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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1
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3answers
25 views

How many Pythagorean triangles which have hypotenuse equal to $2859545$

By using the trail and error, I could find these triangle $$20572, 2859471, 2859545$$$$27056, 2859417, 2859545$$ I couldn't continue to find the others triangles because they need more time. Is ...
0
votes
0answers
9 views

Converting equation into Weierstrass form

I have to convert the equation $y^2 +xy +y=x^3 $ by a change of linear variables to the form $Y^2=X^3+aX+b$ where $a$ and $b$ are rational numbers. So far, by completing the square method I've reduced ...
-1
votes
1answer
13 views

Verification of $F(m)^{d} \pmod n \equiv m$ with very large inputs, where $F(m)=m^e$

Does anyone have the computational power to check whether or not $F(m)^{d} \pmod n \equiv m$, where the values of the variables are found below. According to Wolfram Alpha, I found the result of the ...
2
votes
1answer
30 views

How to find a solution to the elliptic curve

We know that one solution of the given elliptic curve is (2, 1) and we have to find another rational solution such that $x$ is not equal to 2 by drawing a tangent to the curve at (2, 1). ...
0
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0answers
13 views

Primes in two arithmetical progression

For each $x\geq 1$, let $\mathcal{P}$ be the collection of all prime numbers and $$z(x) = \left|\left\{n\in\mathbb{N}:(n\leq x)\wedge\exists k,l\in\mathbb{N}\;\exists p,q\in\mathcal{P}\big((1+3n = ...
2
votes
1answer
37 views

What are the connections between the three Mertens' theorem?

In number theory the three Mertens' theorems are the following. Mertens' $1$st theorem. For all $n\geq2$ $$\left\lvert\sum_{p\leqslant n} \frac{\ln p}{p} - \ln n\right\rvert \leq 2.$$ Mertens' ...
-6
votes
3answers
64 views

Is it possible to find [on hold]

If $$\frac {(a-b)(c-a)}{(b-c)(d-c)}=\frac {2012}{2013}$$ then find the value of $\dfrac {(a-c)(b-d)}{(a-b)(c-d)}$ in terms of numbers Note: $a,b,c,d$ are real numbers
1
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1answer
29 views

Question about problem 53 in Problem Solving and Selected Topics in Number Theory

I solved problem 53 in Problem-solving and selected topics in Number Theory. The problem was: Find the sum of all positive integers that are less than 10,000 and whose square divided by 17 leaves ...
2
votes
0answers
18 views

To calculate what is $\sum_{1 \le m<n ;(m,n)=1} m^2$ or what is the remainder when $\sum_{1 \le m<n ;(m,n)=1} m^2$ is divided by $n$?

For an integer $n >1$ , what is the sum of the squares of all the positive integers that are less than $n$ and relatively prime to $n$ that is I am trying to calculate $f(n):=\sum_{1 \le m<n ...
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0answers
27 views

Probability - Runners in a race [on hold]

Consider a race with N runners, where N is unknown. Each runner is assigned at random a unique number between 1 and N. Suppose a group of n runners is observed crossing the finish line. Let z denote ...
0
votes
1answer
24 views

Least number of weights required to weigh integer weights

In a number theory book, I found the following problems, "What is the least number of weights required to weigh any integral number of pounds up to 63 pounds if one is allowed to put weights in only ...
1
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0answers
24 views

Transcendental solution to system of equations

Suppose $(A)$ $$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and there are functions ...
0
votes
0answers
15 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.
0
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0answers
29 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 ...
0
votes
0answers
26 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
101 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
votes
2answers
42 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 ...
1
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0answers
26 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 ...
5
votes
1answer
70 views

Being $N$ 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 if $x$ is large enough,$$\prod_{p<x}_{p \ \text{prime}}\frac{1}{p^{N+1}-1}>\frac{0.2}{\log^2 x}.$$ Speaking of which, Theorem 6.12, and maybe others, of this paper might be ...
1
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0answers
22 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
votes
1answer
28 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, ...
1
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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
votes
4answers
110 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 $n>1$
0
votes
1answer
17 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
34 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
votes
1answer
61 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
votes
4answers
88 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 $n$
3
votes
0answers
66 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
50 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
48 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
votes
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?
0
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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
votes
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.
0
votes
2answers
28 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
votes
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
votes
0answers
29 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
votes
1answer
63 views

Sign of Ramanujan $\tau$ function

The Ramanujan $\tau(n)$ seemed to have random positive/negative signs: ...
1
vote
1answer
20 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
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3answers
58 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 ...
1
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0answers
33 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
votes
4answers
57 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
votes
3answers
76 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 $\color{blue}{\text{both}}$ $k = 2,4$ ...
1
vote
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
335 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
30 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
votes
0answers
33 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
26 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
22 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 $$A-B\equiv 0\,\,\,\,\, (\!\!\!\!\!\!\mod{9})$$
10
votes
5answers
100 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.