Questions on more advanced topics of number theory, such as quadratic residues, primitive roots, prime numbers, non-linear Diophantine equations, etc. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Help proving ${n \choose k} \equiv 0 \pmod n$ for all $k$ such that $0<k<n$ iff $n$ is prime.

I can prove the $n$ is prime case: If $n$ is prime, then since $k < n$ and $n$ is prime, the factor of $n$ in the numerator won't be cancelled out. So the question boils down to Let an integer ...
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1answer
250 views

Representability as a Sum of Three Positive Squares or Non-negative Triangular Numbers

Let $r_{2,3}(n)$ and $r_{t,3}(n)$ denote the number of ways to write $n$ as a sum of three positive squares (A063691) and as a sum of three non-negative triangular numbers (A008443), respectively. I ...
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1answer
33 views

Prove $∑^{(p−1)/2}_{k=1} \left \lfloor{\frac{2ak}{p}}\right \rfloor \equiv ∑^{(p−1)/2}_{k=1} \left \lfloor{\frac{ak}{p}}\right \rfloor ($mod $2)$

If $p$ is an odd prime number and $a$ is an odd integer not divisible by $p$, then why does $∑^{(p−1)/2}_{k=1} \left \lfloor{\frac{2ak}{p}}\right \rfloor \equiv ∑^{(p−1)/2}_{k=1} \left \lfloor{\frac{...
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1answer
129 views

Extremely difficult: Polynomials $f,g,h$ such that…

I've been trying to solve this for hours and got nowhere, so I can only assume it's a really difficult problem. Problem: Find polynomials $f,g,h$ with integer coefficients such that: $(3n-3)(f^2+g^2+...
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119 views

Suggestion For Books From experts on number theory

Regarding My Background I have covered stuff like 1.Single Variable Calculus 2.Multivariable Calculus (Multiple Integration,Vector Calculus etc) (Thomas Finney) 3.Basic Linear Algebra Course (...
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1answer
169 views

Bounds for Waring's Problem

The question is posed as such: If G(k) = min{ g : every "sufficiently large" natural number can be written as the sum of g kth powers } Then I seek to prove two things. First, to establish the ...
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1answer
52 views

Numbers $n$ with $n,n+2$ coprime to $p_k\#$ on $[1, p_k\#]$

In the context of sieving for twin primes ($p\#$ is the primorial function) the following seems true. The number of $n$ such that $n, (n+2)$ are coprime to $p_k\#$ for $n=1,2,...,~p_k\#$ is $\...
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1answer
75 views

How do I prove the following result in number theory? [closed]

There exist no $(n, m) ∈ \mathbb{N}$ so that $n + 3m$ and $n ^2 + 3m^2$ both are perfect cubes.
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0answers
255 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
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2answers
59 views

Number writable as sum of cubes in $9$ “consecutive” ways

Let's say that a given $n\in\mathbb{N}$ is writable as sum of cubes in $k$ consecutive ways if it can be written as sum of $j,j+1,\ldots, j+(k-1)$ nonzero cubes, for some $j\geqslant 1$. For ...
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2answers
152 views

even and odd perfect numbers existence

Thank for my previous post. Also, thank you so much for this site (m.s.e) 1) If odd perfect numbers there, those numbers can be expressible $12k + 1$ or $324k + 81$ or $468k + 117$. If yes, please ...
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0answers
122 views

$p$ be a prime number , then is it true that $p^2| {pa \choose pb}-{a \choose b} , \forall a,b \in \mathbb N$ ?

Let $p$ be a prime number , then is it true that $p^2| {pa \choose pb}-{a \choose b} , \forall a,b \in \mathbb N$ ?
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3answers
600 views

About some Pythagorean Quadruples

I am trying to find all the Pythagorean Quadruples of the form: $$ 1+(10K+4)^2+(10M+8)^2=(10N+9)^2\qquad K,M,N\in\mathbb{N},M<N $$ Thank you!
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2answers
91 views

Find all the solutions of $6a+9b+20c=-2$

Find all integer solutions of $$6a+9b+20c=-2.$$ I did learn how to show there is no solution. First I find the solution set must also satisfy $$6a\equiv-2\mod 20$$ and so on to find a contradiction. ...
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1answer
52 views

How to prove that every natural number not of the form $4^n(8m+7)$ can be written as $x^2 + y^2 + z^2$?

Every natural number not of the form $4^n(8m+7)$ where $m$ and $n$ are natural numbers, can be represented as sum of three squares.
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3answers
2k views

Extract a Pattern of Iterated continued fractions from convergents

I have been working on an article at https://oeis.org/wiki/Table_of_convergents_constants where I posted a table of "convergents constants" (defined at https://oeis.org/wiki/Convergents_constant) for ...
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1answer
117 views

Solutions to simultaneous Diophantine equations $2y^2-3x^2=-1$ and $z^2-2y^2= -1$

I am looking for integer solutions for the following set of equations: $2y^2-3x^2=-1$ $z^2-2y^2= -1$ I know that there are the solutions (1,1,1) and (-1,-1,-1) for this set of simultaneous ...
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294 views

Prove that this recurrence relation algorithm generates all positive rational numbers, and does so without repetition and in reduced form [closed]

For $n\ge 1$, generate a sequence $\{a_n\}$ such that for any even $n = 2k$: $$ a_n = a_k$$ And for any odd $n=2k+1$: $$ a_n = a_k + a_{k+1}$$ With initial conditions $a_1 = a_2 = 1$ Now, generate a ...
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1answer
193 views

Natural density of solvable quintics

A recent question asked about the topological density of solvable monic quintics with rational coefficients in the space of all monic quintics with rational coefficients. Robert Israel gave a nice ...
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1answer
314 views

Primes as quotients

I ask this question based on a comment of David Speyer in another question. What primes are of the form $$ \frac{p^2-1}{q^2-1} $$ where $p$ and $q$ are prime? The first prime not apparently of this ...
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1answer
354 views

$a^m+k=b^n$ Finite or infinite solutions?

Given positive integers k,a,b, is there a finite or infinite number of solutions in positive integers $m,n>1$, to $a^m+k=b^n$? Pillai's conjecture states that each positive integer occurs only ...
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1answer
233 views

Are $121$ and $400$ the only perfect squares of the form $\sum\limits_{k=0}^{n}p^k$?

I've been looking for perfect squares that can be represented as $\sum\limits_{k=0}^{n}p^k$. Of course, both $n$ and $p$ should be natural numbers larger than $1$. Searching up to $n=100$ and $p=...
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1answer
1k views

a conjectured continued-fraction for $\displaystyle\cot\left(\frac{z\pi}{4z+2n}\right)$ that leads to a new limit for $\pi$

Given a complex number $\begin{aligned}\frac{z}{n}=x+iy\end{aligned}$ and a gamma function $\Gamma(z)$ with $x\gt0$, it is conjectured that the following continued fraction for $\displaystyle\cot\left(...
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1answer
501 views

Fibonacci, compositions, history

There are three basic families of restricted compositions (ordered partitions) that are enumerated by the Fibonacci numbers (with offsets): a) compositions with parts from the set {1,2} (e.g., 2+2 = ...
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0answers
265 views

Dividing the whole into a minimal amount of parts to equally distribute it between different groups.

Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed ...
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5answers
405 views

Alternating sum of multiple zetas equals always 1?

This is more in the category of "recreational math"... I was playing with multiple zetas, in the notation of $\zeta(k),\zeta(k,k),\zeta(k,k,k),\ldots$ as given in wikipedia. Looking at the alternating ...
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2answers
519 views

Distribution of primes?

Do primes become more or less frequent as you go further out on the number line? That is, are there more or fewer primes between $1$ and $1,000,000$ than between $1,000,000$ and $2,000,000$? A proof ...
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4answers
636 views

Clarification of a remark of J. Steel on the independence of Goldbach from ZFC

On page 424 of the following paper: S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, ``Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. ...
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1answer
754 views

Are all subrings of the rationals Euclidean domains?

This is a purely recreational question -- I came up with it when setting an undergraduate example sheet. Let's go with Wikipedia's definition of a Euclidean domain. So an ID $R$ is a Euclidean domain ...
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1answer
630 views

0.246810121416…: Is it a algebraic number?

Is it algebraic the number 0.2468101214 ...? (After point, the natural numbers are juxtaposed pairs).
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1answer
449 views

Primes in a Power series ring

Let $\mathbb Z$ be the ring of rational integers. Consider the power series ring $\mathbb Z[[x]]$. It is known that $\mathbb Z[[x]]$ is unique factorization domain. What are the primes in $\mathbb Z[[...
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1answer
171 views

Are these two sequences the same?

I was browsing OEIS and came across the largely composite numbers, A067128, defined as the natural numbers that have at least as many divisors as all smaller natural numbers. (They are of course ...
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0answers
541 views

Ramanujan and sum of four cubes

This is more a question on History than proof itself. About a decade ago, a college professor and a Math coach told us about this beautiful theorem: Every multiple of 6 can be written as a sum of ...
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1answer
70 views

Induced group homomorphism $\text{SL}_n(\mathbb{Z}) \twoheadrightarrow \text{SL}_n(\mathbb{Z}/m\mathbb{Z})$ surjective?

Let $n, m > 1$. The map $\mathbb{Z} \twoheadrightarrow \mathbb{Z}/m\mathbb{Z}$, of reduction mod $m$, induces a group homomorphism $F: \text{SL}_n(\mathbb{Z}) \to \text{SL}_n(\mathbb{Z}/m\mathbb{Z})...
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242 views

Is there any number greater than 8 of the form $2^{2k+1}$ which is the sum of a prime and a safe prime?

Is there any number greater than 8 of the form $2^{2k+1}$ which is the sum of a prime and a safe prime? While answering @pedja's question about the existence of any such representations I was ...
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2answers
201 views

Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$?

Is $\ p_n^{\pi(n)} < 4^n$ where $p_n$ is the largest prime $\leq n$? Where $\pi(n)$ is the prime counting function. Using PMT it seems asymptotically $\ p_n^{\pi(n)} \leq x^n$ where $e \leq x$
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1answer
466 views

Mysterious number 6174

Kaprekar discovered the Kaprekar constant or 6174 in 1949. He showed that 6174 is reached in the limit as one repeatedly subtracts the highest and lowest numbers that can be constructed from a set of ...
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5answers
353 views

Determine the number of solutions of the equation $n^m = m^n$ [duplicate]

Possible Duplicate: $x^y = y^x$ for integers $x$ and $y$ Determine the number of solutions of the equation $n^m = m^n$ where both m and n are integers.
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1answer
385 views

$F[t]$ has undecidable positive existential theory in the language $\{+, \cdot , 0, 1, t\}$

Consider the ring $F[t, t^{-1}]$ (the polynomials in $t$ and $t^{-1}$ with coefficients in the field $F$). Theorem 1. Assume that the characteristic of $F$ is zero. Then the existential theory ...
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1answer
309 views

Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose m}\left(p-...
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1answer
254 views

Looking for help understanding the Möbius Inversion Formula

I was recently told that the Möbius Inversion Formula can be applied to the Chebyshev Function. Let $\vartheta(x)$,$\psi(x)$ be the first and second Chebyshev functions so that: $$\vartheta(x) = \...
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1answer
390 views

if $m^2 = a^3 - b^3$, then $m$ is the sum of two squares.

(Please read "Edit"s and see this.) How could I prove that : $$\text{If} \space m^2=a^3-b^3\text{ where}\space m,a,b\in\mathbb{N} \rightarrow \exists c,d \in\mathbb{N}\space \text{ such that}\space m=...
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3answers
744 views

Dirichlet's theorem on primes in arithmetic progression

Is there a proof in the spirit of Euclid to prove Dirichlet's theorem on primes in arithmetic progression? (By the spirit of Euclid, I mean assuming finite number of primes we try to construct another ...
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1answer
254 views

Asymptotics of sums of Dirichlet-Characters over prime numbers

Again in relation with some stuff I am currently reading, the authors make use of the following "standard argument in prime number theory": Let $\chi$ be a non-principal Dirichlet-character. Then $$\...
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2answers
251 views

If $x\notin\mathbb Q$, then $\left|x-\frac{p}{q}\right|<\frac{1}{q^2}$ for infinitely many $\frac{p}{q}$?

This appears on problem 1 of chapter 1 in Stein & Shakarchi's Real Analysis: Given an irrational $x$, one can show (using the pigeon-hole principle, for example) that there are infinitely many ...
7
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1answer
140 views

A valuation-like function $w: \mathbb{N}^{+} \rightarrow \mathbb{N}$ is a $p$-adic valuation?

This question is a variant of problem 4, pg. 21, from Birkhoff and Maclane, A Survey of Modern Algebra. Given a function $w: \mathbb{N}^+ \rightarrow \mathbb{N}$ that behaves like a valuation ...
7
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1answer
149 views

Decomposition as a product of factors

For several days, I try to do this excercise. Without success. Prove that for any pair of positive integers $k$ and $n$, there exist $k$ positive integers $m_1, m_2, \dots, m_k$ (not necessarily ...
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4answers
1k views

Connections between number theory and abstract algebra.

I haven't taken abstract algebra yet, but I am curious about connections between number theory and abstract algebra. Do the proofs of things like Fermat's little theorem, the law of quadratic ...
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3answers
2k views

Consecutive non square free numbers

I was thinking to solve this by computer programs but I prefer a solution. How to obtain a list of 3 consecutive non square free positive integers? In general, how to obtain the same kind of list ...
7
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2answers
151 views

Is there a $k$ for which $k\cdot n\ln n$ takes only prime values?

There exist some real $k$ such that $\forall $ integer $ n > 1$ the integer part of $ k *n\ln(n)$ is always prime?