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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0
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9 views

Greatest common divisor of polynomials

let $f(n)=3n(n+p)+p^3$ and $g(n)=n^3+q$. $p$, $q$, $n$ can take any non-negative integer value. $p$ and $q$ are fixed. $$\gcd(f(n), g(n))=d$$ I want to find $n$ such that $d$ is maximized for fixed ...
6
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2answers
39 views

Conjecture about the powers of 2 in the prime factorizations of the series $a_n = n(n+1)(n+2)$

I found the following law and would like to know what do you think about it and if anyone can explain why this is so. Also, is this already known and proven? Consider the following series: ...
3
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0answers
18 views

A Tale of Two Quadratic Identities (Pell-like)

Question is at the end. Let all variables be integers. For some constants $a,b,c,d$, assume we have initial solution {$m,n$} to, $$a m^2 + b m n + c n^2 = d\tag{1}$$ Identity 1: $$a x^2 + b x y + ...
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0answers
16 views

Given $p=(m+n)/(u+v)$, express $p$ in terms of $m/u$ and $n/v$

Given $p=(m+n)/(u+v)$, express $p$ in terms of $m/u$ and $n/v$. My attempt; dividing numerator and denominator by $uv$ we obtain $p=((m/u)*1/v+(n/v)*1/u)/(1/u+1/v)$ but am stuck here
2
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0answers
42 views

Nature of the series $\sum\limits_{n}(g_n/p_n)^\alpha$ with $(p_n)$ primes and $(g_n)$ prime gaps

Let $p_n$ denote the $n$th prime number and $g_n=p_{n+1}-p_n$ the $n$th prime number gap. This is to ask for which values of $\alpha$ the series $S_\alpha$ converges or diverges, where ...
0
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0answers
11 views

What are the number of “minimal 2-complete” partitions for the first 100 natural numbers?

Let λ = (λ0, λ1, . . , λn) be a partition of the natural number m into n+1 parts λi such that 1) m = λ0+ λ1+ . . . . + λn , and 2) λ0 ≤ λ1 ≤ . . . . ≤ λn . The partition λ = (λ0, λ1, . . , λn) of m ...
2
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1answer
29 views

How can I find the Pythagorean hypotenuse which gives a maximum Pythagorean triangles?

The following Pythagorean hypotenuses have many possibilities of triangles. $125$ has three triangles $$35, 120, 125$$ $$44, 117, 125$$ $$75, 100, 125$$ the $365$ has $4$ triangles , $85$ has $4$ ...
0
votes
1answer
102 views

Is this infinite series related to prime and composite numbers convergent? [on hold]

I don't know whether this series converges: $$\frac{1}{4} - \frac{1}{5} + \frac{1}{6} - \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{10} - \frac{3}{11} + \frac{1}{12} - \frac{1}{13} + ...
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0answers
14 views

higher moments of a r.v., combinatorical problem

I'm studying the book of Rick Durrett, I want to understand the proof of the Erdös Kac central limit theorem, so I also need to understand the Lindeberg-Feller theorem: for every $n \in \mathbb{N}$ ...
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0answers
22 views

Proving $3x^{3}+4y^{3}+5z^{3}\equiv 0 (3^{m})$ has nontrivial integer solutions $\forall$ $m\in\mathbb{N}$

I have to prove that $3x^{3}+4y^{3}+5z^{3}\equiv 0 (3^{m})$ has nontrivial integer solutions $\forall$ $m\in\mathbb{N}$ so that: $x_{m+1}\equiv x_{m}(3^{m}) $ $\forall$ $m\in\mathbb{N}$ ...
1
vote
1answer
26 views

Restriction of scalars of a torus

Let $k$ be a number field, $l/k$ be a finite extension, and $T_{/l}$ be a linear algebraic torus over $l$. Is $R_{l/k}(T_{/l})$ a linear algebraic torus over $k$? Here $R_{l/k}$ is the restriction ...
0
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1answer
26 views

Number of positive integer solutions

How many positive integer solutions of the equation: $x_1 + x_2 + \cdots + x_p = n$ where $x_1$ and $x_2$ are odd numbers and other $x_i$'s are even numbers ? Is there any theorm about such equation? ...
3
votes
1answer
58 views

Number theory: $x^y + 1 = y^x$

Today a friend told me the equality: $2^3 + 1 = 3^2$, and i wondered if there exist more solutions to the general problem $$x^y + 1 = y^x$$ where $x$ and $y$ are integers. Some research led me to the ...
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0answers
12 views

Approximate irrational numbers with the same denominator

Let $\alpha$ be a irrational number, then using the continued fraction expansion we can find two sequences $\{p_n\}$ and $\{q_n\}$ with $q_n\rightarrow\infty$ as $n\rightarrow\infty$ such that ...
0
votes
3answers
16 views

Nonzero quadratic residues modulo 101

How many Nonzero quadratic residues are there modulo prime 101 I am lost where to start to my knowledge there is no formula for number of quadratic residues a prime has It will be too much to start ...
3
votes
0answers
24 views

Bounds on eigenvalues of Hecke operator on the Jacobian

Let $p$ be a prime not dividing $N$. Consider the Hecke operator $T_p$ on the Jacobian $\text{Jac}(X_0(N))$. I'm thinking of $T_p$ as coming from a correspondence. If I understand correctly, the ...
1
vote
3answers
93 views

Solving $2^x \equiv x \pmod {11}$

Solve $ 2^x \equiv x \pmod {11}$. I know 2 is a primitive root modulo 11. So. I get $x \equiv \operatorname{ind}_2x \pmod {10}$ And I'm stuck! (Maybe I can $x=1$, $x=2$, $x=3$, and so on... ...
2
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1answer
59 views

Sequences for that $\sum_{n} \frac{1}{x_n}$ is divergent and $\sum_{n} \frac{1}{x_n \ln x_n}$ is convergent

We will denote with $(x_n)$ a given sequence and we introduce the following two series. $$S^* = \sum_{n} \frac{1}{x_n} \quad \text{and} \quad S_* = \sum_{n} \frac{1}{x_n \ln x_n}.$$ We know that if ...
0
votes
1answer
17 views

How can i prove $(p − 1)! ≡ (−1)^{\frac{p-1}{2}} \big(\frac{p-1}{2}!\big)^2 \mod p .$?

I'm trying to prove that $x^2 ≡ -1 \mod p$, so for this I need to know how I can prove this: $$(p − 1)! ≡ (−1)^{\frac{p-1}{2}} \big(\frac{p-1}{2}!\big)^2 \mod p .$$ Thanks!
2
votes
2answers
34 views

Product of consecutive integers

Question 5 Prove that the product of four consecutive positive integers cannot be equal to the product of two consecutive positive integers. So it must equal $n(n+1)(n+2)(n+3)$ hence it must ...
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0answers
17 views

Why do we have to work to prove the surjectivity of the local Artin map (Serge Lang A.N.T., Chapter XI Theorem 3)

I must be misunderstanding something about Artin reciprocity. Let $K/k$ be an abelian extension of number fields with Galois group $G$, $I_k$ the ideles of $k$, and $P$ a prime of $k$ (with $v$ a ...
1
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1answer
37 views

Demostrate: $M_p=2^p-1$

Demostrate: If the number $M_p=2^p-1$ is Composite number, where $p$ is prime, then $M_p$ is a Pseudoprime. This exercise was on a test and I could not do!! Number Pseudoprime: Fermat's little ...
4
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4answers
65 views

Proof $\displaystyle \binom{p-1}{k}\equiv (-1)^k \mod{p}$

Proof that if $p$ is a prime odd and $k$ is a integer such that $1≤k≤p-1$ , then the binomial coefficient $$\displaystyle \binom{p-1}{k}\equiv (-1)^k \mod p$$ This exercise was on a test and I could ...
3
votes
1answer
35 views

evaluate two sums in analytic number theory

How should I evaluate the following sums 1, $\sum_{p\leq t}\frac{log^2(p)}{p}$ where the sum is taken over all prime numbers. 2, $\sum_{n\leq X}\frac{\Lambda^2(n)}{n}$ where $\Lambda(\cdot)$ is ...
0
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1answer
39 views

Least Common Multiple & Arithmetic Sequence

Let set $$ S = \{s\space |\space s=\frac{lcm(a,\space a+d,\space a+2d,\space ...,\space a+10d)}{a+10d}\}$$ Where $a,\space d$ are positive integers, and $lcm$ is the least common multiple ...
2
votes
2answers
36 views

Prove that one integer among $m$ consecutive integers is divisible by $m$

Show that of any $m$ consecutive integers, exactly one is divisible by $m$. I am finding it difficult to prove that there is only one number among $m$ consecutive integers that is divisible by $m$.
2
votes
2answers
27 views

Can every positive integer be expressed as a difference between integer powers?

In mathematical notation, I am asking if the following statement holds: $$\forall\,n>0,\,\,\exists\,a,b,x,y>1\,\,\,\,\text{ such that }\,\,\,\,n=a^x-b^y$$ A few examples: $1=9-8=3^2-2^3$ ...
1
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3answers
63 views

Find all solutions in positive integers of the diophantine equation $w^2+x^2+y^2=z^2$

It's an exercises of the text book Elementary Number Theory and It's Applications 6th Edition by Kenneth H.Rosen. I wanted to solve it using the method in solving the diophantine equation ...
0
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1answer
22 views

Calculating point 2P on an elliptic curve

The equation for the curve is $$y^2=x^3+ax+b$$ and the point in question is $P(x,y)$. We have to verify that the $x$ coordinate of $2P$ is $(x^4-2ax^2-8bx+a^2)/4y^2$. However, the value I get is ...
1
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1answer
23 views

How can I solve these congruences?

I have no idea, how to solve these congruences if you can help me please. Thanks a lot.
1
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2answers
45 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
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2answers
30 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
15 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
34 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
20 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
47 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' ...
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3answers
73 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 integers
1
vote
1answer
31 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
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0answers
22 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 $n$? [duplicate]

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 ...
0
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0answers
32 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
36 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
vote
1answer
48 views

Transcendental solution to system of equations

Suppose $A$ is a set of polynomials:$$P_1(x,y_1,\dots,y_n)=0,$$ $$P_2(x,y_1,\dots,y_n)=0,$$ $$\vdots$$ $$P_k(x,y_1,\dots,y_n)=0$$ is a system of equations with coefficients over $\mathbb{Z}$, and ...
0
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0answers
16 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
35 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
34 views

Solving a system of congruences.

Solve Congruences system $$2x \equiv 1 \mod{5} $$ $$3x \equiv 9 \mod{6}$$ $$4x \equiv 1 \mod{7}$$ $$5x \equiv 9 \mod{11}$$ i dint undertand to my teacher, help me with this excercise step by step.. ...
11
votes
2answers
104 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
43 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
30 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
76 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 ...
2
votes
0answers
28 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 ...