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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2
votes
1answer
10 views

10's Complement of a negative decimal number

What will be 10's complement of ($-417_{10}$) . Should it be $10^{3}-(-417)=1417$ ? No further information in given and I could not find any reference anywhere . EDIT : Number of digits are to ...
2
votes
3answers
44 views

question about prime number

How to prove that there exists a prime number between $n$ and $ n!$,for all $ n> 2$.
1
vote
0answers
12 views

Tree of Simple Pythagorean Triples graph scale infinte series

Consider the tree of primitive Pythagorean triples as seen here: https://en.wikipedia.org/wiki/Tree_of_primitive_Pythagorean_triples Consider the values for c in the triples (a, b, c) and their ...
1
vote
0answers
19 views

On integer $n>1$ and prime $p$ such that $p<n$ , $p$ does not divide $n$ and $n-p$ is a prime

Let $n>1$ be a given integer and $p$ be a prime less than $n$ and not dividing $n$ ; so $p$ and $n$ are co-prime ; hence $n-p$ and $n$ are also co-prime ; I would like to ask when is $n-p$ also is ...
0
votes
1answer
27 views

A question on divisors of $A=(q^n-1)(q^n-q)(q^n-q^2)\cdots(q^n-q^{n-1})$

Let $A=(q^n-1)(q^n-q)(q^n-q^2)\cdots(q^n-q^{n-1}),$ where $q=p^r,\ p$ is prime and for some $r\in \mathbb{N}\cup\{0\}.$ Does $q^s,$ for some positive integer $s,$ divide $A?$ Or what are the divisors ...
-2
votes
1answer
28 views

How to prove this GCD theorem [on hold]

I'm trying to prove the following: Write $a/b$ as $kx/ky$, where a$, b, x,$ and $y$ are positive integers and $k$ represents the greatest common factor of $a$ and $b$. Then $\frac{x}{y} = ...
7
votes
2answers
60 views

Given dividend and divisor, can we know the length of nonrepeating part and repeating part?

$13/92=0.14\overline{1304347826086956521739}$ In this example, the length of nonrepeating part is $3$. The length of repeating part (repeating period) is $21$. I collected some properties related to ...
1
vote
1answer
28 views

A nonprincipal ideal and a nonprime irreducible in $\mathbb{Z}[\sqrt{-17}]$

The problem asks to find a nonprincipal ideal and a nonprime irreducible in $R = \mathbb{Z}[\sqrt{-17}]$. Since $-17 \equiv 3 \pmod 4$, $R$ is the ring of integers of $\mathbb{Q}(\sqrt{-17})$. I ...
1
vote
0answers
78 views

Every natural number is representable as $\sum\limits_{k=1}^{n} \pm k^5$ … if somebody proves it for 240 integers

(This post is inspired by "Is every natural number representable as $\sum\limits_{k=1}^{n} \pm k^3$"? My question is at the bottom.) The problem of whether every natural number $N$ is, ...
1
vote
1answer
21 views

Prove that if $k_1$ and $k_2$ are relatively prime to n, then so is $k_1 *_n k_2$

I think this should be done by contradiction. Let's assume that $k_1 *_n k_2$ is not relatively prime to n. This means that there exist the gcd($k_1 *_n k_2$, n) that is greater than 1. And I am ...
2
votes
1answer
66 views

two questions about primes

I'm very ignorant about results in number theory concerning the primes. Please let me know if these are open conjectures or easy problems: There are infinitely many primes of the form $n!+1$ There ...
-1
votes
1answer
78 views

Can anyone explain the answer? [on hold]

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 ...
3
votes
1answer
31 views

Two questions about divisible

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

Primes in the Fibonacci sequence

Are there infinitely many prime numbers that are Fibonacci numbers as well? Any help will be greatly appreciated.
0
votes
0answers
31 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
vote
0answers
55 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
votes
0answers
20 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
vote
1answer
43 views

Tricky Diophantine Equation

(Komal November B. 4663) Find the integer solutions of the equation $2x^3-y^3=5$. Hint: use modulo / remainders.
5
votes
1answer
115 views

Summation identity involving the floor function

(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
votes
0answers
27 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
votes
2answers
41 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 ...
1
vote
1answer
20 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 ...
-3
votes
2answers
51 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
vote
2answers
34 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
vote
2answers
65 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
votes
1answer
29 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. ...
-2
votes
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?
0
votes
0answers
28 views

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

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
votes
1answer
95 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?
1
vote
1answer
28 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
votes
0answers
24 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 ...
2
votes
1answer
722 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
votes
1answer
101 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 ...
3
votes
5answers
220 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 ...
0
votes
0answers
63 views

Error with Zeta functional equation [duplicate]

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 ...
0
votes
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
votes
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
votes
1answer
37 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
votes
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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votes
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
votes
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
votes
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 ...
0
votes
0answers
142 views
+300

If this inequality 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
votes
0answers
55 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
65 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
votes
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
votes
1answer
108 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| ...
1
vote
0answers
27 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 ...