The congruences tag has no wiki summary.

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### 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**

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**1**answer

148 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 ...

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311 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

**2**answers

122 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

**1**answer

187 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**

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**0**answers

652 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**

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**0**answers

458 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 ...

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**3**answers

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### 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

**2**answers

379 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

**4**answers

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### 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 ...

**12**

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**1**answer

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 ...