# 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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### Calculate φ(36), where φ is the Euler Totient function. Use this to calculate 13788 (mod 36).

Hello I am wondering if any one can help me I am trying to figure out how to Calculate $φ(36)$, where $φ$ is the Euler Totient function. Use this to calculate $13788$ $(mod 36)$. I have an exam ...
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### A diophantine related query

Supposing I give you a multivariate equation $$F\in\Bbb Z[x_1,\dots,x_n]$$ Following is undecidable: 'Is there an $(a_1,\dots,a_n)\in\Bbb N^n$ such that $F(a)=0$?' However is the following always ...
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### Bezout's Identity proof and the Extended Euclidean Algorithm

I am trying to learn the logic behind the Extended Euclidean Algorithm and I am having a really difficult time understanding all the online tutorials and videos out there. To make it clear, though, I ...
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### Show that $r$ is a primitive root (mod $p^k$)

Let $p$ be an odd prime, and let $r$ be a primitive root (mod $p$) such that $r^{p-1} \neq 1$ (mod $p^2$). Show that $r$ is a primitive root (mod $p^k$) for all $k \geq 1$. So I start of by computing ...
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### Suppose $a,b,c > 0$. Then there are finitely many integer $x,y$ with $a^x > cb^y$.

Here is the question: For this question, it says to find finitely many positive numbers pairs of x and y for to fulfill the inequality. My thought is when [A] bigger than 1 or b is smaller than 1, ...
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### Divisibility by 41 and 5-digit number

How to prove that if a $5$-digit number is divisible by $41$,then all the numbers generated from it by cyclic shift are also divisible by $41$
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### How many integers $m$ such that Euler Totient Function $\varphi(m)$ = a given integer?

(I didn't find any other posts related to this but I think this is a weird because my question seems like a very natural one.) Let $\varphi$ denote the Euler Totient Function. Given $m$ an even ...
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### A positive integer a is self-invertible modulo p if and only if a ≡ ±1 (mod p).

What does it mean that "a positive integer is self-invertible modulo p"?
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### Simple NT/Algebra Proof

Prove that if $(p-1)(q-1)\ge 1$, over positive reals greater than 1, then $p+q\ge 4$. In essence, it seems this is proving that if $p+q<pq$, then $p+q\ge 4$, but this doesn't necessarily help. ...
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### Can a carmichael number have consecutive prime factors?

Is there a set $[p_1,p_2,...,p_n]$ of consecutive primes , such that $\prod_{j=1}^n p_j$ is a carmichael number ? For $3$ and $4$ prime factors, I checked upto $p_1\le 10^{10}$ without success. ...