# Questions tagged [modular-arithmetic]

Modular arithmetic (clock arithmetic) is a system of integer arithmetic based on the congruence relation $a \equiv b \pmod{n}$ which means that $n$ divides $a-b$.

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### How do you parameterize simultaneous solutions to equations with expressions like "$x +2 \left\lfloor\frac{x}{3}\right\rfloor + 1 - [3 \mid x]$"?

Let all functions be integer functions herein. I.e. $\Bbb{Z}\to\Bbb{Z}$ or $\Bbb{N}\to\Bbb{Z}$ where appropriate. I found this jewel of floor functions. So that made me wonder whether, we can solve ...
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### number theory, binomial coefficients divisibility

Let $p$ be a prime number greater than $3$ and $n$ a positive integer. Suppose $\nu _p(n) = r$. Prove that $\dbinom{np}{p} - n$ is divisible by $p^{3+r}$ The problem is here: https://poti.impa.br/...
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### how can I solve the equation $19^x+23^y=100z$ for integers $x,y,z$?

I tried to find all solution for integers $x,y,z$ $$19^x+23^y=100z$$ I tried by using $mod$ operation by taking $\mod20$ and I get $$3^y\equiv_{20} (-1)^{x+1}$$ where $\equiv_{n}$ means both sides ...
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### Number of polynomial functions in $\mathbb Z/2\mathbb Z[x_1, \, \ldots, \, x_n]$?

What is the number of polynomial functions in $\mathbb Z/2\mathbb Z[x_1, \, \ldots, \, x_n]$? (Here I define $p \sim q$ iff $p(x) = q(x)$ for all $x$.) What about the case if we allow permutations? ...
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### Polynomial Divisibility by Factorial Plus One [closed]

Find all polynomials with integer coefficients $f$ such that $f(p) \mid (p-1)! + 1$ for all primes $p$. Using Wilson’s theorem we find that $f(p)=p;-p$ satisfy the problem. I also thought about using ...
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### For each integer $k,$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},$ s.t. for each $n,\ p_{n+1}=2p_n- 1$ or $p_{n+1} =2p_n+1?$

For each $k\in\mathbb{N},$ does there exist a $k-$tuple of primes, $(p_n)_{n=1}^{k},\$ such that for each $n,$ the following is satisfied: $p_{n+1} = 2p_n- 1\$ or $p_{n+1} = 2p_n + 1?$ If yes then ...
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### general form for effective magazine size with refunded bullets

This exploration started because of the video game destiny, in which weapons can have essentially magical perks, such as every 3rd shot hit, add one bullet back into the magazine. What I am curious ...
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### distribution of square roots of unity $mod n$ | Factoring with inverse pair

I am writing a proof related to the RSA cryptosystem, specifically showing that given an inverse pair $d, c$ under multiplication mod $\phi(N)$, where $$dc \equiv 1 \pmod{\phi(N)},$$ there exists a ...
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### Confusion about replacing a variable in a expression of parameters

I have the following problem: I have a diophantine equation of the form: $$f(x,a,b)+c=0.........(**)$$ and by reducing it modulo $p^n$ I get a congruence of the form $h(x,a,b)+c\equiv 0\pmod {p^n}$ ...
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### Find two variables from known modulo equation and arithmetic equation.

Given two equations like these: $P \bmod Q = 90$ $P - Q = 9900$ How to find $P$ and $Q$ value? Edit (Using subtition): $P \bmod Q = 90$ $P = 90 \bmod Q$ $9900 + Q = 90 \bmod Q$ Here I'm stuck.....
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### Suppose that there exists infinity prime $p_i$, such that $a$ is congruent to a power of $2 \bmod p_i$, does $a$ need to be a power of $2$?

Let $a$ be an integer. Suppose there exists infinitely many primes $\{p_i\}_{i \in \mathbb N}$ such that $a \equiv 1 \pmod{p_i}$, then it's clear that $a$ has to equal to $1$. Now, suppose there ...
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### Squares in $(\mathbb{Z}/p^n \mathbb{Z})^\ast$ in terms of squares in $(\mathbb{Z}/p \mathbb{Z})^\ast$

Let $p$ be an odd prime and $n$ be a positive integer. For a ring $S$, let $S^\ast$ denote the set of units in $S$. It is known that $x$ is a square in $(\mathbb{Z}/p^n \mathbb{Z})^\ast$ if and only ...
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### Repeated application of the map $x \mapsto 2^x \bmod x$

Recently, I've been playing around with the map $x \mapsto 2^x \bmod x$. I calculated the smallest natural number requiring $n$ application of this map to reach $0$, and here's that sequence: 0, 1, ...
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### Last $4$ nonzero digits of $n!$ in base fourteen

Question: What are the last $4$ nonzero digits of $2025!$ in base fourteen? Note: This is not a contest/homework question. I know how to find the number of trailing zeroes in base fourteen, so $2025!$ ...
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Intuitive Idea: Say we have polynomials modulo positive integers; for instance, $f(x) = 3x + 1\pmod 5$ and $g(x) = 5x^2 - x + 3 \pmod 7$. (Yes, there is some abuse of notation here.) We may "...