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Results tagged with big-list
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user 3106
Questions designed to generate a "big list" of certain results, examples, conjectures, etc. via many individual answers, each contributing one or a few instances. Such a question should typically be in Community Wiki mode (CW); after asking, please, flag for moderators attention requesting the question to be made CW.
137
votes
Suggestions for special lectures at next ICM
How about a lecture on proof assistants/formal proofs?
Most mathematicians are still skeptical of the value of proof assistants, and it's certainly true that proof assistants are still very difficult …
131
votes
Not especially famous, long-open problems which anyone can understand
The lonely runner conjecture. As Wikipedia puts it:
Consider $k + 1$ runners on a circular track of unit length. At $t = 0$, all runners are at the same position and start to run; the runners' sp …
121
votes
Not especially famous, long-open problems which anyone can understand
Gourevitch's conjecture1:
$$\sum_{n=0}^\infty \frac{1+14n+76n^2+168n^3}{2^{20n}}\binom{2n}{n}^7 = \frac{32}{\pi^3}.$$
1Jesús Guillera: About a New Kind of Ramanujan-Type Series; Experimental Mathemati …
121
votes
Nonequivalent definitions in Mathematics
Not a word but a piece of notation: Sometimes I have seen $\subset$ used to mean "is a proper subset of" while other times I have seen it used to mean "is a subset of".
96
votes
Examples of interesting false proofs
$$e^i = (e^i)^{(2\pi/2\pi)} = (e^{2\pi i})^{1/2\pi} = 1^{1/2\pi} = 1.$$
I first saw this one many years ago, written on the wall of a bathroom stall in the Princeton University math department.
86
votes
Accepted
Proofs of the uncountability of the reals
Mathematics isn't yet ready to prove results of the form, "Every proof of Theorem T must use Argument A." Think closely about how you might try to prove something like that. You would need to set up …
- 82.7k
75
votes
How would you have answered Richard Feynman's challenge?
There's a certain gaming/sporting aspect to Feynman's challenge that works in his favor. First of all, as phrased, the challenge gives him a 50/50 shot at being right even if he guesses randomly. Al …
69
votes
Not especially famous, long-open problems which anyone can understand
There are infinitely many primes $p$ such that the repeating part of the decimal expansion of $1/p$ has length $p-1$.
First explicitly asked by Gauss, now generally thought of as a corollary of Artin …
68
votes
What are some reasonable-sounding statements that are independent of ZFC?
Harvey Friedman has devoted a large portion of his career to finding "natural" statements that are unprovable in ZFC. One example is given at the end of Martin Davis's article "The incompleteness the …
60
votes
Examples of common false beliefs in mathematics
False belief: "There are no known sub-exponential time algorithms for NP-complete problems."
This one is tricky for a couple of reasons. The first is that the term "sub-exponential" is sometimes def …
58
votes
Mistakes in mathematics, false illusions about conjectures
Before Erdős and Selberg found an elementary proof of the prime number theorem, G. H. Hardy had predicted that the discovery of such a elementary proof would be cause "for the books to be cast aside a …
57
votes
Old books you would like to have reprinted with high-quality typesetting
I have some experience resurrecting old math books and I want to make a few comments about copyright.
First, it is definitely true that except for very old books, someone owns the copyright. Typicall …
51
votes
Request for examples: verifying vs understanding proofs
Don Zagier has a well-known paper, A one-sentence proof that every prime $p\equiv 1\pmod 4$ is a sum of two squares. An undergraduate mathematics major should be able to verify that this proof is cor …
51
votes
Examples of common false beliefs in mathematics
False belief: Saying that ZFC is consistent is the same as saying that if ZFC proves "there are infinitely many twin primes" (for example) then there really are infinitely many twin primes.
Everybod …
45
votes
Not especially famous, long-open problems which anyone can understand
The Kneser–Poulsen conjecture in dimension 3: An arrangement of (possibly overlapping) unit balls in space is tighter than a second arrangement of the same balls if, for all $i$ and $j$, the distance …