This show is beautifully filmed, with nice shots of the Cambridge University campus. And, it is perhaps "good PR" for the mathematics community. In fact, the other primary protagonist of the show is an NSA agent assigned to monitor the work of mathematicians whose research might impact internet security. She is prone to rambling on about how important mathematics is in the modern world (and therefore how dangerous mathematicians are).

Some other remarks:

• The mathematical details are never discussed, but one is left with the impression that the main character's big idea is that there might be some pattern to the primes. Brooks doesn't seem to know what it is or have any conjectures (let alone theorems). We are apparently to believe that the whole world accepts as dogma that everything about prime numbers is essentially random and that it is tantamount to heresy to propose that one might be able to say anything more specific. At one point, he goes to the library and does a search for "prime number patterns" (or something like that). He finds that there are only a few papers on the subject, and those have all been deleted.
• (Perhaps I'm thinking about it too broadly, but it seems to me that there are lots of things known about the prime numbers which could be called "patterns", and many more conjectures about them which are widely known. I'm thinking about the Twin Primes Conjecture, the Riemann Hypothesis, the Prime Number Theorem, Zsigmondy's Theorem, etc. That's why I think it seems silly that everyone makes such a big deal about Brooks' vague idea. But, I suppose the idea more specifically is that Brooks will find a quick and easy way to identify all prime numbers among the natural numbers. That indeed would be interesting, though I'm sure that many other people have thought of this idea and tried to find such a result.)
• In the third episode, the NSA literally attempts to assassinate Brooks. When the attempt fails, they say that they are changing tactics: instead of trying to keep him from discovering it, they will help him find it so that they can be the ones who have this amazing discovery. Setting aside the question of whether the US is truly so evil, the thing that bothers me about this is that it makes it clear that neither Brooks nor the NSA presently has this miraculous formula, it is just a vague conjecture.
• There are numerous coincidences that I find it difficult to swallow:
  • The wife of Brooks' thesis advisor is an archeologist. She seemingly has no interest in mathematics, despite being married to a mathematician. The recently discovered site that she is in charge of is only of interest to her for its historical significance. When she shows Brooks photos from the site he immediately recognizes that the ceiling decorations reveal knowledge of precisely the sort of pattern to the prime numbers that he is looking for. He later visits the site with her, and we learn that the text on the walls is a description of a mathematical algorithm having to do with identifying prime numbers. What a coincidence that this archeological site was assigned to the wife of the thesis advisor of apparently the only mathematician in the world who is working on that very thing.
  • Also coincidentally, there was someone else who worked on the same idea who was also a student at Cambridge. Safiya Zamil apparently both worked with and was in a relationship with Brooks' advisor decades earlier.
• Safiya Zamil's untimely death was ruled a suicide, but the advisor seems to recognize the danger and urges his current student to stop working on this idea before he himself is also found dead.
• Being a mathematical genius on TV, Brooks is of course a somewhat anti-social person. He's inconsiderate, selfish, and seemingly uninterested in people (though he does end up sleeping with a waiter he meets who seems like a decent guy).
• One thing I find odd is the way the characters on this show sometimes have technology one would expect in the 2020s and at other times they seem not to. When Brooks is kicked out of school (for breaking into his dead advisor's office and trashing the place in a fit of rage), he asks to use a phone on a desk wired to a wall to call his dad. So, I was thinking "Oh, this takes place before cell phones!" But then, we see the NSA agent tracking his advisor driving to the car wash where his body will be found by using his cell phone. Sometimes, Brooks has to go to the library to search for things. And, when he wants to find out what happened to Safiya Zamil he goes to the math department at the university and asks someone behind the desk. However, when he wants to learn about the mysterious "Keplar Institute" he searches for that on his laptop at his apartment. (So, why didn't he do those library searches or search "Safiya Zamil" from his laptop as any normal person would have done?)
• I had to laugh when he went to the math department to ask for Safiya Zamil's thesis. The person working there goes in a back room and when she comes back she says "We, don't have her thesis, but we do have this" and she hands him a box containing some notes Safiya was working on. Yeah, right!
• Another threat to internet security, which hasn't been mentioned on the show AFAIK, is quantum computing. If it ever actually becomes possible to build a quantum computer, then Shor's Algorithm would allow one to break standard public key encryption without a need for Brooks' number theory ideas.