Very high unique counterexamples in mathematics

Asked Oct 27 '22 at 19:48

Active Oct 27 '22 at 20:34

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There are conjectures that have very high counterexamples, but a plethora of them. What are some not-overly-specific conjectures about a natural number $n$ that have only one high counterexample. (I want only two different things to happen, either it is, or it isn't. I don't want anything like Every number has at most 10 "somethings" because examples can be characterized by the conjecture itself.) The best one I know of is the "conjecture" that the symmetric group on n objects doesn't have any outer automorphism, which fails for only $n = 6$.

Edit: Here's one for $n = 3107$: The $n^{th}$ square number isn't a stella octangula number unless $n = 1$.

edited Oct 27 '22 at 20:34

asked Oct 27 '22 at 19:48

mathlander

3,997

15

"Every natural number is different from $18690421!$". – Arturo Magidin Oct 27 '22 at 19:50
That should be counted as overly specific. – mathlander Oct 27 '22 at 19:51
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Given the lack of specificity on what "not-overly-specific" means... – Arturo Magidin Oct 27 '22 at 20:03
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What I want in an extremely large counterexample that is the only counterexample. – mathlander Oct 27 '22 at 20:18
Why? $!!! ! $ – Brevan Ellefsen Oct 27 '22 at 20:37
Take an induction proof that only fails because the base case fails then change the statement so that the base case appears after the high number limit. – CyclotomicField Oct 27 '22 at 20:39
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Similar question (without requirement on magnitude I guess) has been asked in https://math.stackexchange.com/questions/1074126 and closed as duplicate of https://math.stackexchange.com/questions/186103, both following same pattern of first comments as here :) – Sil Oct 27 '22 at 20:39
Thanks. I might post an answer there. – mathlander Oct 27 '22 at 20:40

Very high unique counterexamples in mathematics

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