# Questions tagged [congruences]

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### Solve congruence equation where unknown variable is in both sides of congruent operator

I am trying to solve the following equation: $(a*n + c) \mod (b-n) \equiv 0$ and $n$ must be the lowest value in $[0, b-1]$ for example $a=17$, $c=-59$ and $b=128$, the solution is $n=55$ $n=b-1$ ...
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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 ...
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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  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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### 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 $x$ ...
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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 ...
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### A Wolstenholme type congruence

Consider the following congruence: For $p\geq 5$ prime and every $n,\nu\in\mathbb{N}$ we have \begin{align*} 0\equiv\sum_{k=1\atop p\nmid k}^{pn-1}\frac1k \binom{pn(\nu+1)-k-1}{pn\nu-1} \mod p^{2(\...
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### Density of a set of numbers dividing a fixed number with polynomial exponent

Fix a positive integer $a>1$ and let $f\in\mathbb{Z}[x]$ be a polynomial with positive leading coefficient. We define a set $S$ of positive integers, $$S=\{n\in\mathbb{Z}^+:n\mid a^{f(n)}-1\}.$$ ...
163 views

### Number of odd elements in a vanishing sum of binomial coefficients

Let $n$ be a positive integer, $k$ a non-negative integer and $N(n,k)$ be the number of odd elements among the numbers $\binom{n+k}{j}\binom{-n-k}{n-j}$, $0\le{j}\le{n}$, which sum to $0.$ It seems ...