All Questions
Tagged with congruences reference-request
10 questions
2
votes
1
answer
118
views
Reference request for a proof of the fact that every congruence-permutable variety is semidegenerate
Given an algebra $\mathbf{A}$, a pair of congruences
$ \alpha ,\beta \in Con(\mathbf {A})$ are said to permute when
$ \alpha \circ \beta =\beta \circ \alpha$, and an algebra
$\mathbf{A}$ is called ...
- 21
3
votes
1
answer
439
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 \...
- 33
3
votes
2
answers
710
views
Binomial coefficient congruence modulo $p^n$
I am interested in the following congruence
$$\binom{ap^n}{bp^n}\equiv \binom{a}{b}\pmod{p^n}$$
I am aware that by some reference in a book the above it should actually hold modulo $p^{3n}$; the ...
- 847
8
votes
2
answers
262
views
Congruences of binomial sums
Let $a_n$ is a binomial sum, for example
$$
a_n := \sum_{k} \binom{n-k}{k} \quad \text{or} \quad \sum_{k=0}^n\binom{n+k}{n-k}\binom{2k}{k}
\quad \text{or} \quad \sum_{k=0}^n\sum_{\ell=0}^k\binom{n}{k}\...
- 17k
3
votes
1
answer
80
views
Reference request for a proof of the Mal'cev condition for congruence $n$-permutability
By a theorem of Hagemann and Mitschke, a condition (A) that a variety $\mathcal{V}$ is congruence $n$-permutable, is equivalent to a condition (B) that there exist ternary terms $p_1,\dots,p_{n-1}$ ...
- 133
0
votes
0
answers
180
views
When is $\phi(a^n+b^n+c^n)=0\mod n$?
A corollary Zsigmondy's Theorem leads to the following congruence (one can look to $(24)$),$\phi(a^n+b^n)=0\mod n$ whenever $a, b$ are coprime and $n \neq 2$ and $(a,b)\neq(1,1)$. (Here $\phi$ is the ...
user147204
1
vote
1
answer
218
views
Chinese Remainder Theorem for Remainder Intervals
Given $n$ natural numbers $m_1,\dots,m_n$ and $n$ remainder intervals $[a_1,b_1],\dots,[a_n,b_n]$ holding $a_i < b_i$ for all $i\leq n$ the task is to search for the smallest natural number $x$ ...
- 11
13
votes
1
answer
3k
views
A good reference to the general Chinese Remainder Theorem
I am writing a paper on the topology of the Golomb space and need a good (standard) reference to the following
General Chinese Remainder Theorem. For integer numbers $a_1,\dots,a_n$ and positive ...
- 41.8k
11
votes
2
answers
369
views
Harmonic congruence
There are a number of interesting congruences for harmonic sums, not the least of which is Wolstenholme's theorem: $H_{p-1}:=\sum_{j=1}^{p-1}\frac1j\equiv 0\mod p^2$.
It appears that $\sum_{j=1}^{p-1}...
- 309
5
votes
1
answer
430
views
About Morley congruence
Let $p>3$ be an odd prime and $a$ be a positive integer, is the following congruence true?
$$\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