Factual error: In the beginning of the episode, Charlie is stating that "there is always a solution" and "if there's any limitation it's got to be in the mathematician, not the math". Unfortunately, according to Kurt Godel's incompleteness theorems, this is not true. Simply put, there are mathematical problems that cannot be proven/solved.
Factual error: The Secret Service agent says the bills are counterfeit because you can see the watermark under a uv light, which is completely wrong. The uv reactive strip is not a watermark, it's a plastic strip embedded between the two layers of the bill and fluoresces under uv as a security measure. In older bills, it glows blue as in the film. In newer ones, it glows yellow to orange and is on the right half of the bill. A fake bill would be the one that does not glow.
Factual error: Season 1, Episode 10, "Dirty Bomb": The opening figures refer to "500g of Nuclear Material." These are the wrong units to determine the hazard level in a radiological sample. The "Curie content" describes, roughly, how much radiation will be given off, and is not tied directly to the sample's mass. 500g of Cesium-137 (the isotope being discussed in the episode) could well be less radioactive than 1g of a different isotope.
Add timeRooster of Doom
Factual error: In one episode, there is a chase with the suspect driving an ambulance. The chase ends when the ambulance runs into and flips over a motionless taxi perpendicular to it, killing the suspect. There's no way this could happen, because at those speeds the impact would make the taxi flip over if anything.
Factual error: When Charlie describes the Uncertainty Principle, he is actually describing the Observer Effect, a concept commonly mistaken for the Uncertainty Principle. The Uncertainty Principle actually states that the position and momentum of a particle cannot be accurately known simultaneously.