Why does one counterexample disprove a conjecture? Can't a conjecture be correct about most solutio

ureji1c8r1 2022-05-08 Answered
Why does one counterexample disprove a conjecture?
Can't a conjecture be correct about most solutions except maybe a family of solutions?
For example, a few centuries ago it was widely believed that 2 2 n + 1 is a prime number for any n . For n=0 we get 3 , for n=1 we get 5 , for n=2 we get 17 , for n=3 we get 257 , but for n=4 it was too difficult to find if this was a prime, until Euler was able to find a factor of it. It seems like this conjecture stopped after that.
But what if this conjecture isn't true only when n satisfies a certain equation, or when n is a power of 2 4 , or something like that? Did anybody bother to check? I am not asking about this conjecture specifically, but as to why we consider one counterexample as proof that a conjecture is totally wrong.P.S. Andre Nicolas pointed out that Euler found a factor when n=5, not 4 .
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Answers (1)

Lara Alvarez
Answered 2022-05-09 Author has 11 answers
This is because, in general, a conjecture is typically worded "Such-and-such is true for all values of [some variable]." So, a single counter-example disproves the "for all" part of a conjecture.
However, if someone refined the conjecture to "Such-and-such is true for all values of [some variable] except those of the form [something]." Then, this revised conjecture must be examined again and then can be shown true or false (or undecidable--I think).
For many problems, finding one counter-example makes the conjecture not interesting anymore; for others, it is worthwhile to check the revised conjecture. It just depends on the problem.
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