# Tagged Questions

Questions on congruences, linear diophantine equations, greatest common divisor, divisibility, etc.

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### Is there a formula for $(\frac{3}{p})$?

In number theory we learn when $2$ is a quadratic residue: $\left( \frac{2}{p}\right) = (-1)^{\frac{p^2 - 1}{8}}$ It takes a moment to verify that $\displaystyle \frac{p^2 - 1}{8} \in \mathbb{Z}$ ...
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### construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$?

how might I construct a unit on $\mathbb{Z}[\sqrt[3]{7}]$? Can it be done using pigeonhole principle as with square roots and Pell equation. I had been reading about the Voronoi continued fraction or ...
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### Diophantine equation with binomial coefficient

Suppose that $p$ is a prime number and $p \le q \le p^2$ is an integer. How many solutions are there to the following equation? $$\binom{p^2}{q}-\binom{q}{p}=1$$ This question was proposed ...
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### Find $3^{333} + 7^{777}\pmod{ 50}$

As title say, I need to find remainder of these to numbers. I know that here is plenty of similar questions, but non of these gives me right explanation. I always get stuck at some point (mostly right ...
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### Integer divisibility

Given a (not strictly) decreasing sequence of natural positive numbers $a_1, a_2, \dots, a_n$ prove that $$\prod_{i<j} j-i \quad\big|\quad \prod_{i<j} a_i - a_j - i +j$$ I already know a ...
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### Find the first digit of a huge product in case of multiplication

Let there be many numbers $a_1,a_2,a_3,\dots,a_n$. I want to find the first digit of their product, i.e. of $A=a_1\times a_2\times a_3\times a_4\times \dots\times a_n$. These numbers are huge and ...
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### Compute $(a_n,a_{n+1}) \forall n$ with $a_n$ the fibonacci sequence

Here my attempt: Employ the euclidean algorithm, i.e. $\forall r_0,r_1 \exists q,b: r_0 = q\cdot r_1+r_2, 0\le r_2 \lt r_1$. $q,b$ are determined uniquely. Since the definition of the fibonacci ...
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### Help with finding the remainder of $2^{2^n}$ when divided by 13

I have this problem from an algebra course: Find the remainder of $2^{2^n}$ when divided by 13, $\forall n \in \Bbb N$ It's in a section of Fermat's little theorem and Chinese Remainder Theorem ...
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### Compute $(a^{2^m}+1,a^{2^n}+1)$

I need your uncaring objective eyes. The hint for the problem was: show that for a given $m>n$ $A_n|A_m-2$. So here my attempt: $A_m-2=a^{2^m}-1$, additionally I can write $m=n+k$ and therefore I ...
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### Let $a \in \Bbb Z$ such that $gcd(9a^{25}+10:280)=35$. Find the remainder of $a$ when divided by 70.

I'm stuck with this problem from my algebra class. We've recently been introduced to Fermat's little theorem and the Chinese Remainder Theorem. Let $a \in \Bbb Z$ such that $gcd(9a^{25}+10:280)=35$...
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### Lower bound for $\text{ord}_{pq} a$

Let $p$ and $q$ be distinct odd prime numbers and $\gcd(a, pq)=1$. Is there an expression for lower bound of order of $a$ modulo $pq$? I know that with the given conditions we havea^{\text{lcm}(p-1,...
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### There are many numeral systems. Why do we only use the $0-9$ Hindu-Arabic numeral system?

Here is a list of other systems: Babylonian numerals Egyptian numerals Aegean numerals May numerals Chinese numerals These system are far older than the current system. How did it get to be known ...
Is there a name of or a reference to the following topology on $X=\mathbb N$: $A\subseteq X$ is closed if and only if $n\in A\wedge m|n\implies m\in A$?
### Prove that $\forall k = m^2 + 1. \space m \in \mathbb{Z}^+$, if $k$ is divisible by any prime then that prime is congruent to $1, 2 \pmod 4$.
Prove that $\forall k = m^2 + 1. \space m \in \mathbb{Z}^+$, if $k$ is divisible by any prime then that prime is congruent to $1, 2 \pmod 4$. I am unable to realize why it can't have $2$ prime ...