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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1answer
75 views

Can anyone explain the answer?

Question: The numbers of subset $\{x,y\}$ of the first $50$ natural numbers such that $x^2-y^2$ is divisible by $7$, is _____. Answer: Consider 7 subsets leaving remainders $0,1,2,3,4,5,6$ when ...
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0answers
21 views

Two questions about divisible

I have two questions. 1) Why the relation $a^n\equiv a^{n+4k}(mod ~10)$ is true? 2) Let $(a,90)=1$. Which number can be $ x$ in $ x|a^4-1$? Answer is $240$. Why?
2
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0answers
45 views

Primes in the Fibonacci sequence

Are there infinitely many prime numbers that are Fibonacci numbers as well? Any help will be greatly appreciated.
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0answers
25 views

Are there way of proving that polynomials are relatively prime using number theory or abstract algebra?

This question is inspired by question A5 from the Putnam Mathematical Competition: Let $$P_n(x) = 1 + 2x + 3x^2 + \cdots + nx^{n-1}.$$ Prove that polynomials $P_j(x)$ and $P_k(x)$ are ...
1
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0answers
52 views

How to calculate the $(3)$ and $(4)$?

In Gérald Tenenbaum's book "Introduction to Analytic and Probabilistic Number Theory" Cambridge University Press 1995, On the page of 97-98, I Can calculate the $(1)$ and $(2)$, but I do not know how ...
2
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0answers
18 views

Finding partitions $\{A,B\}$ of the set $\mathbb N_n:=\{1,…,n\}$ such that $\Big|\prod_{i \in A}p_i-\prod_{i\in B}p_i\Big|=1$

For a fixed $n$ , for what partitions $\{A,B\}$ of the set $\mathbb N_n:=\{1,...,n\}$ do we have $\Big|\prod_{i \in A}p_i-\prod_{i\in B}p_i\Big|=1$ ? , where $p_m$ denotes the $m$th prime for ...
1
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1answer
41 views

Tricky Diophantine Equation

(Komal November B. 4663) Find the integer solutions of the equation $2x^3-y^3=5$. Hint: use modulo / remainders.
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0answers
76 views

Summaton of Factors

(Kömal November B. 4666) Prove that $\sum_{k=1}^n (2k-1) [\frac{n}{k}]=\sum_{k=1}^n [\frac{n}{k}]^2$ for every positive integer $n$, where $[n]$ is the largest integer greater than or equal to $n$.
0
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0answers
24 views

Sum of Numerator and Denominator [on hold]

(AIME 2014 Problem 3) Find the number of rational numbers $r$, $0<r<1$, such that when $r$ is written as a fraction in lowest terms, the numerator and the denominator have a sum of $1000$.
2
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2answers
36 views

A monotonic multiplicative integer functional equation.

Let $ f:\mathbb N \to \mathbb N $ be such that $ f (x)> f (y)$ if $x> y$. $ f (xy)=f (x) f (y) $. $ f (3) \geq 7$. Find the smallest value of $ f (3) $ My attempt:if we can define the ...
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1answer
19 views

Hamming (5-smooth) numbers

Until quite recently, I was not aware of the idea of "smooth" numbers. This is perhaps better expressed as "N-smooth" numbers (i.e., integers where the largest prime factor is <= N). 5-smooth ...
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2answers
48 views

Squares in number theory [on hold]

Show that the square of a positive integer plus one can not be the square of some positive integer. It seems obvious and I suspect it may require a proof by induction.
1
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2answers
33 views

Algebraic solution to the Broken Weight Problem

Here is a problem I was sent, which it turns out was first posed by Claude Gaspard Bachet de Méziriac in a book of arithmetic problems. The problem is as follows: A few years ago, a King's ...
1
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2answers
57 views

Exponential congruence

Hi All am a bit stuck on some revision that I am trying to do. Firstly (part a) I must calculate the inverse of 11 modulo 41, which I have done and believe it to be 15. The next part is to: Now use ...
2
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1answer
28 views

Is a unit conversion factor ever legitimately zero?

I was writing a unit converter for an industrial setting. To ensure that $\frac 0 0$ and $\infty$ never show up in the user interface I made a rule that no unit conversion factor can ever be zero. ...
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2answers
78 views

How prove $\sqrt{2}+\frac{1}{\sqrt{5}+\sqrt[3]{5}}$ is irrational? [on hold]

How prove that $\sqrt{2}+\frac{1}{\sqrt{5}+\sqrt[3]{5}}$ is irrational?
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0answers
27 views

What will be the form of the $k^{th}$ component of $x^{(i)}$?

Suppose that we index the components of the elements of $\mathbb{Z}_p$ by subscripts. Indexing the terms of the sequence by superscripts in parentheses $x^{(i)}$ is a term of the sequence, and ...
7
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1answer
94 views

Is $(n+\ell)^{-1}\binom{kn}{n}$ an integer for only $(\ell,k)=(1,2)$?

Find all pairs $(\ell,k)$ of natural numbers, such that the number $\dfrac1{n+\ell}\dbinom{kn}{n}$ is an integer for all natural $n$. Is $(\ell,k)=(1,2)$ the only solution?
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1answer
27 views

Lowest consecutive number that is the result of an addition of 2 different integers

I need to know what the lowest consecutive number would be that is possible by simply adding 2 numbers any times necessary. I came up with a simple formula for numbers with greatest common divisor 1: ...
4
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0answers
22 views

Number of valid NxN Takuzu Boards a.k.a 0h h1 (details inside)?

Takuzu a logic puzzle which has a NxN grid filled with zero's and one's following these rules: 1) Every row/column has equal number of 0's and 1's 2) No two rows/columns are same 3) No three ...
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1answer
697 views

Trick to find if number is composite or prime

I was doing some maths and required a function which mimics the following function: $$ f(k,c) = \mid \sin(k/2) \sin(k/3) ... \sin(k/c) \mid $$ So that I can evaluate (say $ f(k,3.5) $) or is there ...
4
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1answer
98 views

Additive function $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ is zero everywhere.

Let $f: \mathbb{Z}^\infty \rightarrow \mathbb{Z}$ be an additive function ($f(x+y)=f(x)+f(y)$ for every $x,y \in \mathbb{Z}^\infty$). In addition for every $x=(0,\dots, 0,1,0, \dots)$ we have ...
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5answers
215 views

How are Zeta function values calculated from within the Critical Strip?

We note that for $Re(s) > 1$ $$ \zeta(s) = \sum_{i=1}^{\infty}\frac{1}{i^s} $$ Furthermore $$\zeta(s) = 2^s \pi^{s-1} \sin \left(\frac{\pi s}{2} \right) \Gamma(1-s) \zeta(1-s)$$ Allows us to ...
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0answers
62 views

Error with Zeta functional equation

I was trying to prove $$1 + 2 + 3 +\cdots = -\frac{1}{12}$$ Using the following $$\zeta(s) = \sum _{i=1}^{\infty} \left [\frac{1}{i^s} \right]$$ Thus: $$\zeta(-1) = \sum _{i=1}^{\infty}\left [i ...
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0answers
34 views

Induction: Fibonacci / Lucas Numbers [duplicate]

From Andrews' Number Theory, Chapter 1, Section 1, Problem 15: Prove, by induction, that $F_{2n} = F_nL_n$ where $F_n$ denotes the nth Fibonacci number and $L_n$ denotes the nth Lucas ...
4
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0answers
83 views

An infinite series that gives $f(s)=s$. How could it be explained more easily?

This question loosely builds this one. Equate the following two infinite series for $\zeta(s)$: $$\displaystyle \zeta(s) = \frac{1}{4\,(s-1)} \left(1+s+\sum _{n=1}^{\infty } \left( {\frac ...
2
votes
1answer
29 views

Sum of numbers between consecutive multiple numbers of $N$ proof

I need to see if I can generalize a proof: whether the sum of all numbers between two consecutive numbers multiples of $N$, being $N$ a natural number such that $N > 2$ is a multiple of $N$. I ...
1
vote
3answers
42 views

Mean of a vector

Be a set of numbers $v=(a_1, a_2, \ldots, a_n)$ I want to form the following average vector $\mu = (\frac{\sum a_i}{n}, \frac{\sum a_i}{n}, \ldots, \frac{\sum a_i}{n})$. If I do it iteratively step ...
4
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1answer
34 views

What is the connection with quadratic map

While reading Prof. Tao's Wordpress blog. I noticed he mentioned a different function $\displaystyle\Lambda_2(n):= \sum_{d|n}\mu(d)\log^2(n/d)\ldots(\ast)$ and said that this function vanishes ...
2
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2answers
55 views

Chebyshev's first function prime count

How is Chebyshev's first function $$\vartheta(N)=\sum_{p\leq N}\log p$$ useful in counting primes? Can it alone be used to analytically derive the prime number theorem?
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0answers
34 views

Proof writing exercises [on hold]

I am looking for exercises on writing proofs, especially number theory, since I am quite new to writing proofs. My only experience is a little Euclidean geometry and set theory (very, very limited). ...
5
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1answer
34 views

Can PA prove very fast growing functions to be total?

The Goodstein-sequence is a total function, but PA cannot prove this. Is this true for any other function with growth rate at least $f_{\epsilon_0}$ or are there functions growing at least as fast ...
5
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2answers
138 views

Powers containing every digit equally often

There are several nontrivial powers containing every digit equally often, for example $32043^2$ $2158479^3$ $69636^4$ $643905^5$ $3470187^6$ A necessary condition for a power with the desired ...
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115 views
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If this inequality involving prime numbers holds for $n$ larger than some $N$, what is an upper bound to $N$?

Let $p_n$ be the $n$-th prime and set $k_1=10;k_2=6;k_3,k_4=4;k_5,k_6=3;k_7,\ldots ,k_{n+1}=2$. For large enough $n$, prove or disprove that ...
0
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0answers
52 views

Sum and product of certain prime sequences

Let $p_k$ is the $k$th prime. What are good estimates for the following? $$S_k=1+\frac{1}{\log_23}+\frac{1}{\log_25}+\cdots+\frac{1}{\log_2p_{k-1}}+\frac{1}{\log_2p_k}$$ ...
5
votes
1answer
64 views

How does the size of the set $A(R) = \{(a,b) \; | \; a,b \in N \times N, \; \gcd(a,b) = 1, \; a^2 + b^2 \leq R^2\}$ grow?

How does the size of the set $$A(R) = \{(a,b) \; | \; a,b \in \mathbb{N} \times \mathbb{N}, \; \gcd(a,b) = 1, \; a^2 + b^2 \leq R^2\}$$ grow as a function of $R$? My try: It's clear that $|A(R)| ...
5
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3answers
46 views

Sum of $2$ equal squares also a square

Is there an integet solution to $a^2 + a^2 = b^2$? Because there's this universift that has this logo of the pytagorean theorem where the two squares are equal, but I don't think it's possible.
6
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1answer
107 views

On solutions of an equation in $\mathbb{Z}_3$

For integer numbers $x_1, x_2, y_1, y_2, y_3$ suppose that $$ x_1 + x_2 \equiv y_1 + y_2 + y_3 \pmod 3. $$ For $k=0, 1, 2$ define $$ s_k = \Big| \{ y_i \,|\, y_i \equiv k \pmod 3 \} \Big| - \Big| ...
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0answers
26 views

Extension of valuation

This is a question from Milne's 'Algebraic Number Theory'. Let $K$ be a valued field with absolute value $|\cdot|$ and $L=K(\alpha)$ a finite separable extension of $K$. Let $\hat{K}$ be the ...
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2answers
74 views

For a Diophantine equation $x^2+py^2=z^2$ show that $z$ is necessarily odd.

For a Diophantine equation $x^2+py^2=z^2$ where $p$ is a prime of the form $p\equiv 1(mod4)$ and $(x,y,z)=1$. Show that $z$ is necessarily odd.
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1answer
23 views

Lagrange's theorem in number theory

I'm trying to understand proof of Lagrange's theorem in wiki. In proof it says: we can compute $g(k)$ either directly, i.e. by plugging in (the residue class of) $k$ and performing arithmetic ...
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0answers
37 views

Find height h in a cubical box [on hold]

Given a cubical closed box of edge ‘$a$’ cm with liquid filled inside it up to height H. Now some liquid is being removed from the box such that only liquid up to h height is left in the box.Suppose ...
6
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0answers
55 views

Explaining the difference between the number theoretic Langlands program and geometric Langlands program to a graduate student.

I am a graduate student who just took a course introducing some notions in algebraic number theory and algebraic geometry (officially, it was a course on an introduction to the Langlands program). ...
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0answers
118 views

Are these known telescoping series for $\zeta\left(\frac12\right)$?

There are many known telescoping series for $\zeta(s)$ and I was playing with the following two: $$\displaystyle \zeta(s) = \frac{1}{(s-1)} \left(\sum _{n=1}^{\infty } \left( {\frac {n}{(n+1)^{s}}} - ...
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3answers
323 views

Other representations of factorial

I have a little question: which other representation of factorial $n!$ without using the factorial? Is there any definition of factorial as a series? or any other way? Thanks in advanced.
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6answers
1k views

Digit sequence that is not prime in any base

Is there a sequence of base-$b$ digits of length greater than one with all digits $\ne 0$ that does not represent a prime number in any base? Example: $12_{10}=12$ is not prime, but $12_3=5$ is.
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0answers
27 views

Oddness about Odd numbers [duplicate]

I know it's a little opinion based question but I'd like to get a couple of perspectives on their perception of why ODD numbers are termed ODD. Just in general why are ODD numbers a group of numbers ...
1
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1answer
28 views

Does ${\frac{k}{2\left(1-H\right)}} + \frac{1}{H}\in Z$ when $H$ is irrational and $k \in Z^{+}$?

While working on something, I have stumbled across the following expression $$\frac{\Gamma \left({\frac{k}{2\left(1-H\right)}} + \frac{1}{2H}\right)}{\Gamma \left(\frac{1}{2H}\right)}$$ where $0 < ...
0
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1answer
56 views

A function f(n), that for any n gives the sum of 1^2 up to n^2?

I was solving Project Euler's problem # 6, where one has to find the sum of square of numbers from 1 to 100. I solved it using code and downloaded the Overview provided, to learn from it but I can't ...
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2answers
61 views

Size of prime products

How many primes are there that are $n$ bits long? How many primes are there that are atmost $n$ bits long? What is the number of bits in product of all primes that are $n$ bits long (with carry ...