Questions tagged [congruences]
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138 questions
38
votes
4
answers
7k
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What did Yu Jianchun discover about Carmichael numbers?
There's a news story going around (see for example [1]; other accounts are even more breathless) about an amateur mathematician, Yu Jianchun, finding an "alternative method to verify Carmichael ...
- 9,114
35
votes
2
answers
2k
views
Is the sum of digits of $3^{1000}$ divisible by $7$?
Is the sum of digits of $3^{1000}$ a multiple of $7$?
The sum of the digits of $3^{1000}$ can be computed using a computer. It is equal to $2142$, so the answer is positive.
Is there a short proof ...
- 453
22
votes
1
answer
1k
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
$$ \binom{2n}{n} \equiv 2\pmod p ? $$
...
- 23k
22
votes
0
answers
1k
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 ...
- 1,241
19
votes
2
answers
1k
views
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,
$$ a(n) =\frac{1}{2n^2}\...
- 50.8k
18
votes
2
answers
3k
views
Binomial supercongruences: is there any reason for them?
One of the recent questions, in fact
the answer
to it, reminded me about the binomial sequence
$$
a_n=\sum_{k=0}^n{\binom{n}{k}}^2{\binom{n+k}{k}}^2,
\qquad n=0,1,2,\dots,
$$
of the Apéry ...
- 13.5k
18
votes
0
answers
755
views
Two curious series for $1/\pi$
On Jan. 18, 2012 I conjectured that for any prime $p>3$ we have
$$R_p^2\equiv\frac1{10}\left(512\left(\frac{10}p\right)-27\left(\frac{-15}p\right)-475\right)\pmod p,$$
where $(\frac{\cdot}p)$ ...
- 15.6k
13
votes
1
answer
3k
views
A good reference to the general Chinese Remainder Theorem
I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following
General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive ...
- 41.8k
13
votes
1
answer
584
views
A congruence for a product of binomial coefficients?
For every prime $p\geq 5$ one seems to have the congruence
$$(-1)^{(p-1)/2}\prod_{k=0}^{p-1}{p-1\choose k}\equiv 1-p+\frac{3}{2}p^2-\frac{7}{6}p^3\pmod{p^4}\ .$$
(I have checked all primes up to $5000$...
- 17.9k
13
votes
1
answer
1k
views
On cubic reciprocity for $x^3+y^3+z^3 = 996$?
I. The Diophantine equation,
$$x^3+y^3+z^3 = 3w^3\tag1$$
with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
- 12.6k
12
votes
4
answers
3k
views
Prove that $\sum_{n = 1}^{p - 1} n^{p - 1} \equiv (p - 1)! + p \pmod {p^2}$ with $p$ being an odd prime
First, I have to admit that I have already asked the same question on MSE several days ago. If I am bending any rules, I apologize for that and moderator can delete or close this question without ...
- 223
12
votes
1
answer
838
views
Numbers $k$ with $\{\binom nk:\ n\in\mathbb N\}$ dense in $\mathbb Z_p$ for any prime $p\le k$
Let $k$ be a positive integer and let $p$ be a prime. In my 2011 PAMS paper joint with my former student W. Zhang [Proc. Amer. Math. Soc. 139(2011), 1569-1577], we studied when $$S(k)=\left\{\binom nk:...
- 15.6k
11
votes
2
answers
1k
views
Ramanujan's tau function, $691$ congruence, and $\eta(z)^{12}$
Let $q = e^{2\pi i\,z}$.
I. 24th power
The Ramanujan tau function $\tau(n)$ is given by the expansion of the Dedekind eta function $\eta(z)$'s $\text{24th}$ power. Then
$$\begin{aligned}\eta(z)^{...
- 12.6k
11
votes
2
answers
369
views
Harmonic congruence
There are a number of interesting congruences for harmonic sums, not the least of which is Wolstenholme's theorem: $H_{p-1}:=\sum_{j=1}^{p-1}\frac1j\equiv 0\mod p^2$.
It appears that $\sum_{j=1}^{p-1}...
- 309
11
votes
1
answer
643
views
A conjecture on binomial coefficients and roots of unity
Is the following true?
Let $p$ be a prime and let $w$ be a $(p-1)$st root of unity (not necessarily primitive). Then
$$\binom{w}{n}=\frac{w(w-1)\cdots(w-n+1)}{n!}$$ is $p$-integral; i.e., it can be ...
- 17k
11
votes
1
answer
625
views
A congruence conjecture regarding $(r-s)^4-1 \equiv 0\!\pmod{4r^2s}$
Is the following conjecture true?
Conjecture. If $r > s \ge 1$ are relatively prime integers such that
\begin{equation}
(r-s)^4-1 \equiv 0\!\pmod{4r^2s}, \tag{1}
\end{equation}
then $r-s = 1$ ...
- 1,055
10
votes
2
answers
1k
views
Is there a fixed integer $x>1$ satisfing ${\sigma}^{k}(x)\equiv 0\pmod{x}$ for all positive integers $k$?
This question related to this question from SE. I'm interested to know if there exists an integer $x>1$ that satisfies $${\sigma}^{k}(x)\equiv 0\pmod{x}$$
for all positive integers $k$.
Note. $\...
- 414
9
votes
2
answers
276
views
For which values of $k$ is it known that there are infinitely many $n$, such that $2^{n+k}\equiv 1\pmod{n}$?
I know that there are no solutions to $2^n\equiv 1\pmod{n}$ for $n>1$ and I can prove that there are infinitely many $n$ such that $2^{n+1}\equiv1\pmod{n}$.
My question is:
Do we know other ...
- 3,866
9
votes
1
answer
357
views
On the permanent $\text{per}[i^{j-1}]_{1\le i,j\le p-1}$ modulo $p^2$
Let $p$ be an odd prime. It is well-known that
$$\det[i^{j-1}]_{1\le i,j\le p-1}=\prod_{1\le i<j\le p-1}(j-i)\not\equiv0\pmod p.$$
I'm curious about the behavior of the permanent $\text{per}[i^{j-...
- 15.6k
9
votes
0
answers
315
views
congruences of level 1 and level p modular forms
I've been carrying out some experiments on the computer and I noticed the following congruence phenomenon: fixing a prime $p$, it seems that any modular form over $SL_2(\mathbb{Z})$ and of weight $k \...
- 597
8
votes
1
answer
286
views
On the determinant $\det[\gcd(i-j,n)]_{1\le i,j\le n}$
In Sept. 2013, I investigated the determinant
$$D_n=\det[\gcd(i-j,n)]_{1\le i,j\le n}$$
and computed the values $D_1,\ldots,D_{100}$ (cf. http://oeis.org/A228884). To my surprise, they are all ...
- 15.6k
8
votes
1
answer
338
views
Subgroups of Sp(2g,Z) that map onto all Sp(2g,Z/m)
I stumbled into the following problem. I apologize for being a bit naive.
For $g\geq 3$, consider the group $\mathrm{Sp}(2g,\mathbb{Z})$ of symplectic square matrices of order $2g$ with integral ...
8
votes
2
answers
262
views
Congruences of binomial sums
Let $a_n$ is a binomial sum, for example
$$
a_n := \sum_{k} \binom{n-k}{k} \quad \text{or} \quad \sum_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k}
\quad \text{or} \quad \sum_{k=0}^n\sum_{\ell=0}^k\binom{n}{k}\...
- 17k
8
votes
1
answer
407
views
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
8
votes
0
answers
351
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 ...
- 15.6k
7
votes
1
answer
1k
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 $a \in \mathbb{Z}/l\mathbb{Z}-\{0,1\},$ $P_l(x)$...
- 495
7
votes
2
answers
373
views
Is there a nonzero solution to this infinite system of congruences?
Is there a triple of nonzero even integers $(a,b,c)$ that satisfies the following infinite system of congruences?
$$
a+b+c\equiv 0 \pmod{4} \\
a+3b+3c\equiv 0 \pmod{8} \\
3a+5b+9c\equiv 0 \pmod{16} \\
...
- 413
7
votes
2
answers
779
views
Could a nice principle be extracted from this lemma of Gauss
I asked the following question in the math SE, with a bounty of 200 pts, without result.
question:
To prove the quadratic reciprocity law, Gauss needed the following lemma:
If $p$ is a prime number ...
- 687
7
votes
1
answer
185
views
2-adic valuation of $L(0,\chi)$ for a Dirichlet character
Let $\chi : (\mathbb Z/f\mathbb Z)^\times \to K = \mathbb Q(\mu_{\phi(f)})$ be a primitive Dirichlet character. Assume moreover that it is not quadratic, that is, $\chi^2$ is not the trivial character....
- 7,736
7
votes
1
answer
290
views
On $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i+j}(2i+j)$ and $\prod^{(p-1)/2}_{i,j=1\atop p\nmid 2i-j}(2i-j)$ modulo a prime $p>3$
QUESTION: Is my following conjecture true?
Conjecture. Let $p>3$ be a prime and let $h(-p)$ be the class number of the imaginary quadratic field $\mathbb Q(\sqrt{-p})$. Then
$$\frac{p-1}2!!\prod^{...
- 15.6k
7
votes
1
answer
860
views
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
7
votes
1
answer
342
views
Corollaries of the halo conjecture that do not involve the eigencurve
In the theory of p-adic modular forms there is a certain construction called the Coleman-Mazur eigencurve. The spectral halo conjecture roughly states that if you remove a closed subdisc of the weight ...
user145520
7
votes
1
answer
421
views
On $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$
Euler's totient function $\varphi$ is multiplicative, and it plays important roles in number theory.
QUESTION: Is it true that for each integer $m>6$ we have $\varphi(m)\varphi(n)\equiv0\pmod{m+n}$...
- 15.6k
7
votes
2
answers
780
views
Some equalities involving prime powers
Let $p,a,b,x,y$ be positive integers where $p$ is an odd prime; $x$ and $y$ are odd; $p,x$ and $y$ are all coprime. I'm interested in finding examples of such numbers that satisfy this equation:
\...
- 11.2k
7
votes
0
answers
183
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....
- 15.6k
7
votes
0
answers
213
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 ...
- 15.6k
7
votes
0
answers
217
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
$$\sum_{k=1}^\infty\frac{3H_{k-1}^2+4H_{k-1}/k}{k^2\binom{2k}k}=\frac{...
- 15.6k
6
votes
2
answers
245
views
Congruence equation for Apery numbers
Does the system of congruence equations
\begin{eqnarray}
A_{17k}&\equiv& 0 \pmod {17^2}, \nonumber \\
A_{17k+1}&\equiv& 0 \pmod {17^2}, \tag{1}
\end{eqnarray}
has solutions other ...
- 16.5k
6
votes
3
answers
2k
views
Finite subgroup of $Gl(n,\mathbb Z)$ and congruences
Suppose we have an invertible matrix q in a finite subgroup $Q$ of
$Gl(n,\mathbb Z)$, the group of all invertible integer matrices. Now I want to
find all $x\; mod\; \mathbb Z^n$ for which
$(q+q^2+q^...
6
votes
1
answer
576
views
Are there finitely many primes $x$ such that for a fixed odd prime $p$, $n=x^{p-1}+x^{p-2}+\dotsb + x+1$ is composite and $x \mid \phi(n)$?
Let
\begin{equation} n =x^{p-1}+x^{p-2}+\dotsb + x+1
\end{equation} where $x$ and $p$ are odd primes.
If $p$ is set to $5$, it appears $x=5$ is the only prime $x$ such that $n$ is composite and $x \...
- 319
6
votes
2
answers
645
views
Number of solutions of $am \equiv bn \pmod{q}$
Let $q$ be a (large) prime. Let $N$ be a positive integer of size${}\approx \sqrt{q}$. Let $\mathcal{M}$ be an arbitrary subset of $\{1, \dots, q\},$ such that $\mathcal{M}$ has cardinality $N$. ...
- 2,197
6
votes
0
answers
192
views
A conjecture involving $P_n=\prod_{k=1}^np_k$
For each positive integer $n$ let $P_n=\prod_{k=1}^n p_k$, where $p_k$ is the $k$th prime.
Question. Is my following conjecture true?
Conjecture. For any integer $n>1$, there are $k,m\in\{1,\...
- 15.6k
6
votes
0
answers
217
views
Two conjectural congruences for Franel numbers
Recall that the Franel numbers are given by
$$f_n:=\sum_{k=0}^n \binom{n}{k}^3\ \ \ (n=0,1,\ldots).$$
Question. How to prove my following conjecture?
Conjecture. For each odd prime $p$, we have
$$\...
- 15.6k
5
votes
2
answers
510
views
"Efficient" way to build a table of multiplicative orders modulo $p$ of a fixed integer $a$
Given an integer $a$, I would like to build a table of entries $(p, \text{ord}_p(a))$, where $p$ runs over the prime numbers not dividing $a$ and not exceeding a fixed parameter $P$, and $\text{ord}_p(...
- 53
5
votes
2
answers
287
views
Modulo $x^2 + y^2 - 1$, is every homogeneous polynomial that is a square of a polynomial, necessarily of sum of squares of homogeneous polynomials?
I am hoping this question is alright for Math Overflow. I didn't get a definitive solution in Math Stack Exchange.
Let $f(x, y) \in \mathbb{R}[x, y]$ be a homogeneous polynomial with real coefficients ...
- 331
5
votes
1
answer
329
views
Reversing the CRT: Is $5$ tough?
Given odd primes $p\ne q$, by the CRT we can find an integer $x$ such that $x\equiv 2^{p-1}\pmod q$ and $x\equiv 2^{q-1}\pmod p$. Can this procedure be reversed?
For which integers $x$ there exist ...
- 550
5
votes
1
answer
270
views
Up to $2000$, $A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$
A145722 is
Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.
Using the pari program and offset 0, up to $2000$...
- 25.4k
5
votes
4
answers
374
views
When does the following congruence identity hold?
Let $m$,$l$ be coprime integers where $m,l\geq 2$. For any integer $a$ and positive base $b \ (b\geq 2)$, let
$
[a]_b
$ denote the element of $\{0,\ldots, b-1\}$ that satisfies the equivalence
$[a]...
5
votes
1
answer
430
views
About Morley congruence
Let $p>3$ be an odd prime and $a$ be a positive integer, is the following congruence true?
$$\binom{p^a-1}{\frac{p^a-1}{2}}\equiv(-1)^{\frac{p^a-1}{2}}4^{p^a-1}\pmod{p^3}.$$
When $a=1$, this is ...
- 51
5
votes
2
answers
348
views
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