Tagged Questions

Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Find all even numbers that can be represented as a difference of squares in only two ways

I am currently working on this proof. I am looking to find (with proof) all even numbers that can be represented as a difference of squares in only two ways. My thoughts thus far. I examined the ...
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Need help with the proof of a theorem about Gaussian integers

Theorem 6-3. If $\alpha$ and $\beta$ are integers of $Z[i]$, and $\beta \neq 0$ then there are $\kappa$ and $\rho$ in $Z[i]$ such that $$\alpha =\beta\kappa+\rho, \text{ } N_\rho < N_\beta$$ ...
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Notation for representing ANY number?

i'm working on a mathematics/number-manipulation program, and i was wondering if you could practically have a representation that could holds the value of any number. This would need to include ...
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Deriving the sequence if the generating function is irreducible?

I am trying to better understand generating functions and how they can be derived / manipulated / etc. Right now I am operating on this identity, slightly modified from the answer here: For a ...
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lower bound for sum of distinct n-th roots of unity

Given a positive integer $n$, define $\zeta = e^{2\pi i/n}$ and define $s: \mathbb Z^n \to \mathbb C$$s(\vec x) = \sum_{k=0}^{n-1} x_k \zeta^k$$ Let us consider the set$S = \{ |s(\vec x)| : \vec x \...
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Find all the integer pairs $(r,s)$ that satisfy $s= (r^2 +3r +8) / (r^2 +r -2)$?

I have been trying to solve this question but struggling to see where to start. Examples I've seen that works are the pairs: $(-3,2) , (4,2), (0,-4)$
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Multivaritate version of Fermat's little theorem

If $f$ is irreducible over $\mathbb{F}_p[x]$ of degree $d$, then $$g(x)^{p^d} \equiv g(x) \bmod f(x)$$ and $p^d - 1$ is the order of the cyclic group $\left( \mathbb{F}_p[x]/f(x) \right)^{\times}$. ...
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Can the 290 Theorem be refined/sharpened to include special conditions?

The 290 theorem states If a positive-definite quadratic form with integer coefficients represents the twenty-nine integers $1$, $2$, $3$, $5$, $6$, $7$, $10$, $13$, $14$, $15$, $17$, $19$, $21$, ...
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Decimal expansion of an irrational number not ending in a particular sequence

Consider the set $N$ of natural number, $N$={1,2,3,4,5,6,7,8,9,10,11...} and consider the subsequence {$N_i$} ,$i \in N$ Each $N_i$ consists of elements in ascending sequence greater than $i$ $N_1$...
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Can you Prove or Disprove this?

In $a^n+b^n=c^n$ , $(a<b<c)$ , $a,b,n$ belongs to natural numbers, If $n>=b/2$ , $c$ lies between $(b,b+1)$. Also, only for $n=1,2$ , $c=b+1$.
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$\sqrt[n]{m}$ is irrational if $m$ is not the nth power of an integer

I'm reading the book A Classical Introduction to Modern Number Theory by Ireland and Rosen, and I think that there is an exercise which is false. The exercise says: Prove that " $\sqrt[n]{m}$ is ...
$a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$? [closed]
As the question title suggests, how do I see that $a \in \mathbb{Z}[i]$ is a unit if and only if $a$ divides every element of $\mathbb{Z}[i]$?
Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$? [closed]
Suppose that $a$, $b \in \mathbb{Z}[i]$ satisfy $a \mid b$ and $b \mid a$. Does it necessarily follow that $a = ub$ for some unit $u \in \{\pm1, \pm i\}$?