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Consider the example $f=x_1^2+x_2^2+x_3^2$, $g=x_1^2$. For any prime $p\equiv 3\mod 4$, we have $f\equiv g\equiv 0\mod p\Leftrightarrow x_1\equiv x_2\equiv x_3\equiv 0\mod p$. So for this example $$ \ …

Yes, the statement is true. For any positive integer $m$ there are $2^m$ subsets of $\{1^k,2^k,\ldots,m^k\}$. Each subset has sum bounded by $m^{k+1}$, so by the pigeonhole principle $$ \max_{1\leq i …

It is known that for $p$ a prime, the following conditions are equivalent: ${2p-1\choose p-1}\equiv 1\mod{p^4}$ $p^3$ divides the numerator of $\sum_{n=1}^{p-1}\frac{1}{n}$ $p$ divides the numerator …

As far as I know this quantity does not have a name. There is a mention of a general family of $p$-adic limits, of which $S_{k,p}$ is a special case, on p. 31 of [1]. The quantity $S_{k,p}$ can be ex …

Let $n$ be a positive integer, $a_1,\ldots,a_{n-1}\in\mathbb{Z}$. Suppose that for every $u\in(\mathbb{Z}/n\mathbb{Z})^\times$, $$ \tag{$\star$} \sum_{i=1}^{n-1} i a_{(ui\!\!\!\mod n)} = 0. $$ Then th …

There is some number field $K$ containing $z$, and there we have a prime factorization $$ (z)=\frac{\mathfrak{p}_1\ldots\mathfrak{p}_a}{\mathfrak{q}_1\ldots\mathfrak{q_b}} $$ of fractional ideals, whe …

We should be able to construct such a real number $x$. Let $\epsilon\in(0,1/2)$ be fixed. If $S\subset \mathbb{R}_{>0}$ and $r>0$, we write $S^r$ for the set of positive $r$-th powers of elements of $ …

This is not a complete answer, but the following result might useful. Basically it says that if the poles of $f(x)$ are negative integers, then the time to compute $\sum_{k=0}^{p-1}f(k)$ is bounded by …

Not an answer, but an argument that your sum is between $(1/4+o(1))p\log^2(p)$ and $(3/4+o(1))p\log^2(p)$. Write $S(p)$ for your sum. Separate the sum into pieces according to the integer part of $(a …

For each $n$, the differences $a_{n+1}-a_n$, $a_{n+2}-a_{n+1}$, and $a_{n+2}-a_n$ can only be divisible by powers of $2$ and primes less than or equal to $c$. Since $$ \frac{a_{n+2}-a_{n+1}}{a_{n+2}- …

I am considering the following two isomorphisms: First, if $X$ is a reasonably nice topological space, then $X$ has a normal covering space which is maximal with respect to the property of having an …

Here are some bounds on the stable value $r(n)$, as well as the number of terms of the sequence that need to be calculated. The short version is that $$ \frac{\sqrt{2}}{2}\sqrt{n}+O(1)\leq r(n)\leq \f …

The group algebra $\mathbb{Z}_p[[\mathbb{Z}_p\times \mathbb{Z}_p]]$ has the structure of a complete Hopf algebra, where $\mathbb{Z}_p\times \mathbb{Z}_p$ consists of precisely the group-like elements. …

The usual power series for $\log(1+x)$ determines an injection from $1+(x_1,x_2)\subset\mathbb{Z}_p[[x_1,x_2]]$ into $\mathbb{Q}_p[[x_1,x_2]]$. A power series $f\in 1+(x_1,x_2)$ is in $\langle 1+x_1,1 …

The matrix entry $*$ is $\log(u^{1-p})/p$ (where $\log$ is the $p$-adic logarithm). This can be obtained from the description of the inverse of Frobenius given in sections 2.9-2.10 of [1]. You can fin …