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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15
votes
1answer
234 views

An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime generating polynomials of a particular form. Kindly look at the questions given below it. $$\begin{array}{cccc} \text{#} & P(n)=an^2+bn+c\,; & d = ...
2
votes
1answer
49 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
16 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!
4
votes
4answers
55 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 ...
0
votes
2answers
28 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 ...
2
votes
2answers
28 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 ...
0
votes
1answer
32 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 ...
0
votes
0answers
9 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 ...
3
votes
1answer
29 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 ...
26
votes
1answer
2k views

Books about the Riemann Hypothesis

I hope this question is appropriate for this forum. I am compiling a list of all books about the Riemann Hypothesis and Riemann's Zeta Function. Here is my list: The Riemann Hypothesis: A Resource ...
1
vote
1answer
34 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 ...
2
votes
2answers
31 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 $m$.
-6
votes
3answers
71 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
0
votes
0answers
31 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.. ...
1
vote
2answers
46 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 ...
23
votes
2answers
407 views
+50

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 ...
0
votes
1answer
26 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
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: $1=9-8=3^2-2^3$ ...
0
votes
1answer
16 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
vote
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
vote
1answer
47 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 ...
2
votes
3answers
87 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
34 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 ...
-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 ...
1
vote
1answer
49 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 ...
2
votes
1answer
33 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
votes
0answers
17 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 = ...
0
votes
3answers
287 views

Inverse bit in Chinese Remainder Theorem

I need to solve the system of equations: $$x \equiv 13 \mod 11$$ $$3x \equiv 12 \mod 10$$ $$2x \equiv 10 \mod 6.$$ So I have reduced this to $$x \equiv 2 \mod 11$$ $$x \equiv 4 \mod 10$$ $$x ...
2
votes
1answer
44 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' ...
2
votes
0answers
176 views

Every Cauchy sequence converges

SENTENCE: The p-adic numbers are complete with respect to the p-norm, ie every Cauchy sequence converges. PROOF: Let $(x_i)_{i \in \mathbb{N}}$ a Cauchy-sequence in $\mathbb{Q}_p$. We want to show ...
1
vote
1answer
30 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
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
votes
1answer
75 views

Iterating over every permutation of factors of 800 OR possible isomorphism types of abelian groups of orders 74, 147, 666, 800 and 1221

this is self learning This may smell of homework but I am doing this http://homepages.warwick.ac.uk/~masdf/alg1/p4.pdf worksheet I found, you can see this question is in the "practice" section and ...
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 ...
4
votes
2answers
121 views
+100

the first $2k$ terms of the 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
0answers
29 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 ...
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 ...
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
1answer
146 views

What are the necessary and sufficient conditions for a cubic equation to have integers roots

Let's start with Fermat equation with the lowest power, $x^3 + y^3 = z^3$. Now let's set $y = x + a, z = x + b$ with $b > a$ and $a,b$ integers. then the equation becomes $$x^3 + (3a-3b)x^2 + ...
0
votes
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
votes
0answers
33 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 ...
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
vote
0answers
29 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 ...
1
vote
0answers
25 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
36 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
vote
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: ...
1
vote
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 ...
5
votes
3answers
113 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 $n>1$
5
votes
2answers
187 views
+50

$\sum_{p \in \mathcal P} \frac1{p\ln p}$ converges or diverges?

We will denote the set of prime numbers with $\mathcal P$. We know that the sum $\sum_{n=1}^{\infty}\frac1n$ and $\sum_{n=2}^{\infty}\frac1{n\ln n}$ diverges. It is also known that $\sum_{p \in ...
4
votes
2answers
297 views

Positive integer solutions of $x^2+21y^2=z^4 $

Can one find all positive integer solutions of $$x^2+21y^2=z^4 ?$$ I am not sure if this is possible. I just saw this problem and this problem came to my mind.