Dietrich Burde
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27 answers
86 votes
11k views
Modern Mathematical Achievements Accessible to Undergraduates
57 votes

Primes are in P. The proof is indeed accessible, see for example the article "Primes are in P: A breakthrough for "Everyman", http://www.ams.org/notices/200305/fea-bornemann.pdf‎. The idea is really ...

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3 answers
41 votes
18k views
Which integers can be expressed as a sum of three cubes in infinitely many ways?
35 votes

For $n\equiv \pm 4\pmod{9}$ there is no solution to $(1)$. Otherwise, for $n\ge 1$, it is conjectured that there are always solutions, even infinitely many. There are no analytic results, but ...

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3 answers
18 votes
2k views
Algebraic Groups in Characteristic p
20 votes

Over the complex numbers, connected linear algebraic groups correspond to Lie algebras in the usual way. This Lie correspondence breaks down over number fields, and breaks down even more over fields ...

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25 answers
30 votes
6k views
Mathematicians who made important contributions outside their own field?
18 votes

Werner Nahm not only made important contributions to conformal field theory, but also conducted research about the Mayan civilization and their astronomy. In his Mayan research, he also worked with ...

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2 answers
15 votes
1k views
Are there only finitely many associative algebras of fixed dimension?
18 votes

Even for commutative associative algebras it is not true. The article of Björn Poonen "Isomorphism types of commutative algebras of finite rank over an algebraically closed field" gives a ...

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4 answers
13 votes
4k views
How many three dimensional real Lie algebras are there?
17 votes

There are already uncountably many isomorphism classes of $3$-dimensional real Lie algebras. In fact, there are $1$-parameter families of $3$-dimensional solvable Lie algebras. The classification has ...

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4 answers
12 votes
2k views
Partitions-sum of divisors identity
16 votes

2.) There is a proof, due to P. Erdös, in the Annals of Mathematics (2), 43, 1942, pp. 437-450, which does not use the generating function, but rather proves the identity $$ np(n)=\sum_{m=1}^n \sum_{k=...

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2 answers
7 votes
715 views
On finite groups with same complex-valued character table
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13 votes

Finite groups have the same complex character tables if and only if their group algebras are isomorphic as quasi-Hopf algebras (if and only if the group algebras are twisted forms of each other as ...

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2 answers
8 votes
752 views
Kaplansky's idempotent conjecture for Thompson's group F
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13 votes

Thompson's group $F$ satisfies the idempotent conjecture, because it is torsion-free and orderable. For torsion-free groups it is known that the zero-divisor conjecture for group rings implies the ...

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4 answers
20 votes
1k views
Does Anyone Know Anything about the Determinant and/or Inverse of this Matrix?
13 votes

The determinant formula for $\det (1/(x_i+y_j)^2)_{i,j}$ is due to Borchardt, see Krattenthaler's article given in Steve's answer, which contains a $(q)$-deformation of it as well. I want to mention, ...

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22 answers
74 votes
12k views
Are there proofs that you feel you did not "understand" for a long time?
13 votes

The first proof of quadratic reciprocity I read in a book was using some figure, with several lines in it, and some lattice points. I did not really understand the argument and had the feeling that ...

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1 answers
8 votes
286 views
Generating function of $p(25n + 24)$
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12 votes

Hirschhorn and Hunt have published a note M. D. Hirschhorn; D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math., 326 (1981), 1-17., where they give a proof (...

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1 answers
13 votes
1k views
What is the arithmetic Nullstellensatz?
12 votes

One "arithmetic version" of the Nullstellensatz states that if $f_1, ..., f_s$ belong to $\mathbb{Z}[X_1,...,X_n]$ without a common zero in $\mathbb{C}^n$, then there exist $a \in \mathbb{Z} \setminus ...

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2 answers
14 votes
2k views
sum of three cubes and parametric solutions
12 votes

I found a proof of the following fact in the article of G. Payne and L. Vaserstein, "Sums of three cubes", contained in the book "The arithmetic of function fields" (1992): The set of integral ...

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67 answers
239 votes
117k views
Awfully sophisticated proof for simple facts
12 votes

One can also show with Fermat's last theorem that $\sqrt{2}$ is irrational - the answer of mt did $2^{1/n}$ for $n\ge 3$. Suppose that $\sqrt{2}$ is rational. Then there is a right-angled triangle ...

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12 answers
12 votes
4k views
Obscure Names in Mathematics
11 votes

The Killing-Hopf theorem (yes, Hopf is already dead). The ugly duckling theorem. The no free lunch theorem. The Cox-Zucker machine.

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1 answers
8 votes
975 views
Would Elliott-Halberstam conjecture follow from GRH?
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11 votes

The Elliott-Halberstam conjecture is not known to follow from GRH. Even the weak version of EH (which is with $Q=x^{1/2+\epsilon}$ for any fixed $\epsilon>0$) does not follow from GRH. On the ...

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5 answers
11 votes
2k views
Groups as Union of Proper Subgroups: References
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11 votes

The mentioned result of Cohn has been further extended. Let us write $σ(G) = n$ whenever $G$ is the union of $n$ proper subgroups, but is not the union of any smaller number of proper sub- groups. ...

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3 answers
10 votes
701 views
History of profinite groups, when was it first mentioned? What was the original definition?
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11 votes

Profinite groups were first called "Groups of Galois type", see J.P. Serre's book "Cohomologie Galoisienne" of $1964$. The term "profinite" comes from Serre (if I am not mistaken). Of course, some ...

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2 answers
10 votes
2k views
When is a Baumslag-Solitar group linear?
11 votes

The metabelian groups $BS(n,1)\simeq BS(1,n)=\langle a,b\mid aba^{-1}=b^n \rangle$ are also linear (this seems not mentionened in the Wikipedia article). A faithful, linear representation $BS(1,n)\...

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2 answers
13 votes
480 views
Why is every deformation of the universal enveloping algebra of a complex semisimple Lie algebra trivial?
Accepted answer
10 votes

The article Deformation par quantification et rigidite des algebres enveloppantes by M. Bordemann, A. Makhlouf, T. Petit addresses these questions. They call Lie algebras $\mathfrak{g}$ with $HH^2(U(\...

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1 answers
11 votes
698 views
When is the integral group ring Noetherian?
10 votes

As far as I know the only groups known to have a Noetherian integral group ring are polycyclic-by-finite groups. This is often discussed in connection with the so-called "Zero Divisor Conjecture" for ...

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1 answers
7 votes
561 views
Is $\mathcal M _{g,n}$ anabelian?
10 votes

Grothendieck expected the moduli spaces $\mathcal{M}_{g,n}$ over $\mathbb{Q}$ to be the basic examples of anabelian varieties (besides hyperbolic curves, which was proved by Mochizuki, even over ...

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4 answers
9 votes
854 views
Applications of n-dimensional crystallographic groups
10 votes

One of the well-known applications of crystallographic groups is the classification of flat complete Riemannian manifolds by their fundamental group, which is a torsion-free crystallographic group (...

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3 answers
12 votes
814 views
Which nilpotent Lie algebras appear as nilradicals of parabolic subalgabras?
10 votes

The subclass of nilpotent Lie algebras formed by arbitrary ideals of parabolic subalgebras consisting of nilpotent elements in reductive Lie algebras has been classifed in the article Yu.B. ...

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2 answers
9 votes
624 views
Square root in complex reductive groups
9 votes

As the comment shows the answer is negative in general. Perhaps it is worth to mention that for connected compact Lie groups the answer is yes, because its exponential map is surjective. In general, ...

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2 answers
9 votes
342 views
Which finite p-groups occur as commutators of finite p-groups?
9 votes

There are some results for special cases. Burnside has proved in $1912$ that, if $G$ is a non-metabelian $p$-group, then the centre of the derived group of $G$ cannot be cyclic. In particular, a non-...

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3 answers
6 votes
661 views
Rigid nilpotent Lie algebras
9 votes

Vergne's conjecture is still open. It says that there is no complex $n$-dimensional nilpotent Lie algebra which is rigid in the variety $\mathcal{L}_n(\mathbb{C})$ of all $n$-dimensional complex Lie ...

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3 answers
7 votes
807 views
Affine structures
9 votes

Kostant and Sullivan proved that the Euler characteristic of a compact complete affine manifolds must vanish, affirming the Chern conjecture in the complete case (Bull. AMS 81 (1975)). Benzecri proved ...

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2 answers
3 votes
944 views
Question about ring of integers of cyclotomic field
9 votes

The ring of integers $\mathbb{Z}[\zeta_p]$ is an UFD if and only if the class number of $\mathbb{Q}[\zeta_p]$ is $1$. This is the case if and only if $p\le 19$. For bigger primes $p$ the class numbers ...

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