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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

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=...

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 ...

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 ...

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, ...

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 ...

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 (...

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 ...

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 ...

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 ...

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

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 ...

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. ...

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 ...

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)\...

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(\...

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 ...

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 ...

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 (...

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. ...

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, ...

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-...

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 ...

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 ...

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 ...