Timeline for Theorems that are 'obvious' but hard to prove
Current License: CC BY-SA 2.5
10 events
| when toggle format | what | by | license | comment | |
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| Jan 10, 2011 at 15:59 | comment | added | Gottfried Helms | Couldn't correct the previous comment. Surely it should be written: "primefactors of $c^q$ must occur..." . Later I meant: "it is required, that its exponent must be $q$ ". Sorry... | |
| Jan 10, 2011 at 15:55 | comment | added | Gottfried Helms | Some obviousness for the FLT. If you look at the statement of $a^q - b^q = c^q$ then it is firstly obvious that all primefactors of $c$ must occur with the exponent $q$. But from the little theorem of fermat and some checking around it becomes "obvious", that the form of the lhs $ \frac{a^q - b^q}{a-b} $ in the most cases allows primefactors only with low exponent: the number of occurences of high values of a so-called fermat-quotient is somehow reciprocal to its height, but it is required, that it must be $q$. So, few simple (well: not trivial) heuristics make FLT (a bit more) "obvious"... | |
| Jan 10, 2011 at 4:44 | comment | added | Pete L. Clark | I also don't see any reasonable sense in which FLT is obvious. Let me just restrict attention to FLT(3). This problem is equivalent to the following: you are given a rational elliptic curve which has three obvious (!!) torsion points on it. You look by trial and error for a while and don't find any further rational points. What is the chance that there are no further rational points on this curve? Current wisdom suggests: precisely $\frac{1}{2}$. So that there are no further rational points even on this one curve is far from obvious: one should not even bet on it at unequal odds. | |
| Jan 9, 2011 at 15:04 | comment | added | Qiaochu Yuan | Wadim, for what it's worth I also disagree that the Jordan curve theorem is obvious. It's certainly not any more obvious than the nonexistence of space-filling curves. | |
| Jan 9, 2011 at 14:16 | comment | added | Wadim Zudilin | Zen, I guess you are not the one who tells me who could tell me that the Jordan curve theorem is obvious (how many students in your class would agree?). The word "obvious" is too subjective. The things which are obvious to you may be not obvious to others. Therefore, we have a perfect voting system. Everybody feel free to vote down! I have to go to bed to get rid of this nightmare. English language lessons are too much for Sunday. G'night. | |
| Jan 9, 2011 at 14:06 | comment | added | Zen Harper | Wadim, sorry for the downvote; but I think you're misunderstanding the English language here. Although the meaning of the statement is obvious, its truth is definitely not. Suppose you'd never heard of it before, and someone asked you to guess the truth or falsity of the statement, in less than an hour, say for a bet of 100 dollars or similar. I cannot believe that any human being could possibly guess "true" and have enough "reasonable" confidence to want to make the bet. | |
| Jan 9, 2011 at 13:52 | comment | added | Wadim Zudilin | Citing the dictionary, "Obvious: easily perceived or understood; clear, self-evident." Citing the OP, "I'm interested in examples of theorems that are 'obvious', and known to be true." | |
| Jan 9, 2011 at 13:48 | comment | added | Wadim Zudilin | It was already obvious to Fermat! ;-) There are different ways to interpret the word "obvious". Not so many people doubt it's true. And just a few understand its proof. | |
| Jan 9, 2011 at 13:40 | comment | added | Denis Serre | Why is Fermat's last theorem 'obvious' ? | |
| Jan 9, 2011 at 13:36 | history | answered | Wadim Zudilin | CC BY-SA 2.5 |