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I have found a proof of more general formula in the book Postnikov, A. G. Introduction to analytic number theory American Mathematical Society, 1988, (section 4.2). This proof is simple but it has a s …

Simple algoritm based on continued fractions was proposed by Rödseth, O. J. On a linear Diophantine problem of Frobenius J. Reine Angew. Math., 1978, 301, 171-178 All algorithm are described in Ramre …

Rödseth formula for Frobenius numbers is good not only for computation. It allows to find weak asymptotic for Frobenius numbers with three arguments and density function for normalized Frobenius numbe …

Conway's Soldiers. And an interesting special case Reaching row 5 in Solitaire Army.

Let $$S_a(N)=\sum_{n\le N}\frac{\varphi(an)}{n^2}.$$ The usual machinery gives an asymptotic formula $$S_a(N)=\frac1{\zeta(2)}\cdot\frac{a^2}{\varphi_+(a)}\log N+C(a)+O(N^{-1+\varepsilon}a^{1+\varepsi …

They can be deduced via Dirac Belt Trick, see Understanding Quaternions and the Dirac Belt Trick by Mark Staley. This Trick is also known as Plate Trick.

Did anybody consider addition law for elliptic curves of the form $$x^2y^2+a(x+y)+b=0\,?$$ Does this form have any specific name?

It was done by Euler . Here you can see original text and english tranclation. $\smile$ Now it is known as Euler-Chasles correspondence.

Also a partial answer. Partial quotients in periods of quadratic irrationals satisfy the Gauss-Kuz’min law (in average over quadratic irrationals with the length $\le N$), see Spin chains and Arnold …

The formula $|\alpha-(2m+1)/(2n+1)|<\epsilon/(2n+1)$ is equivalent to $\|\alpha m+\frac{\alpha-1}{2}\|<\epsilon/2,$ where $\|x\|$ is a distance from $x$ to nearest integer. This problem is covered by …

It is not an answer but some numercal data for $A_2(N)$. Here digits are ordered according to their frecuences ($2$, $4$ and $6$ look like most frequent): $$ \begin{array}{rl} N=1000: & 2, 1, 4, 6, 8, …

This result is classical, but unfortunately many usual textbooks (e.g. Davenport's "The higher arithmetic") don't give a direct refference to the original paper. This theorem belongs to Évariste Galo …

Corresponding sums of Legendre symbols are know as Jacobsthal sums $$J(u)=\sum_{x\mod p}\left(\frac {x^3+ux}p\right).$$ They are equal to $0$ for prime $p\equiv 3\pmod4$. If $p\equiv 1\pmod4 $, $\le …

What are the most simple examples which can explain the meaning of Yang–Baxter equation? Is there any way to explain this mysterious object to a person who is not a professional in quantum groups? Il …

Numerical results give random picture. If it is really the case then usual heuristic arguments confirm your conjecture. ListPlot[Table[FractionalPart[Sqrt[PartitionsP[i]]], {i, 1, 5000}]] Distance …