Although the hype is overblown, there is nothing particularly wrong with the article (except for the hype), which is why it was accepted into _Historia Mathematica._

A good introduction to Old Babylonian mathematics (the broader context for this paper) is in a paper by (surprise) Knuth, called _Ancient Babylonian Algorithms._ (You can find PDF links here: https://scholar.google.com/scholar?cluster=10887370978433539...) It's a very nice paper, and an example of Knuth's scholarly approach: despite not being a historian in this paper he produced the first translation of many old Babylonian tablets into English, by comparing the published German and French translations and then looking up Akkadian and Sumerian dictionaries to resolve differences! (Unfortunately he misread one of the secondary sources and later published a retraction; ignore the first column on page 672, about sorting (everything about Inakibit-Anu.)

Now to Plimpton 322: it has, for various triples (a, b, c) satisfying a^2 + b^2 = c^2, columns containing a, c, and either a^2/b^2 or c^2/b^2. This has been known for decades; what's been unclear is what this table was used for. Some have proposed this was number-theoretic (just a listing of Pythagorean triples), but this doesn't answer why these specific triples were chosen. Some have proposed this was trigonometric, but the concept of an angle is not otherwise known in Babylonian mathematics.

Enter Wildberger: for many years now he's championed "rational trigonometry" (trigonometry using only rational numbers). So instead of the sine and cosine he proposes using their squares; instead of angles there's something called "spread", etc. It all works reasonably fine, is not significantly harder to do than regular trigonometry (he claims it's easier, but it's about the same), and you get to work with only rational numbers. (This is also related to a broader project within mathematics which rejects infinite numbers -- see finitism, ultrafinitism.)

So the Plimpton 322's fractions of the form a^2/b^2 are exactly like the functions of "rational trigonometry", and the authors of this paper (http://www.sciencedirect.com/science/article/pii/S0315086017...) make the argument that these tables were used for such a purpose. In one of the recent discussions, a commenter here pointed out this demonstration they (the commenter) made: https://teacher.desmos.com/activitybuilder/custom/59a05b5f50...

If you ignore the "more exact fractions" (which is true under a particular interpretation, i.e. numbers that can be written as finite sequences of integers in the "floating-point" representation), and "more useful than current mathematics" hype from the newspapers, the paper itself is sound and an argument about the nature of this table that is as plausible as any other speculation. They address previous interpretations and discussions, and have a coherent theory of how this table could have been generated or what it could have been used for. It even addresses the objection that Babylonians didn't have the concept of an angle: using exactly the objections raised against the trigonometric interpretation to say they used rational-trigonmetry-like functions instead. :-)

The paper is a good example of one aspect of the history of mathematics, the way arguments are adduced for speculation on topics where there's little evidence (have you read the rhetoric in Robson's paper?), and the fun of constructing explanations out of little data. Here it is again: http://www.sciencedirect.com/science/article/pii/S0315086017...