# 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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### Approximating $\pi$ by an expression of the form $\sqrt{\sqrt{ \cdots \sqrt{ n!! \cdots !}}}$

Here is a problem that appeared as a prize challenge in a periodical for science students, back when I was a student: Find an approximation of $\pi$ formed of the numbers $0$ through $9$, each used ...
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### Why does the elliptic curve for $a+b+c = abc = 6$ involve a solvable nonic?

The curve discussed in this OP's post, $$\color{brown}{-24a+36a^2-12a^3+a^4}=z^2\tag1$$ is birationally equivalent to an elliptic curve. Following E. Delanoy's post, let $G$ be the set of rational ...
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### How do you find the smallest legitimate encryption exponent when you are only give a p and q value in a given range?

I have been given this as an assignment question but I'm not sure approach it. EDIT: Sorry I should have added more details. It is a cryptosystem using the RSA scheme. p and q are both old prime ...
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### $p+q\neq 1+pq$ for distinct odd primes $p$ and $q$

I'm trying to show that that $\sqrt p + \sqrt q$ cannot be written as a linear combination of $1$ and $\sqrt{pq}$ with rational coefficients, and I have boiled it down to showing that $p+q \neq 1+pq$ ...
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### Finite sums of integers and similar problems: book request

I recently learned about Faulhaber's formula, which says that for each integer $p \geq 1,$ we can simplify the finite sum $\sum_{k \in \mathbb{N}}[k<n]k^p$ so that it becomes an (integer-valued) ...
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### Finding Numbers where modulo is k

I have given a number $A$ where $1\le A\le 10^6$ and a number $K$. I have to find the all the numbers between $1$ to $A$ where $A\%i=k$ and $i$ is $1\le i\le A$. Is there any better solution than ...
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### If $a$, $b$, $z$ and $y$ are integers such that $\gcd(a,b)=1$ and $x^a=y^b$, show that $x=n^b$ and $y=n^a$ for some integer $n>1$

I first noted that the set of primes dividing $x$ will be the same as the set of primes dividing $y$. Then i assumed $x=p^l$ and $y=p^m$ (where $p$ is any prime dividing $x$ and $y$ and $p^l$ is the ...
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### Counting non-decreasing integer sequences with a condition

I am having difficulty framing this properly. How many non-decreasing integer sequences are there of length $n$, where each element is bound between $1$ and $m$ inclusive, such that the longest ...
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### Periodic tribonacci-like sequence

How to prove that if $a_n =[(t_{n-3} + 2t_{n-2} + t_{n-1}) a_{1} + (t_{n-3} + t_{n-2} + 2t_{n-1})] \quad (\text{mod}10)$ and $a_{1}, a_{2}, a_{3}$ are consecutive numbers and $t_{1}=0, t_{2}=1$ and ...
The following corollary of Krasner´s Lemma says: Let k be a global field and p a prime of k. Then $(\overline{k})_p=\overline{k_p}$. Im wondering if $(\overline{k})_p$ means the completion of ...
### Cover $\{1,2,…,100\}$ with minimum number of geometric progressions?
In another question, posted here by jordan, we are asked whether it is possible to cover the numbers $\{1,2,\ldots,100\}$ with $20$ geometric sequences of real numbers. Naturally, we would like to ...