A top hat for Moser's four mathemagical rabbits

If the equation 1^k+2^k+...+(m-2)^k+(m-1)^k=m^k has an integer solution with k>1, then m>10^{10^6}. Leo Moser showed this in 1953 by remarkably elementary methods. His proof rests on four identities he derives separately. It is shown here that Moser's result can be derived from a von Staudt-Clausen type theorem (an easy proof of which is also presented here). In this approach the four identities can be derived uniformly. The mathematical arguments used in the proofs were already available during the lifetime of Lagrange (1736-1813).Comment: 7 pages. Meanwhile MacMillan and Sondow showed that Lagrange (1736-1813). can be replaced by Pascal (1623-1662

The Erd\H{o}s--Moser equation 1k+2k+...+(m−1)k=mk1^k+2^k+...+(m-1)^k=m^k revisited using continued fractions

If the equation of the title has an integer solution with $k\ge2$, then $m>10^{9.3\cdot10^6}$. This was the current best result and proved using a method due to L. Moser (1953). This approach cannot be improved to reach the benchmark $m>10^{10^7}$. Here we achieve $m>10^{10^9}$ by showing that $2k/(2m-3)$ is a convergent of $\log2$ and making an extensive continued fraction digits calculation of $(\log2)/N$, with $N$ an appropriate integer. This method is very different from that of Moser. Indeed, our result seems to give one of very few instances where a large scale computation of a numerical constant has an application.Comment: 17 page

The critical polynomial of a graph

Let $G$ be a connected graph on $n$ vertices with adjacency matrix $A_G$. Associated to $G$ is a polynomial $d_G(x_1,\dots, x_n)$ of degree $n$ in $n$ variables, obtained as the determinant of the matrix $M_G(x_1,\dots,x_n)$, where $M_G={\rm Diag}(x_1,\dots,x_n)-A_G$. We investigate in this article the set $V_{d_G}(r)$ of non-negative values taken by this polynomial when $x_1, \dots, x_n \geq r \geq 1$. We show that $V_{d_G}(1) = {\mathbb Z}_{\geq 0}$. We show that for a large class of graphs one also has $V_{d_G}(2) = {\mathbb Z}_{\geq 0}$. When $V_{d_G}(2) \neq {\mathbb Z}_{\geq 0}$, we show that for many graphs $V_{d_G}(2)$ is dense in ${\mathbb Z}_{\geq 0}$. We give numerical evidence that in many cases, the complement of $V_{d_G}(2)$ in ${\mathbb Z}_{\geq 0}$ might in fact be finite. As a byproduct of our results, we show that every graph can be endowed with an arithmetical structure whose associated group is trivial