In characteristic $p$, the so-called biologists' rule $$(a+b)^p = a^p + b^p$$ (which got its name by mathematics students that worked as teaching assistants for "mathematics for biologists") is ...

Citing Graham's answers 1 and 2 to two other questions: Definition: A polynomial $f(x)\in \mathbb C[x]$ is indecomposable if whenever $f(x)=g(h(x))$ for polynomials $g$, $h$, one of $g$ or $h$ is ...

hilbertthm90 and Maxime Bourrigan mentioned already in the comments to the question that the Henstock-Kurzweil integral offers a good alternative to the Riemann integral (see also the lecture notes ...

How about "The classification of finite quasithin groups" by G. Mason from 1980? The classification of finite simple groups was announced when G. Mason was still working on this important case and he ...

Ed Dean linked to this story in a comment, but I think it is too nice to stay hidden there: On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they ...

Hall's universal group is a countable locally finite group that contains every countable locally finite group (see these lecture notes).

The classification of finite simple groups was announced 1983 when Geoff Mason was still working on the quasithin case. I've heard somewhere that he lost his motivation then and never finished his 600+...

Let $N$ be normal in $G\times H$. For $n=(n_1, n_2) \in N$ and $(g, 1) \in G\times H$ follows $([n_1, g], 1) = (n_1^{-1}n_1^g, 1) = n^{-1}\cdot n^{(g, 1)} \in N$ (taking the notations used in group ...

The paper "A Personal View of Average-Case Complexity" by Russell Impagliazzo considers five different worlds depending on the average case complexity of NP-complete problems, one of them ("...

Take a look at these papers from Dan Bernstein. It's not quite what you are looking for, but he does even more than you need in time $n(\lg n)^{2+o(1)}$ where $n$ = number of bits of $N\cdot d$ (one ...

How about this alternative approach to answer Q1: Step 1: Get some room by defining first the diagonal of $*$ (which is mapped to numbers divisible by $5$): $z*z := 5z$ for $z > 0$ and $z*z := 5z-...

A good source for highly efficient algorithms and implementations for this kind of problems is Dan Bernstein's homepage. There I found an algorithms that might be useful for weeding out all the small ...

If you need your random sequence unpredictable in a cryptographic sense, then take a look at the article How to Encipher Messages on a Small Domain: Deterministic Encryption and the Thorp Shuffle ...

Every finite Borel measure defined on a Polish space is regular, see e.g., Lemma 26.2 in Heinz Bauer: Measure and Integration Theory.

[Slightly too long for a comment, so I post it community wiki answer.] The kernel of the epimorphism $\quad\varphi : G\times G \to H\times H\quad$ is a normal subgroup of $G\times G$, for which by an ...

You could take a look in Lectures on Riemann surfaces by Otto Forster (Springer, Graduate Texts in Mathematics 81). The book starts at a moderate level (you just need to know basic complex analysis ...