Questions tagged [congruences]
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48 questions with no upvoted or accepted answers
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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
18
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0
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755
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Two curious series for $1/\pi$
On Jan. 18, 2012 I conjectured that for any prime $p>3$ we have
$$R_p^2\equiv\frac1{10}\left(512\left(\frac{10}p\right)-27\left(\frac{-15}p\right)-475\right)\pmod p,$$
where $(\frac{\cdot}p)$ ...
- 15.6k
9
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0
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315
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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 \...
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8
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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
7
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0
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183
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Some conjectural congruences involving Domb numbers
The Domb numbers are given by
$$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$
Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....
- 15.6k
7
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0
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213
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Does Morley's congruence characterize primes greater than $3$?
In 1895 Morley showed that $$\binom{p-1}{(p-1)/2}\equiv(-1)^{\frac{p-1}2}4^{p-1}\pmod{p^3}$$
for any prime $p>3$.
In 2009, I formulated the following conjecture concerning the converse of Morley's ...
- 15.6k
7
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0
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217
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How to prove the identity $\sum_{k=1}^\infty\frac{3H_{k-1}^2+4H_{k-1}/k}{k^2\binom{2k}k}=\frac{\pi^4}{360}$?
For each $n=0,1,2,\ldots$, the harmonic number $H_n$ is given by
$$H_n:=\sum_{0<k\le n}\frac1k.$$
In 2016 I conjectured that
$$\sum_{k=1}^\infty\frac{3H_{k-1}^2+4H_{k-1}/k}{k^2\binom{2k}k}=\frac{...
- 15.6k
6
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0
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192
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A conjecture involving $P_n=\prod_{k=1}^np_k$
For each positive integer $n$ let $P_n=\prod_{k=1}^n p_k$, where $p_k$ is the $k$th prime.
Question. Is my following conjecture true?
Conjecture. For any integer $n>1$, there are $k,m\in\{1,\...
- 15.6k
6
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0
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217
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Two conjectural congruences for Franel numbers
Recall that the Franel numbers are given by
$$f_n:=\sum_{k=0}^n \binom{n}{k}^3\ \ \ (n=0,1,\ldots).$$
Question. How to prove my following conjecture?
Conjecture. For each odd prime $p$, we have
$$\...
- 15.6k
5
votes
0
answers
541
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Two conjectures for primes $p\equiv 1\pmod 8$
Motivated by my paper Quadratic residues and quartic residues modulo primes [Int. J. Number Theory 16 (2020), 1833-1858], here I pose two new conjectures for primes $p\equiv1\pmod8$ based on my ...
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5
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Number of solutions of linear congruence with bounded variables
Fix some nonzero integers $a_1, \dots, a_k$, with $k \geq 2$, and some real numbers $c_1, \dots, c_k \in (0,1)$.
For every positive integer $m$, let $N(m)$ be the number of $k$-tuple of integers $(x_1,...
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5
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Is an algebraic number satisfying certain super-congruences a root of unity?
Let $K|\mathbb{Q}$ be a number field, $D$ its discriminant and $\mathcal{O}$ the ring of integers in $K$. Let $x\in K$ (or maybe $\in \mathcal{O}[\frac 1D]$) such that for all primes $p$ in $\mathbb{Q}...
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Differential duality: Triangular codes vs. VT codes / Single-substitution vs. Single-deletion
Here is the introduction to my problem:
Codes correcting single-deletion. Let $q$ and $n$ be non-negative integers, and let $\mathbf{x}= \left ( x_{1}, x_{2}, \ldots, x_{n} \right )\in\mathbb{Z}_{q}^{...
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Congruences between hypergeometric functions coming from modular forms
Consider the formal power series
$${}_2F_1(1/12,5/12;1;z)^{24}-{}_2F_1(1/12,7/12;1;z)^{24}=-z/3-z^2/2-(320293/559872)z^3-\cdots$$
It follows from a theorem on modular forms that for $p\ge5$ the ...
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4
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Is it true that $|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|=(1-e^{-1})p+O(\sqrt{p})\ ?$
For each prime $p$, let us define
$$w_p:=|\{k^{k+1}+(k+1)^k\pmod p:\ k=0,\ldots,p-1\}|,$$
where $a\pmod p$ denotes the residue class $a+p\mathbb Z$.
Based on my computation, I conjecture that
$$w_p=...
- 15.6k
4
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0
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407
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Congruence for the product of quadratic residues + the product of quadratic non-residues
My question has been here on MSE for a long time, but it has not received a full answer. I bring it here:
Find a prime $p$ such that $p \equiv 1 \bmod 4$ and such that the
product in the range $[...
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4
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0
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Computing the density of a set of multiples
Erdős and his coauthors often wrote about problems relating to the densities of sets of multiples. I have a computational question about the same topic. I have a finite* set $A=a_1<\cdots<a_r$ ...
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3
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0
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62
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Congruences regarding $4n$-dimensional lattices
A sequence of integers $(a_n)_{n\geq 1}$ satisfies Gauss congruence if
$$\sum_{d\mid n}\mu(d)a_{n/d}\equiv 0\pmod{n}$$
for every $n\geq 1$. Such sequences are also called Dold sequences, Newton ...
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3
votes
1
answer
439
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Chinese remainder theorem for target interval
Given $n$ pairwise coprime natural numbers $m_{1}, \dots, m_{n}$ with remainders $y_{i}$, for all $i \leq n$. Furthermore, we have a target interval $I := \left[ a, b \right]$, with $1 \leq a < b \...
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3
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0
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329
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When are modular forms linearly independent modulo $p$?
Let $M_2(\Gamma_0(N))$ be the space of weight $2$ modular forms for $\Gamma_0(N)$ and let $f_1, \dots, f_r$ be a basis of normalized eigenforms for $M_2(\Gamma_0(N))$. Given a rational prime $p$, I'll ...
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3
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0
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183
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Congruences of modular forms modulo other modular forms
Congruences between modular forms are certainly a big topic in number theory, maybe with
$$E_{p-1}\equiv 1 \mod p \qquad \text{for a prime }p\geq 5$$
as the easiest example. Sometimes, $p$ might be ...
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3
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299
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An attempt to get a variant of Agoh–Giuga conjecture
The idea of this post is an attempt to explore a variant of the so-called Agoh–Giuga conjecture. In past days, and today, I tried to think about variants of this conjecture exploring congruences about ...
2
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64
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Solution distribution in a quadratic congruence
Fix a modulus $M$. For varying $m$, consider the (possibly empty) set $R_{m,M}$ of roots of $x^2\equiv m^2+2 \pmod{M}$.
Are there good known bounds on
$$\sum_{m\le M^{1/100}}\sum_{r\in R_{m,M}}\min(M^{...
- 21
2
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0
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152
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On the smallest solution of a linear congruence
I have the following question. First, consider the following congruence for primes $p\geq 5$:
$24x\equiv -1\;(\mbox{mod}\;p)$.
The smallest $x$, that is, $1\leq x\leq p-1$ for which the above ...
- 21
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Is it possible to deduce statements for odd perfect numbers from the convolution sums involving divisor functions or other arithmetic functions?
Dividing and using some identities of [1] I've deduced the following facts, see also the remarks below. After these introductory paragraphs, to motivate our question, I am asking if we can deduce some ...
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$\varphi(m+n)\mid n$ for some positive integer $n$
Let $\varphi$ be Euler's totient function.
If $p$ is a prime, then $\varphi(1+n)=n$ for $n=p-1$.
Question. Is it true that for each integer $m>1$ there is a positive integer $n\le m^2-m$ such that ...
- 15.6k
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0
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"Close" roots of polynomials and Pillai property
A sequence of integers $a(n)_{n \geq 0}$ has the Pillai property if there exists an integer $G \geq 2$ such that for all integers $k \geq G$ there exists an integer $n \geq 0$ such that none of the $k$...
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0
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Monoid prime ideals and prime congruences
I was wondering what the connection is between the notion of "prime congruence" on a monoid, and the notion of "prime ideal" in a monoid. Starting from a prime ideal $P$ in a monoid $M$, one can ...
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2
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0
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488
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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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Curious congruences modulo $4$ involving primes
We define
$$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2}
\sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$
(Searching the OEIS yielded no results.)
For $n>2$ we have the following experimental observations (...
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A new Conjecture at OEIS sequence A376842
Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review.
Let $b \in \mathbb{Z}^+$ and assume that $n$ is an integer greater than $1$ and not a multiple ...
- 1,451
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0
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Some $p$-adic congruences involving permutations
Motivated by my study of determinants and permanents, here I present several conjectures on $p$-adic congruences involving permutations.
As usual, we let $S_n$ be the symmetric group consisting of all ...
- 15.6k
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The congruence $\mathrm{per}[|j-k|]_{1\le j,k\le p}\equiv-1/2\pmod p$ with $p$ an odd prime
For a matrix $[a_{j,k}]_{1\le j,k\le n}$, its permanent is given by
$$\mathrm{per}[a_{j,k}]_{1\le j,k\le n}=\sum_{\tau\in S_n}\prod_{j=1}^na_{j,\tau(j)}.$$
Let $p$ be an odd prime. I have proved the ...
- 15.6k
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0
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Probability of satisfying the congruent mod equation
I'm wondering about the probability of picking three different numbers $x,y,z$ out of the set $[50]=\left\{ 1,2,3,...,50\right\}$ satisfying the equation: $$xyz\equiv \gcd(x,y,z)\mod 7$$ I started out ...
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An explicit solution to the congruence $x^2\equiv 14(\frac 3p)-(\frac p3)-12\pmod {p}$?
Let $p>3$ be a prime, and let $(\frac{\cdot}p)$ be the Legendre symbol. Then
$$14\left(\frac 3p\right)-\left(\frac p3\right)-12=\begin{cases}1&\text{if}\ p\equiv1\pmod{12},
\\-25&\text{if}\ ...
- 15.6k
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Efficient scissors congruence between efficiently describable convex polytopes and simplex?
Is there a convex polytope in $\mathbb R^n$ describable by only $O(poly(\log n))$ half-plane inequalities with positive volume (so at least $n+1$ vertices) such that the standard simplex has a ...
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Wieferich primes and arithmetic prgressions
Let $p$ be an odd prime number. Let $K$ be a number field with Galois group $G$ and $H$ be a subgroup of $G$ stable under conjugation. Then the Cebotarev density theorem gives that $$\mathcal{L}=\{\...
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Equivalent condition for Poincare polynomial
I have found a statement in the introduction of the paper 'Sets of Recurrence and Generalized Polynomials' by Bergelson & Haland, which is
Result: Given a polynomial $p \in \mathbb{R}[x]$ such ...
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1
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Presentation of amalgamated sum as a quotient of the direct sum
I am currently reading Arthur Ogus' "Lectures on Logarithmic Algebraic Geometry" (https://math.berkeley.edu/~ogus/preprints/log_book/logbook.pdf).
I'm trying to understand why the amalgamated sum of ...
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Computational number theory
Suppose that $p$ is prime and $q$ is an even number divides $p-1$, such that $q<\frac{p-1}{q}$ and $u$ has order $q$ modulo $p$. Let $S$ be the subgroup of $Z^*_p$ consisting of the powers of $u$. ...
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"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" ...
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Factorial quadratic residue
I would like to find all positive integers $n$ and $m$ such that $n^2 \equiv m! \ ( \text{mod } 2024)$. I see that for $m=1$ there is $n=45$ such that the relation holds.
I think that there is no ...
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Defining the number of rightmost frozen digits of Graham's number
It is well-known that (in radix-$10$) Graham's number, $G$, can be expressed as a tetration with base $3$ and a very large hyperexponent $\tilde{b}$. Thus, we can write that $\exists! \hspace{1mm} \...
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Which consequences can be deduced from this peculiar property of tetration?
Recently (assuming radix-$10$), I showed that, for any $a \in \mathbb{N}_{0}$ that is not a multiple of $10$, there exists a unique value $V(a) \in \mathbb{N}_{0}$ which corresponds to the number of ...
- 1,451
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0
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When is $\phi(a^n+b^n+c^n)=0\mod n$?
A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
user147204
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Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?
let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$
and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^...
0
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0
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Systems of linear modular equations with unknowns in the moduli
I am interested in systems of linear modular equations, where the unknowns also appear in the moduli. The general form would be:
$A \vec{x}= \vec{b} \;\textrm{mod} \; (C \vec{x}+\vec{d})$
where A ...
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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