What are some examples of when Mathematics 'accidentally' discovered something about the world? I do not remember precisely what the equations or who the relevant mathematicians and physicists were, but I recall being told the following story. I apologise in advance if I have misunderstood anything, or just have it plain wrong. The story is as follows.


  A quantum physicist created some equations to
  model what we already know about sub-atomic particles. His equations
  and models are amazingly accurate, but they only seem to be
  able to hold true if a mysterious particle,
  currently unknown to humanity, exists. 
  
  More experiments are run and lo and
  behold, that 'mysterious particle' in actual fact exists! It was found to be a quark/dark-matter/anti-matter, or something of the sort.


What similar occurrences in history have occurred, where the mathematical model was so accurate/good, that it 'accidentally' led to the discovery of something previously unknown? 

If you have an answer, could you please provide the specific equation(s), or the name of the equation(s), that directly led to this?

I can recall one other example.


  Maxwell's equations predicted the existence of radio waves, which were
  then found by Hertz.
The planet Neptune's discovery was an example of something similar to this. It was known that Newtons's Equations gave the wrong description of the motion of Uranus and Mercury. Urbain Le Verrier sat down and tried to see what would happen if we assumed that the equations were right and the universe was wrong. He set up a complicated system of equations that incorporated a lot of ways contemporary knowledge of the universe could wrong, including the number of planets, the location and mass of the planets, and the presences of the forces other than gravity. He would eventually find a solution to the equations where the dominating error was the presence of another, as of yet undetected, planet. His equations gave the distance from the sun and the mass of the planet correctly, as well as enough detail about the planet's location in the sky that it was found with only an hour of searching.

Mercury's orbit's issues would eventually be solved by General Relativity.
I did a quick scan and saw no mention of Riemannian manifolds here. Historically, it might've been the exact high-precision mathematical machinery that Einstein needed; from this  this wikipedia link:


  Albert Einstein used the theory of Riemannian manifolds to develop his
  general theory of relativity. In particular, his equations for
  gravitation are constraints on the curvature of space.