes a source in Earth orbit anyway.

Steve has now looked at the data outside the automated mode, and it doesn’t seem to be a program that somebody with a warped sense of humor put into the computer. The region of the sky we’re looking at includes Vega, which is an A-zero main sequence dwarf star. It’s not exactly like the Sun, but it’s only twenty-six light-years away, and it has the prototype stellar debris ring. There are no known planets, but there certainly could be planets we don’t know anything about around Vega. We’re setting up a proper motion study to see if the source is well behind our line of sight to Vega, and we should have an answer in - what? - a few weeks if we’re restricted on our own, a few hours if we do some long-baseline interferometry.

Finally, what’s being sent seems to be a long sequence of prime numbers, integers that can’t be divided by any other number except themselves and one. No astrophysical process is likely to generate prime numbers. So I’d say - we want to be cautious of course - but I’d say that by every criterion we can lay our hands on, this looks like the real thing."
 

[excerpt from p.85-87]

Although there had been a range of contingency plans for the public release of any findings, the actual circumstances had caught them largely unprepared. They dreafted as innocuous a statement as they could and released it only when they had to. It caused, of course, a sensation.

They had asked the media’s forbearance, but knew there would be only a brief period before the press would descend in force. They had tried to discourage reporters from visiting the site, explaining that there was no real information in the signals they were receiving, just tedious and repetitive prime numbers. The press was impatient with the absence of hard news. "You can only do so many sidebars on ‘What is a prime number?’" one reporter explained to Ellie over the telephone.

Television camera crews in fixed-wing air taxis and chartered helicopters began making low passes over the facility, sometimes generating strong radio interference easily detected by the telescopes. Some reporters stalked the officials from Washington when they returned to their motels at night. A few of the more enterprising had attempted to enter the facility unobserved - by beach buggy, motorcycle, and in one case on horseback. She had been forced to inquire about bulk rates on cyclone fencing.

Immediately after der heer arrived, he had received an early version of what was by now Ellie’s standard briefing: the surprising intensity of the signal, its location in very much the same part of the sky as the star Vega, the nature of the pulses.

"I may be the President’s Science Adviser," he had said, "but I’m only a biologist. So please explain it to me slowly. I understand that if the radio source is twenty-six light years away, then the message had to be sent twenty-six years ago. In the 1960’s, some funny-looking people with pointy ears thought we’d want to know that they like prime numbers. But prime numbers aren’t difficult. It’s not like they’re boasting. it’s more like they’re sending us remedial arithmetic. Maybe we should be insulted."

"No, look at it this way," she said smiling. "This is a beacon. It’s an announcement signal. It’s designed to attract our attention. We get strange patterns of pulses from quasars and pulsars and radio galaxies and God-knows-what. But prime numbers are very specific, very artificial. No even number is prime, for example. It’s hard to imagine some radiating plasma or exploding galaxy sending out a regular set of mathematical signals like this. The prime numbers are to attract our attention."

"But what for?" he had asked, genuinely baffled.

"I don’t know. But in this business you have to be very patient. maybe in a while the prime numbers will turn off and be replaced by something else, something very rich, the real mesage. We just have to keep on listening."

This was the hardest part to explain to the press, that the signals had essentially no content, no meaning - just the first few hundred prime numbers in order, a cycling back to the beginning, and again the simple binary arithmetic representations: 1, 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31,... Nine wasn’t a prime number, she’d explain, because it was divisible by 3 (as well as 9 and 1, of course). Ten wasn’t a prime number because 5 and 2 went into it (as well as 10 and 1). Eleven was a prime number because it was divisible only by 1 and itself. But why transmit prime numbers? It reminded her of an idiot savant, one of those people who might be grossly deficient in ordinary social or verbal skills but who could perform mind-boggling feats of mental arithmetic - such as figuring out, after a moment’s thought, on what day of the week June first in the year 11,977 will fall. It wasn’t for anything; they did it because they liked doing it, because they were able to do it.

She knew it was only a few days after receipt of the message, but she was at once exhilerated and deeply disappointed. After all these years, they had finally received a signal - sort of. but its content was shallow, hollow, empty. She had imagined recieving the Encyclopedia Galactica.

We’ve only achieved the capacity for radio astronomy in the last few decades, she reminded herself, in a Galaxy where the average star is billions of years old. The chance of receiving a signal from a civilization exactly as advanced as we are should be miniscule. If they were even a little behind us, they would lack the technological capability to communicate with us at all. So the most likely signal would come from a civilization much more advanced. Maybe they would be able to write full and melodic mirror fugues: The counterpoint would be the theme written backwards. No, she decided. While this was a kind of genius without a doubt, and certainly beyond her ability, it was a tiny extrapolation from what human beings could do. Bach and Mozart had made at least respectable stabs at it.

She tried to make a bigger leap, into the mind of somene who was enormously, orders of magnitude, more intelligent than she was, smarter than Drumlin, say, or Eda, the young Nigerian physicist who had just won the Nobel Prize. But it was impossible. She could muse about demonstrating Fermat’s Last Theorem or the Goldbach Conjecture in only a few lines of equations. She could imagine problems enormously beyone us that would be old hat to them. But she couldn’t get into their minds; she couldn’t imagine what thinking would be like if you were much more capable than a human being. Of course. No surprise. What did she expect? It was like trying to visualize a new primary color or a world in which you could recognize several hundred acquintances individually only by their smells...She could talk about this, but she couldn’t experience it. By definition, it has to be mighty hard to understand the behaviour of a being much smarter than you are. But even so, even so: Why only prime numbers?"