# Tagged Questions

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### How many solutions to prime = $2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$

Let $a,b,c$ be integers, no sign restriction. Let $p$ be a given prime. How to find the number of solutions to $p = 2 b^2 c^2 + 2 c^2 a^2 + 2 a^2 b^2 - a^4 - b^4 - c^4$ ? Note, from Heron's ...
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### How many solutions to prime = $a^3+b^3+c^3 - 3abc$

Let $a,b,c$ be integers. Let $p$ be a given prime. How to find the number of solutions to $p = a^3+b^3+c^3 - 3abc$ ? Another question is ; let $w$ be a positive integer. Let $f(w)$ be the number of ...
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### How many solutions to prime = $(d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$?

Let $a,b,c,d$ be integers $>-1$. Let $p$ be a given prime. How to find the number of solutions to $p = (d^2-2ad+b^2-2ab+2a^2)(d^2-2cd+2c^2-2bc+b^2)$ ? I assumed that this polynomial above does not ...
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### primes of type norm($a+b\sqrt{-c}$) = primes of type $1$ $mod$ $c 2^{n-1}$?

Let $a,b,c,n$ be strictly positive integers where $c$ is a prime and such that the normed ring $a+b\sqrt{-c}$ is a UFD. I noticed the primes of norm($a+b\sqrt{-1}$) are $1 \bmod 4$. Also the ...
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### Question about rings and modulo multiplication tables

Why is 2=0 in K for part a)? Also I don't understand what part b) is asking you to do - what does it mean by alpha = [X], so M=etc. Could someone please explain the question and the solutions to part ...
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### Finding an integer that satisfies a congruence class equation

We have integers $a = 1231940$, $b = 9935$ and $n = 3999831$ and the ring $\mathbb{Z}/n\mathbb{Z}$. Now we should find an integer $x$ that satisfies the equation $[a] \odot_n [x] = [b]$. How can such ...
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### Chinese Remainder Theorem result varies

Sorry if this question is lame. First post! I was going through this book Abstract Algebra Theory and Applications Thomas W. Judson Stephen F. Austin State University In the Chapter ...
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### What is the difference between ring $\mathbb{Z}_n$ and congruence to modulo $n$.

What is the difference between ring $\mathbb{Z}_n$ and congruence to modulo $n$. Are they same thing or what are significant differences between them except we can use integers greater than $n$ for ...
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### When are quotient maps induced by equivalence relations surjective and injective?

Let $\sim$ be an equivalence relation on a set $X$. Also, there is a natural function $p:X\to \tilde X$ where $\tilde X$ is a set of all equivalence classes. (Equivalence classes are defined as, ...
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### Multiplication structure for finite abelian rings of order $p^2$.

Consider an odd prime $p$ and a positive integer $a$ as close to 2 as possible that is not a quadratic residue $mod$ $p$. If we extend the ring $mod$ $p$ with the element $b = a^{\frac{1}{2}}$ then ...
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### Solving a polynomial modulo an integer

Say I have a polynomial $F$ of degree $n$ with coefficients in $Z_m$ and I wish to find $x$ such that $F(x)=0$ (mod $m$). For instance if $F(x)=x^{2}-a$ the solution would be the modulo $m$ squareroot ...
The question rephrased and compressed: Let $F=F_2[a]$ be a finite extension field of the field of two elements $F_2$. We are given a polynomial $R(X)\in F[X]$, and pairwise coprime irreducible ...
### Prove the sum of all numbers that do not have a multiplicative inverse mod $n$
I understand that for a number $a$ to have a multiplicative inverse in mod $n$, it must be coprime to $n$; therefore, all numbers that do not have a multiplicative inverse mod $n$ must share a factor ...