The tag has no wiki summary.

learn more… | top users | synonyms

0
votes
0answers
1 views

Cardinality of the prime divisor set of a k-power sum

Let $a_{1},\dots,a_{n}$ positive natural numbers $(n>2)$ , $a_{i}\neq a_{j}$ if $i\neq j$ . I want to prove that$$\left|\left\{ p\, prime:\, p\mid\overset{n}{\underset{i=1}{\sum}}a_{i}^{k},\, ...
7
votes
1answer
538 views

Roots of a polynomial in a finite field related to Fermat's Last Theorem

In my class, we proved the following condition: define the polynomial $P_l(x)$ as $$P_l(x) = \sum_{r=1}^{l-1}{\frac{1}{r}x^{l-1-r}}$$ Then if for all $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ ...
15
votes
0answers
387 views

Does $\binom{2n}{n} \equiv 2 \pmod p$ ever hold?

Well, the title does not tell the whole story; the complete question is: Are there any primes of the form $p=2n(n-1)+1$, with integer $n\ge 1$, such that $$ \binom{2n}{n} \equiv 2\pmod p ? $$ ...
5
votes
1answer
322 views

A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$

Is the following conjecture true? Conjecture. If $r > s \ge 1$ are relatively prime integers such that \begin{equation} (r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1} \end{equation} then $r-s = 1$ ...
0
votes
2answers
284 views

What are all positive integers n for which the congruence $a^{n+1} \equiv a (mod n)$ holds? [closed]

Fermat's little theorem says that the congruence $a^p \equiv a (mod p)$ if $p$ is a prime number. $a^{n+1} \equiv a (mod n)$ works for all integers $a$ and some positive integers $n$, how can we ...
1
vote
0answers
47 views

“embedding” various matrix equivalences into the equivalence of particular linear map

Consider the square matrices over a (local) ring $R$, up to conjugation, $A\rightarrow UAU^{-1}$, where $U$ is an invertible matrix over $R$. Such an equivalence embeds into the "left-right" ...
1
vote
1answer
179 views

Does the congruence $a^p \equiv 1 \pmod{b^p}$ with prime $p \ge 5$ force $b \le p$?

I'm considering the congruence in the title, i.e., $$a^p \equiv 1 \pmod{b^p},$$ where $a \ge b \ge 1$ are positive integers and $p$ is an odd prime. For $p=3$, a brute-force computer search found ...
4
votes
1answer
131 views

Congruence equation for Apery numbers

Does the system of congruence equations \begin{eqnarray} A_{17k}&\equiv& 0 \pmod {17^2}, \nonumber \\ A_{17k+1}&\equiv& 0 \pmod {17^2}, \tag{1} \end{eqnarray} has solutions other ...
8
votes
0answers
193 views

congruences of level 1 and level p modular forms

I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k ...
-1
votes
1answer
208 views

how to solve system of congruence with multivariables [closed]

There n variables x1,x2,...,xn represented as X, n equations whose coefficient matrix (n*n) is represented as A, and this system ...
0
votes
0answers
329 views

Solutions to a quadratic congruence

Fix an odd prime $p$. Let $\alpha = (\alpha_0,\dots,\alpha_k)$ be a solution to the congruence $\sum_{i=0}^{k} \alpha_i^2 \equiv x \mod p$. Now consider the number $N_\alpha$ of solutions to the ...
1
vote
2answers
125 views

Number of solutions to $mx^2+ny^2 \equiv k\pmod{p}$

I need a reference for the result which gives the number of solutions to the congruence $mx^2+ny^2 \equiv k\pmod{p}$. This result seems to be something that would be discussed in Gauss' ...
2
votes
1answer
198 views

perfect shuffle of 2n cards

permutation is given by f(i) = 2i if i<=n and 2(i-n)-1 if i>n where i denotes the position of cards. eg pack of cards (1,2,3,4)-->(3,1,4,2) Basically trying to find the multiplicative order of 2 ...
18
votes
0answers
683 views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
2
votes
0answers
465 views

An elementary question in modular arithmetic

Let us fix a positive natural number $N$. When $i$ is a natural number smaller than $N$, coprime with $N$, we let $\mu(i)$ be the unique number in $\{1, \ldots, N-1\}$ that is the multiplicative ...
6
votes
3answers
1k views

Finite subgroup of $Gl(n,\mathbb Z)$ and congruences

Suppose we have an invertible matrix q in a finite subgroup $Q$ of $Gl(n,\mathbb Z)$, the group of all invertible integer matrices. Now I want to find all $x\; mod\; \mathbb Z^n$ for which ...
1
vote
2answers
393 views

Proving Congruence Without Leech Lattice

Let $\sigma_{11}(n)$ denote the sum of the 11th powers of the positive integral divisors of the positive integer n. Let $\tau(n)$ denote Ramanujan's tau function, which is the coefficient of $q^n$ in ...
2
votes
4answers
2k views

Best way to introduce the Chinese Remainder Theorem (to a high school student)

What do you think to be the most effective way to teach the Chinese remainder theorem to a smart high school student, which is supposed to only have a soft idea about how modular arithmetic works, and ...
13
votes
1answer
1k views

Binomial supercongruences: is there any reason for them?

One of the recent questions, in fact the answer to it, reminded me about the binomial sequence $$ a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2, \qquad n=0,1,2,\dots, $$ of the Apéry ...