The Goldbach conjecture is probably one of the best known. The conjecture is that every even number greater than 2 can be expressed as a sum of two primes. T. Oliveira e Silva has confirmed the conjecture for number up to 4*10^18 but, despite many years of effort, the conjecture has not been proved.
A counter example is a statement that shows conjecture is false.
A counter-example.
pie tree
hypothesis
It's a counterexample.
Counter example
A conjecture should be testable. You test it and if it fails the test, it is a false conjecture.
The Poincaré Conjecture.
Counter Example
a conjecture is disproved if it is shown to be false. this can be done by providing a single concrete example (e.g. with actual numbers, functions, etc) that shows the conjecture's premise does not necessarily lead to its conclusion. alternatively, a conjecture could be shown to be false (i.e. disproved) by demonstrating that IF it were true then a logical consequence would be a clearly wrong statement (e.g. 2 + 2 =5)
A counterexample.
a proof