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 $a-b$.
13,378 questions
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$3^x\times 7+475=2^y$ has no integer solutions?
I wanted to solve the equation :
$3^x\times 7+475=2^y$ where $(x,y) \in \mathbb{N^2}$
My attempt :
If $x$ is even, $x=2k \Rightarrow 3^x = 9^k \equiv 1 \pmod 8$
so $7 \times 3^x +475 \equiv 7\times 1+...
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$2^n-1$ can't be a powerful number?
If $2^n - 1 = a^b$ with $b>1$ , then by Mihailescu's Theorem such $(n, a, b) > 1$ would not exist.
Moreover , can we always find a prime $p$ such that $v_p(2^n-1)=1$ ?
i.e : $2^n-1$ can't be ...
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Radium Rabbit Conjecture, version 3.0: The fractional part of the square of the area of a triangle with odd-integer sides is $\frac{3}{16}$. [closed]
Why Version 3.0?
In the earlier versions of this conjecture, I focused on triangles whose side lengths are distinct prime numbers.
Through the discussion that followed, it became clear that the ...
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Improve strategy for Goldbach's conjecture?
Consider the Goldbach conjecture that states there is always a pair of primes $p_1$ and $p_2$ such that
$$ p_1 + p_2 = 2m$$
Where m is an integer $>1$.
One way of reformulating this (via prime ...
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Minimal odd $k$ for which $y^k=y$ stabilizes in the $n$-adics
We consider fixed-point equations of the form $y^k = y$ ($k \in \mathbb{N}$) in $\mathbb{Z}_n$, where here $\mathbb{Z}_n$ denotes the ring of $n$-adic integers (i.e., the projective limit $\...
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When three translates of a cubic monomial hit every remainder modulo a prime?
The polynomial $x\mapsto x^3$ is not a permutation polynomial modulo 7, for example $1^3\equiv 2^3\pmod{7}$. However, for every integer $n$, at least one of three congruences $x^3\equiv n\pmod{7}$, $x^...
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Is there an official name for a modulo-like function with result in the range $-m/2$ to $m/2$? [duplicate]
(NB: edited after the criticism of the first answer)
The standard result will be
$$
n \ \text{modulo} \ m
= n - m \left\lfloor\frac{n}{m}\right\rfloor\
\ \in \ \{0,\, \dots,\, m\!-\!1\}
$$
and ...
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Solutions to $x^n \equiv x \pmod{n}$ for composite $n$.
By Fermat's Little Theorem we know that if $p$ is some prime number, the congruence $x^p \equiv x \pmod{p}$ is solved by any integer $x$, but can we say something about the solutions to $x^n \equiv x \...
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What is the least possible value of p+q? [duplicate]
Problem: $$n^{3pq}- n$$ is a multiple of $3pq$ for all positive integers n. Find the least possible value of p + q?
My Progress:
If $$ n^{3pq} \equiv n \pmod{3pq} $$ for all n, it hints at using ...
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The numerator of $ C_{n>4} = \frac{1}{3+\frac{5}{7+\frac{11}{13+\frac{17}{19+\frac{23}{29+\dots p_n}}}}} $ is always divisible by $17$?
For $n^{th}$ odd prime $p_n$,
We define the following fraction:
$$
C_n = \frac{1}{3+\frac{5}{7+\frac{11}{13+\frac{17}{19+\frac{23}{29+\dots p_n}}}}}
$$
We also define $N_n$ as the numerator of the $\...
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Some subset of $n$ numbers has the same sum $\!\bmod n$ as the whole set [duplicate]
If a sum of n numbers has remainder 1 after division by n. Does there exist a sub-sum of those numbers that have remainder 1 after division by n.
Put more formally:
Given a sequence $(a_1, a_2,...,a_n)...
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Deduce no integer solutions for $y^2 = x^3 + 7$ using prime factors. [duplicate]
I am working on an exercise that involves proving that the equation $$y^2 = x^3 + 7$$ has no integer solutions $(x, y)$. I am thinking of using modular arithmetic and congruency. Starting with LHS:
...
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Adding a number to a multiple of nine does preserve its digital root. [duplicate]
If we add any number to a multiple of nine, then the summation of the digits of the addent (till it comes to single digit)is equal to the summation of the digits of the result(till it comes to single ...
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Is the reverse bit shift map times $3^n$ guaranteed to have sequences with no further multiples of $3$?
Question
The dynamical system:
Let $f(x)=\begin{cases}(x+1)/2&&x\textrm{ odd}\\x/2&&x\textrm{ even}\end{cases}$
is the the bit shift map with binary strings reversed. It terminates ...
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Order of a polynomial $P\pmod{n}$
Let $P$ be a polynomial in $\mathbb{Z}[X]$ and $n \geq 1$ be an integer.
Consider the vector $\left(P(0), P(1),\ldots,P(n-1) \right) \pmod{n}$
Now apply $P$ again, and again, pointwise to the vector, ...
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