Questions tagged [congruences]

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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 ...
William DeMeo's user avatar
18 votes
0 answers
746 views

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 $(\frac{\cdot}p)$ ...
Zhi-Wei Sun's user avatar
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9 votes
0 answers
305 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 \...
Nadim Rustom's user avatar
8 votes
0 answers
316 views

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 ...
Zhi-Wei Sun's user avatar
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7 votes
0 answers
179 views

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....
Zhi-Wei Sun's user avatar
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7 votes
0 answers
206 views

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 ...
Zhi-Wei Sun's user avatar
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7 votes
0 answers
210 views

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{...
Zhi-Wei Sun's user avatar
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6 votes
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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,\...
Zhi-Wei Sun's user avatar
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6 votes
0 answers
212 views

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 $$\...
Zhi-Wei Sun's user avatar
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5 votes
0 answers
500 views

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 ...
Zhi-Wei Sun's user avatar
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5 votes
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200 views

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,...
Erik4's user avatar
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272 views

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}...
LFM's user avatar
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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=...
Zhi-Wei Sun's user avatar
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4 votes
0 answers
391 views

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 $[...
René Gy's user avatar
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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 $A=a_1<\cdots<a_r$ ...
Charles's user avatar
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3 votes
0 answers
315 views

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 ...
Adithya Chakravarthy's user avatar
3 votes
0 answers
179 views

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 ...
Lennart Meier's user avatar
3 votes
0 answers
282 views

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 ...
user142929's user avatar
2 votes
1 answer
379 views

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 \...
Samdney's user avatar
  • 23
2 votes
0 answers
61 views

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^{...
quser86341's user avatar
2 votes
0 answers
151 views

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 ...
Jimoni's user avatar
  • 21
2 votes
0 answers
66 views

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 ...
user142929's user avatar
2 votes
0 answers
100 views

$\varphi(m+n)\mid n$ 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 ...
Zhi-Wei Sun's user avatar
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2 votes
0 answers
110 views

"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 $k$...
MacNamara's user avatar
2 votes
0 answers
173 views

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 ...
THC's user avatar
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2 votes
0 answers
487 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 ...
Mathoverflow's user avatar
1 vote
0 answers
173 views

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 ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
0 answers
138 views

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 ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
0 answers
164 views

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 ...
Marek Kryspin's user avatar
1 vote
0 answers
151 views

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}\ ...
Zhi-Wei Sun's user avatar
  • 14.4k
1 vote
0 answers
48 views

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 ...
VS.'s user avatar
  • 1,816
1 vote
0 answers
67 views

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}=\{\...
Zakariae.B's user avatar
1 vote
0 answers
84 views

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 ...
Surajit's user avatar
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1 vote
0 answers
245 views

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 ...
gmp's user avatar
  • 65
1 vote
0 answers
242 views

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 $u$. ...
shamym shamim's user avatar
1 vote
0 answers
63 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" ...
Dmitry Kerner's user avatar
0 votes
0 answers
121 views

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} \...
Marco Ripà's user avatar
  • 1,119
0 votes
0 answers
43 views

A question about algebraic indicator functions

Let $f \in \mathbb{Z}[x]$ and $m,k \in \mathbb{Z}$. Consider the indicator function $g_f : \mathbb{Z} \to \{1,0\}$ given by \begin{align*} g_f(n) = \begin{cases} 1 &\text{if there exists $r \in \...
matt stokes's user avatar
0 votes
0 answers
126 views

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 ...
Marco Ripà's user avatar
  • 1,119
0 votes
0 answers
179 views

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 ...
user avatar
0 votes
0 answers
90 views

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^...
zeraoulia rafik's user avatar
0 votes
0 answers
588 views

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 ...
Arnaud's user avatar
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0 votes
0 answers
407 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 ...
Gary's user avatar
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