I'm preparing a talk for school children (9th year and older) where I would like to explain why proofs are necessary in math. I like to show something that seems "obvious" or can at least be confirmed "without reasonable doubt" with lots of (numerical) examples but is still wrong. My main workhorse for this task is the Mertens conjecture. It is a good example, but maybe it's a tad too complicated and takes too much time to explain. Any ideas for simpler ones? It doesn't have to be number theory but it should be something that doesn't need much background knowledge. Maybe something from graph theory?
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Bill Dubuque
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Frunobulax
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3This one is easier than Mertens: You would believe that $n^{17}+9 \text{ and } (n+1)^{17}+9 \text{ are relatively prime}$. The first counterexample is $n=8424432925592889329288197322308900672459420460792433$ – Dietrich Burde Sep 09 '23 at 09:38
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Do you only want to present surprising counterexamples (like Skewes number) ? Or shall the children also understand why the counterexample is surprising ? Then , I think the suggested examples are still too difficult. – Peter Sep 09 '23 at 11:28
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Recently , a solution of $x^3+y^3+z^3=42$ in integers was found. One would probably have conjectured there is none. – Peter Sep 09 '23 at 11:29
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@Peter It would be sufficient if there was simply something like "overwhelming evidence" for some easy-to-explain conjecture. I think the question mine was considered a duplicate of has indeed some good examples. – Frunobulax Sep 09 '23 at 15:46
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2For many interesting examples see R.K. Guy's papers on the strong law of small numbers. – Bill Dubuque Sep 09 '23 at 19:43
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1An important example is that many students think that it is obvious and needs no proof that prime factorizations are unique. But this is easily corrected by nonunique prime factorizations like $,9\cdot 49=441,,$ i.e. $,3^2 7^2 = (3\cdot 7)^2,$ in the Hilbert numbers $1+4\Bbb N$. General examples of this type of error are in Unique steps leading to a non-unique answer. – Bill Dubuque Sep 09 '23 at 20:05
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If you are looking for examples other than the "extremely large" examples in the linked dupe then please clarify that in the question so we can reopen it. – Bill Dubuque Sep 09 '23 at 20:11