In the fourth episode, Professor Marcus du Sautoy concludes his investigation into the history of mathematics with a look at some of the great unsolved problems that confronted mathematicians in the 20th century. After exploring Georg Cantor's work on infinity and Henri Poincare's work on chaos theory, he sees how mathematics was itself thrown into chaos by the discoveries of Kurt Godel and Paul Cohen, before completing his journey by considering some unsolved problems of maths today, including the Riemann Hypothesis.

The Pythagorean Theorem is simple: x2 + y2 = z2. In this form, the equation can be solved. But what if the 2 is replaced with any positive integer greater than 2? Would the equation still be solvable? More than 300 years ago, amateur mathematician Pierre de Fermat said no, and claimed he could prove it. Unfortunately, the book margin in which he left this prophecy was too small to contain his thinking. Fermat's Last Theorem has since baffled mathematicians armed with the most advanced calculators and computers. Andrew Wiles methodically worked in near isolation to determine the proof for this seemingly simple equation.

Questions are raised about the search for intelligent life beyond the Earth, with UFOs and other close encounters refuted in favor of communications through SETI and radio telescope such as the Arecibo Observatory. The probability of technically advanced civilizations existing elsewhere in the Milky Way is interpreted using the Drake equation and a future hypothetical Encyclopedia Galactica is discussed as a repository of information about other worlds in the galaxy. The Cosmos Update notes that there have been fewer sightings of UFOs and more stories of abductions, while mentioning the META scanning the skies for signals.

There could be an undiscovered species on Earth unlike anything we’ve ever known. Not in the ocean, but the ocean itself! Its body spans thousands of miles; its heart beats with a one-thousand-year pulse. It could even have an immune system capable of annihilating all other life on earth. Just as our bodies function through the interaction of water with individual cells, including bacteria and other microorganisms, the ocean’s residents might collectively form a super-organism. A recent discovery suggests the ocean is a living being capable of thought. If so, what is the ocean thinking about us?

A featured documentary that explores the rise of a new Internet. Decentralized, encrypted, dangerous and beyond the law. The Deep Web, also called the Deep Net, Invisible Web, or Hidden Web is the part of the World Wide Web not indexed by search engines. The deep web was several orders of magnitude larger than the surface web. The majority of the information is hidden or locked inside databases and is growing exponentially at a rate that cannot be quantified. In 2008, to facilitate users of Tor hidden services in their access and search of a hidden .onion suffix, Aaron Swartz designed Tor2web, a proxy application able to provide access by means of common web browsers. Using this application, deep web links appear as a random string of letters followed by the .onion TLD.

Beneath Jupiter's swirling clouds lie our solar system's deepest secrets: from its violent youth, through the birth of life to the death of the sun. Now, scientists are unlocking these secrets and discovering that every living thing exists thanks to Jupiter.