Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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10 views

Similarities/differences between multivariate polynomials and integers

There are a few questions on this site that asks for similarities between integers and univariate polynomials. I am wondering if multivariate polynomials have any related analogies with integers.
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0answers
20 views

Natual density inside a subsequence

Let $S \subset \mathbb N$ be a subset. The natural density is defined as $$D(S) = \lim_{n \to \infty} \frac{|E \cap \{1, \cdots, n\}|}{n}$$ whenever this limit exists. So question is the ...
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0answers
23 views

Solving a system of congruences.

Solve Congruences system $$2x \equiv 1 \mod{n} $$ $$3x \equiv 9 \mod{m}$$ $$4x \equiv 1 \mod{m}$$ $$5x \equiv 9 \mod{m}$$ i dint undertand to my teacher, help me with this excercise step by step.. ...
11
votes
2answers
86 views

How prove this diophantine equation $x^2+y^2+z^3=n$ always have integer solution

show that: For any postive ineteger $n$,then the equation $$n=x^2+y^2+z^3$$ always have integer solution My idea: such as $n=1$,then we have $$1=0^2+0^2+1^3$$ $$2=0^2+1^2+1^3$$ ...
2
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2answers
34 views

Prove if $3$ does not divide $n$, then $n^2=1+3k$ for some integer $k$

I am proving by cases but am getting confused. I am not sure if this leads to a contradiction or not. Here's what I have so far: Direct Proof. Suppose $3$ does not divide $n$. Case 1: remainder ...
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0answers
19 views

proof $x \equiv a \mod{n} $, $x \equiv b \mod{m}$

proof that the congruences. $$x \equiv a \mod{n} $$ $$x \equiv b \mod{m}$$ have the same solution if and only if $ gcd ( n , m ) | a-b $ ; also if you have solution , show that is single mod ...
3
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0answers
25 views

Being $\gamma$ the Euler-Mascheroni constant and $N$ a constant $>0$, show $\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}$.

Related. Show that $$\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this paper might be useful. If $N$ cannot be ...
1
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0answers
17 views

multiples (of primes) coverage formula

I apologize in advance if my explanation is not clear. Please let me know if clarification is required and I will do my best to fix it! I am attempting to find an explicit formula (in terms of ...
4
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1answer
23 views

Do primes modulo k form a normal sequence?

For some $k>2$, form a sequence whose nth term is the nth prime that is not a divisor of $k$ modulo $k$. e.g. for $k=4$ the sequence would be 1,3,1,3,3,1,1,3,3,1,3,1... Is this sequence normal, ...
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1answer
32 views

a possible period of 124 for the sign of Ramanujan $\tau(3^n)$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
5
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4answers
105 views

How can I prove the last two digits of $1+2^{2^{n}}+3^{2^n}+4^{2^n}$ always are $54$

How can I prove the last two digits of $$1+2^{2^{n}}+3^{2^n}+4^{2^n}$$ are $54$ when $n$ is a positive integer number if $n>1$
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1answer
16 views

Prove that $\Bbb F_p^\times$ is equal to Miller–Rabin primality test for prime number

I want to prove, that $\Bbb F_p^\times = MRP(p)$. I think, that I have to start with this statement: $\{a \in \Bbb F_p^\times | a^2 = 1 \} = \{1; -1\}$ But I do not know how to continue this idea.
1
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0answers
30 views

What is the 'Hom-description'?

I am trying to learn about the 'Hom-description' of the class group $Cl(A)$ of an $R_K$-order $A$ in $K[G]$ where $K$ is a number field with ring of integers $R_K$ and $G$ is a finite group. I've ...
2
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1answer
57 views

Finding the least prime of the form $6^{6^6}+k$

I try to find the least prime number of the form $6^{6^6}+k$. I sieved out the candidates by trial division upto $10^6$, but there are still many candidates left upto $k=10000$ How can I further ...
5
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4answers
87 views

Prove that $13\vert(3^{n+1} +3^{n} +3^{n-1})$

Prove that $3^{n+1} +3^{n} +3^{n-1}$ is divisible by $13$ for all positive integral values of $n$
2
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0answers
57 views

Why does $n$ always divide this sum?

If we assume $m=p_1^{a_1}\cdots p_s^{a_s}, n=p_1^{b_1}\cdots p_s^{b_s}p_{s+1}^{b_{s+1}}\cdots p_t^{b_t}$, where $0<a_i<b_j$, $p_j$ are different primes($i=1,\cdots,s; j=1,\cdots, t$). Then ...
5
votes
3answers
43 views

Eisenstein integers and applications to Diophantine equations

Solve the equation $7\times 13\times 19=a^2-ab+b^2$ for integers $a>b>0$. How many are there such solutions $(a,b)$? I know that $a^2-ab+b^2$ is the norm of the Eisentein integer $z=a+b\omega$, ...
1
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1answer
41 views

Number of coprimes of $n$ divisible by 3

I'm looking for a formula for $C(n)$ := the number of coprimes of $n$ in the range $[1, n]$ divisible by 3, where $n$ is any positive integer. The formula should be quick to compute, preferably at ...
3
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2answers
34 views

What natural numbers are not equal to the sum of the sum and the product of two natural numbers

What natural numbers $n$ do not satisfy the equation $$n = (x+y)+xy$$ where $x$ and $y$ are both natural numbers?
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0answers
25 views

Integer, sum of odd and even.

Is there a unique way to tell whether an integer is always the sum of precisely one odd integer and one even integer? Emphasis on unique.
2
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3answers
40 views

Prove that if $p\mid(aq)^2$ and $(a,p) = 1$ then $p = q$ where $p,q$ are primes.

There is a theorem that if $p\mid aq$ and $(p,a) = 1$ then $p|q$ but I don't know how to use this theorem to solve the problem.
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2answers
26 views

a basic question about the natural Density

There is the same question about the irrational natural density, but I can't find the explicit form about the answer. Are there any sets of natural numbers with irrational natural density? I.e., does ...
0
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1answer
26 views

Is binary isomorphic to decimal representation?

My friend and I were just talking about whether decimal representation isomorphic to binary one. He said that it is true since there is a obvious 1-1 relationship between them. But how about ...
2
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0answers
23 views

Asymptotics for the Alternating Mertens Function

Are there any tight bounds, or any nontrivial ones actually, known for, with the lack of a better name, the Alternating Mertens Function: $$ S(n) = \sum_{k=1}^{n} \left((-1)^k \mu\left(k ...
4
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1answer
63 views

Sign of Ramanujan $\tau$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
1
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1answer
19 views

Demostrate,, $σ_{t}(n)=(\frac{p1^{t(k_{1}+1)}-1}{p1^t-1})…(\frac{p1^{t(k_{1}+1)}-1}{p1^t-1})$ [on hold]

If $n=p1^{k1}p2^{k2}...pr^{kr}$ is the prime factorization. $$σ_{t}(n)=(\frac{p1^{t(k_{1}+1)}-1}{p1^t-1})...(\frac{p1^{t(k_{1}+1)}-1}{p1^t-1})$$
1
vote
3answers
57 views

Are there in pure mathematics ensembles of number's which not divided by them self except $0$?

In pure mathematics we know well that each number divided by him self except $0$ , the question that let me confused is: Is there a proof in pure mathematics show to us that there are others ...
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0answers
32 views

Theorem for Equal Sums of Like Powers $x_1^8+x_2^8+x_3^8+\dots$

Kindly see the question at the end of post. Solutions to the system of three equations, $$\begin{aligned} a^2+b^2+c^2+d^2\, &= e^2+f^2+g^2+h^2\\ a^4+b^4+c^4+d^4\, &= e^4+f^4+g^4+h^4\\ abcd\, ...
2
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4answers
56 views

What does it mean to say breaking RSA generically is equivalent to factoring?

I am giving a one hour presentation on the RSA crypto-system as one of the requirements for Masters degree. I just want to get some facts straight here. I was told casually by a professor that RSA is ...
2
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1answer
49 views

Sinha's Theorem for Equal Sums of Like Powers $x_1^7+x_2^7+x_3^7+\dots$

Sinha’s theorem can be stated as, excluding the trivial case $c = 0$, if, $$(a+3c)^k + (b+3c)^k + (a+b-2c)^k = (c+d)^k + (c+e)^k + (-2c+d+e)^k\tag{1} $$ for $\color{blue}{\text{both}}$ $k = 2,4$ ...
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2answers
18 views

$f(x)\equiv_n 0$ has a solution iff $f(x)\equiv_{p_i ^{a_i}} 0$ has a solution

Let $f(x)\in \mathbb Z [x]$ and $n=p_1 ^{a_1}\cdot...\cdot p_t ^{a_t}$ prime factorization. show that $f(x)\equiv_n 0$ has a solution iff $f(x)\equiv_{p_i ^{a_i}} 0$ has a solution for each ...
22
votes
2answers
293 views

Estimate for $n$th prime

A good approximation I have found for $p_{n}$ is \begin{align} \int_{2}^{n}\log (x \log (x \log (x)))\ dx\\ \end{align} and seems to be a better estimate than $n \log (n)$. The error term seems to ...
2
votes
1answer
27 views

Can we tell if a number is prime by the number of its partition ?

Can we tell if a number is prime by the number of its partition ? Or in general, how much can we know about a number itself from its partition function ? I understand that Ramanujan has some ...
2
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0answers
32 views

Find all primes p such that $\frac{2^{p-1}-1}{p}$ is a perfect square.

Find all primes $p>2$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square. My try: Since $p>2$, we can write $\frac{2^{p-1}-1}{p}=\frac{(2^{\frac{p-1}{2}}-1)(2^{\frac{p-1}{2}}+1)}{p}=n^2$ ...
1
vote
1answer
25 views

A question about the natural density from number theory

Take a set $ S \subseteq N $, define the sequence $ x_{n}=\#(A\cap [1,n])/n $, and then if $ \lim x_{n} $ exists, call it $ D(A) $ , the (natural) density of A on N. Prop: If for any natural number ...
0
votes
1answer
35 views

Show that x = (66B − 65a) mod 143.

For each natural number $m$ we define $J_m = \{0, 1, . . . , m − 1\}$, the set of all possible residues modulo $m$. Let $x \in J_{143}$. Define $a \equiv x \pmod{11}$, $B \equiv x \pmod{13}$ Show ...
0
votes
2answers
19 views

same digit for two numbers A and B, proof A-B \equiv 0 \mod 9

Consider $$A=\overline{a_0a_1\cdots a_k}$$ $$B=\overline{b_0b_1\cdots b_k}$$ $$\{a_0,a_1,\ldots ,a_k\}=\{b_0,b_1,\ldots ,b_k\}$$ How do I prove that $$A-B\equiv 0\,\,\,\,\, (\!\!\!\!\!\!\mod{9})$$
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5answers
98 views

$4^{2^n}+2^{2^n}+1$ is Divisible by $7$

I have one question. How do I prove that $$4^{2^n}+2^{2^n}+1$$ is Divisible by $7$ ? thanks in advances.
1
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4answers
36 views

If $(a,c)=1$ and $(b,c)=1$, prove $(ab,c)=1$.

I'm posed with the problem in the title, Let $a,b,c\in\mathbb{Z}$. Then if $(a,c)=1$ and $(b,c)=1$, prove $(ab,c)=1$. (By the way, $(a,c)=1$ means that the greatest common divisor of $a$ and $c$ ...
1
vote
2answers
56 views

Primes of the form $an^2+bn+c$?

Wondering if this has been proven or disproven. Given: $a,b,c$ integers $a$, $b$, and $c$ coprime $a+b$ and $c$ not both even $b^2$-$4ac$ not a perfect square are there infinite primes of the ...
2
votes
1answer
44 views

“Half-primitive root”?

I've made a topic related to this (but containing a different question) and it got no responses, so I was wondering if I've stumbled on something new or if it's obvious and I'm just not seeing it at ...
1
vote
1answer
24 views

Number of solution pairs $(i,j)$ such that $i+jk \leq l$

I have show that the number of solutions $\left(\, i,j\,\right)$ of non-negative integers to $i + jk \leq l$ is $$ \left(\,\left\lfloor\, l \over k\,\right\rfloor +1\,\right) {2l + 2 - k\left\lfloor\, ...
0
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0answers
12 views

Number of solutions of a diophantine inequality

In my problem i am looking for the number of the nonnegative integer pairs $(x,y)$ which satisfy the inqueality $x+my\leq n$, where $m$ and $n$ are coprime integers. The answer in the book is given ...
2
votes
0answers
17 views

Find n and k so that there is an positive integer that is dividable by n and the sum of his digits is k

So n is a positive integer which is not dividable by 3 and $k\ge n$ is also a positive integer. Prove that there exists a positive integer so that it is dividable by n and the sum of his digits is k.
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0answers
33 views

Trying to understand a strange regularity found for a ratio of repeated products

I was considering an alternate simplification of $\binom {2 n} {n} $ by pairing the components of one of the denominator factorials with the even terms in the numerator and pairing the other ...
5
votes
1answer
117 views

Solve $3^a-5^b=2$ for integers a and b.

So I have got that (a,b)=(1,0),(3,2) are solutions for the eqations, and maybe the only one.
-1
votes
0answers
25 views

Non-archimedean balls are/aren't abelian groups

I was reading a book and the author says the following, after defining $B_n(a)=\lbrace x\in\mathbb{Q}_p:|x-a|_p\le\tfrac{1}{p^n}\rbrace$ and $S_n(a)=\lbrace ...
1
vote
0answers
17 views

Number of solutions to $x^n \equiv a \pmod{2^b}$.

I've been trying to prove the following statement: Let $m \in \mathbb{N}$ and let $2^k$ be the highest power of 2 that divides $m$. Further, let $a$ be an odd integer such that $x^m \equiv a ...
1
vote
1answer
52 views
+100

power series of $\sec x + \tan x$ at $x=-\pi/2$

We know the power series of $\sec x+\tan x$ is as follows, $f(x)=\sum_{n\geq 0}\frac{E_n}{n!}x^n$, where $E_n$ is Euler Zigzag numbers and clearly the radius of convergence of $f(x)$ is $\pi/2$. ...
0
votes
1answer
51 views

Number theory practice exam questions

Looking for some help for these two practice problems for my exam. I'll explain to you what I have so far and my ideas. So for this problem, I solved part (a) using induction, it wasnt too tricky. ...