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Results tagged with big-list
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user 28104
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.
119
votes
Examples of unexpected mathematical images
One can obtain a nice picture showing somewhat unexpected patterns by marking
all rational points on the unit sphere whose coordinates have denominator
less than some upper bound, and projecting this …
73
votes
17
answers
9k
views
Mathematical research published in the form of poems
The article
Friedrich Wille: Galerkins Lösungsnäherungen bei monotonen Abbildungen,
Math. Z. 127 (1972), no. 1, 10-16
is written in the form of a lengthy poem, in a style similar to that
of the work …
37
votes
PhD dissertations that solve an established open problem
The thesis of Martin Hertweck answered the at that time 60-years-old
isomorphism problem for integral group rings in the negative, by constructing
a counterexample. That is, a pair of non-isomorphic f …
21
votes
When has the Borel-Cantelli heuristic been wrong?
The Borel-Cantelli heuristic suggests that for any odd $n \in \mathbb{N}$, there is some
$k \in \mathbb{N}$ such that $n+2^k$ is prime -- and for small $n$ this is in fact true
(in particular, for any …
- 19.6k
12
votes
Taking a theorem as a definition and proving the original definition as a theorem
A very basic example is the field $\mathbb{C}$ of complex numbers. -- It can be defined as the field one obtains when adjoining the square root of -1 to $\mathbb{R}$, in which case it needs to be prov …
10
votes
When is 2 qualitatively different from 3?
Let $G$ be a finite group. If the order of $G$ is not divisible by 2, then $G$ is solvable; if it is not divisible by 3, then no such conclusion can be drawn.
7
votes
Examples of eventual counterexamples
Let $Q(n), n \in \mathbb{N}$ denote Hofstadter's Q sequence --
i.e. $Q(1) = Q(2) = 1$, and $Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2))$ for $n > 2$.
Then we have:
$Q(3 \cdot 2^0) = 2$,
$Q(3 \cdot 2^1) = 4$,
$Q …
7
votes
Examples of eventual counterexamples
Assertion: Every integer greater than 1 can be written as the sum of
a prime number and a perfect power of a nonnegative integer.
The smallest (and maybe only?) counterexample to this assertion is
$11 …
5
votes
1
answer
493
views
Nice diophantine equations with large smallest solutions
Given a polynomial $P$ with integer coefficients in finitely many variables,
we denote by $v(P)$ the product of the absolute values of the non-zero coefficients
and the non-zero total degrees of the m …
4
votes
Examples of unexpected mathematical images
Explanation:
Let $r(m)$ denote the residue class $r+m\mathbb{Z}$, where $0 \leq r < m$.
Given disjoint residue classes $r_1(m_1)$ and $r_2(m_2)$, let the class
transposition $\tau_{r_1(m_1),r_2(m_2 …
3
votes
The half-life of a theorem, or Arnold's principle at work
An example which shows that even MathOverflow does not always prevent
a theorem from being forgotten and rediscovered later is the following
assertion which has been proved by G. A. Miller [1] in 1900 …
3
votes
Examples of eventual counterexamples
$1223$ is the smallest odd prime which does not divide any Carmichael number
with $3$ prime factors -- cf. e.g. here.
3
votes
Examples of eventual counterexamples
While there is no known counterexample to the assumption that the probabilistic
Baillie–PSW primality test is actually a proper primality test, there is strong
evidence that there exist such counterex …
2
votes
Tweetable Mathematics
Sz(q) has two orbits more than PSL(2,q) under the action of its automorphism group - see https://doi.org/10.1081/AGB-120004501, Thm. 3.4.
1
vote
Breakthroughs in mathematics in 2021
The negative answer by counterexample to the Modular Isomorphism Problem
for group rings (that is, the question whether for $p$-groups $G$ and $H$,
the group rings ${\mathbb F}_p G$ and ${\mathbb F}_p …