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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24 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
13
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1answer
160 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
20 views

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

Let $λ = (λ_0, λ_1, . . , λ_n)$ be a partition of the natural number $m$ into $n+1$ parts $λ_i$ such that: 1) $m = λ_0+ λ_1+ . . . . + λ_n$ 2) $λ_0 ≤ λ_1 ≤ . . . . ≤ λ_n$ The partition $λ = ...
2
votes
1answer
33 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
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1answer
183 views

Is this infinite series related to prime and composite numbers convergent?

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} + ...
2
votes
1answer
30 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 $n \in \mathbb{N}$ ...
3
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1answer
70 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
28 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
37 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
83 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 ...
1
vote
1answer
33 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
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3answers
26 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 ...
5
votes
1answer
63 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 ...
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3answers
115 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
66 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
21 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 ...
3
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1answer
101 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
vote
1answer
43 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
71 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
39 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
votes
1answer
43 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
44 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
34 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
102 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 ...
1
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1answer
27 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
26 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.
2
votes
2answers
63 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
2answers
37 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
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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
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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). ...
2
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1answer
61 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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votes
3answers
105 views

Is it possible to find [closed]

If $$\frac {(a-b)(c-d)}{(b-c)(d-a)}=\frac {2012}{2013}$$ 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
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1answer
34 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
votes
1answer
38 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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1answer
57 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 ...
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0answers
18 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
39 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 ...
11
votes
2answers
108 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
46 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
31 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
84 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 ...
4
votes
1answer
41 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
33 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: ...
6
votes
3answers
128 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
19 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
45 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
66 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 ...