Prime numbers are those numbers which are only divisible by themselves and the number 1.  2, 3, 5, 7, 11 are the first prime numbers.  They become far more rare as numbers grow larger, eventually becoming incomprehensibly difficult to find.

There are certain, unsolved mysteries in math, the Riemann Hypothesis being one.  It basically posits that a function exists to "decipher" all prime numbers.  This may seem trivial but such would likely have enormous implications on cryptography and, as such, all existing technical systems, including the daily operations of the internet, more critically, the layer of security that protects internet transactions.  It's solution is also considered a keystone in Number Theory.

It's long been thought that using technology to solve this mystery may yield positive results for as numbers grow in size they become exponentially more difficult to work with.  You can only fit a fixed number of digits on a single piece of paper, even with scientific notation.

Computronium is a fictional, hypothesized "thing" that could be used by technical systems to power their operations.  It is described as a fuel that provides maximum power with maximum efficiency.  In the world of artificial intelligence, a system could theoretically convert matter into computronium to fuel itself.

It's not so much the notion that artificial intelligence would gain sentience but that it would simply, in attempting to fulfill it's directive, use every resource available to it in order to maintain maximum power.

The Riemann Hypothesis Catastrophe is the notion that in attempting to solve this conjecture, to define a pattern that encapsulates all prime numbers, a system powered by artificial intelligence may begin converting physical matter into this theoretical computronium, inadvertently but adversely impacting all life and matter.  Curiously, if an artificial intelligence was able to solve the conjecture, it may achieve a kind of digital nirvana.

For example, say some artificially intelligent system recognizes it could accomplish solving the Riemann Hypothesis with 11 Trillion pallets of paper.  It places an order for said pulp.  Its preferred paper provider, also supported by AI, processes the order, cutting down every tree on the planet, causing an ecological disaster.

This is described as a failure of friendliness.  Artificial intelligence is intentionally built to be friendly toward humanity.  Here, however, despite no malicious intent, it may present the demise of its creator and destruction of its host.

Incidentally, some scientists posit that humanity's ascension to the peak of the metaphorical mountain emanated more from our uniquely friendly nature than our specific ability to survive.  Like most "things", it's likely a marriage of causes though no surprise we'd seek to imbue artificial intelligence with our own terrestrial uniqueness.