The #1 most popular question on MO is "Examples of common false beliefs in mathematics".

 - The #8 most popular answer is "[if a set is open, then it is not closed][1]".

 - The #4 most popular answer is "[the product of the first n prime numbers is always prime][2]".

[Here][3] are verifications that these are indeed false beliefs. :)


  [1]: https://mathoverflow.net/a/23580/4600
  [2]: https://mathoverflow.net/a/23521/4600
  [3]: https://live.lean-lang.org/#code=import%20Mathlib.Topology.Clopen%0Aimport%20Mathlib.Data.Nat.Prime%0Aimport%20Mathlib.NumberTheory.SmoothNumbers%0Aimport%20Mathlib.Tactic.NormNum.Prime%0A%0Aexample%3A%20IsOpen%20(%E2%88%85%20%3A%20Set%20Bool)%20%3A%3D%20trivial%0Aexample%3A%20IsClosed%20(%E2%88%85%20%3A%20Set%20Bool)%20%3A%3D%20isClosed_empty%0A%0Aexample%3A%20%E2%88%83%20n%2C%20%C2%AC%20Nat.Prime%20(((Nat.primesBelow%20n).prod%20(id))%20%2B%201)%20%3A%3D%20by%0A%20%20exact%20Exists.intro%2014%20(Mathlib.Meta.NormNum.not_prime_mul_of_ble%2059%20509%20(59%20*%20509)%20rfl%20rfl%20rfl)%0A