Questions tagged [congruences]
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11 questions
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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 ...
- 1,241
19
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2
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A congruence involving binomial coefficients
The following open problem was shown to me by Maxim Kontsevich. I
state it in a different but equivalent form. Let $a(n)$ be the sequence
at http://oeis.org/A131868, that is,
$$ a(n) =\frac{1}{2n^2}\...
- 50.8k
7
votes
1
answer
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Is $n=6$ the only integer satisfies ${\sigma}_x(n) \equiv 0\bmod{n}$ for every odd integer $x > 0$ and $2 (\bmod n)$ if $x$ is even integer? [closed]
After a few computations in wolfram alpha about the divisor function for some values of $n$ to look the behavior of $\sigma_x(n)\bmod n$ for $\,n=6,\,$ i got this result : $\sigma_x(6)=0 \bmod 6$ for $...
- 414
18
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2
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Binomial supercongruences: is there any reason for them?
One of the recent questions, in fact
the answer
to it, reminded me about the binomial sequence
$$
a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2,
\qquad n=0,1,2,\dots,
$$
of the Apéry ...
- 13.5k
8
votes
0
answers
351
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A hypergeometric series for $\sqrt3\pi$ with converging rate $1/9$
Recently, I found a (conjectural) new series for $\sqrt3\pi$:
$$\sum_{k=1}^\infty\frac{(8k-3)\binom{4k}{2k}}{k(4k-1)9^k\binom{2k}k^2}=\frac{\sqrt3\pi}{18}.\label{1}\tag{1}$$
The series converges fast ...
- 15.6k
8
votes
1
answer
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Cardinality of the image of a polynomial modulo $p^n$
Let $f \in \mathbb{Z}[x]$ be a nonconstant polynomial and let $p$ be a prime number. I'm looking for results about $$N_f(p^k) := \#\{(f(n) \bmod p^k) : n \in \mathbb{Z}\},$$ as $k \to +\infty$, where $...
- 81
6
votes
1
answer
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views
Are there finitely many primes $x$ such that for a fixed odd prime $p$, $n=x^{p-1}+x^{p-2}+\dotsb + x+1$ is composite and $x \mid \phi(n)$?
Let
\begin{equation} n =x^{p-1}+x^{p-2}+\dotsb + x+1
\end{equation} where $x$ and $p$ are odd primes.
If $p$ is set to $5$, it appears $x=5$ is the only prime $x$ such that $n$ is composite and $x \...
- 319
5
votes
2
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Permutations $\pi\in S_{p-1}$ with $\frac1{\pi(1)\pi(2)}+\frac1{\pi(2)\pi(3)}+\cdots+\frac1{\pi(p-2)\pi(p-1)}+\frac1{\pi(p-1)\pi(1)}\equiv0\pmod{p^2}$
A well known congruence of Wolstenholme states that
$$\frac1{1^2}+\frac1{2^2}+\cdots+\frac1{(p-1)^2}\equiv0\pmod{p}$$ for any prime $p>3$. For each $n=3,4,\ldots$ we clearly have
$$\frac1{1\times2}+...
- 15.6k
4
votes
0
answers
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Computing the density of a set of multiples
Erdős and his coauthors often wrote about problems relating to the densities of sets of multiples. I have a computational question about the same topic. I have a finite* set $A=a_1<\cdots<a_r$ ...
- 9,114
4
votes
2
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A conjecture about 433
Motivated by Question 405105, I found the following
Conjecture. $2^{2n}-2^n+1 \equiv 0 \pmod {433}$ and $n=4m$ iff
$$\phi(m) = \phi(i + j) = \phi(i) + \phi(j) ,$$
and
$$\phi(m) = \phi(ik) = \phi(i)\...
- 884
4
votes
2
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Non-torsion part of the abelianisation of congruence subgroups
I've posted this question on math.stackexchange, but haven't gotten any responses so I'm trying here instead.
Let $A = F_q[T]$ be the ring of polynomials in one variable with coefficients in a finite ...
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