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Q: What is a convincing argument that logically shows why a conjecture is true?

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It's called a proof

a proof

A counter example is a statement that shows conjecture is false.

A conjecture should be testable. You test it and if it fails the test, it is a false conjecture.

A counter-example.

pie tree

It's a counterexample.

hypothesis

Counter example

a way that shows clear sound reasoning

a way that shows clear sound reasoning

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.

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