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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6
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0answers
53 views

Primes $p$ for which $2p \pm 1$ are also primes

Out of curiosity and trouble sleeping, I decided to look at the distribution of primes $p$ for which $2p \pm 1$ are also primes. I looked at the first 25,910,000 primes and counted the number of ...
2
votes
0answers
25 views

Closure of Set of Fractions with Lowest Terms Condition

Suppose I have a set of rational numbers where elements have denominators are odd and numerators and denominators are co-prime. I need to show that the set is closed under addition. It is clear that ...
0
votes
0answers
24 views

Test on representing prime using eight variate quadratic form

Assume $f_i(x_1,\dots,x_8)$ are linear forms in $\Bbb Z[x_1,\dots,x_8]$ where $i\in\{1,\dots,m\}$. Assume $c_i\in\Bbb Q$ are constants where $i\in\{1,\dots,m\}$. Consider quadratic form ...
1
vote
0answers
20 views

Rate of large composite numbers, which are strong probable prime to the bases $2,3$ and $5$

Here http://primes.utm.edu/glossary/xpage/StrongPRP.html is the definition and some useful informations about strong probable primes. For higher numbers, lets say near $10^{50}$, strong probable ...
-2
votes
2answers
30 views

Rank of an $m$ by $n$ matrix?

Can anyone state, in plain English, how to find the rank of an $m$ by $n$ matrix? Is it necessary to perform Gaussian elimination first, or translate it into upper triangle form (or however it is ...
0
votes
0answers
73 views

Is there more than one looping sequence in the Collatz conjecture? [on hold]

Is it known whether there is more than one loop in the Collatz conjecture? Following advice and warnings on meta, I try below to claim that there is only one looping sequence in all the sequences ...
1
vote
3answers
49 views

Is there an error in this GRE question?

I was doing a Manhattan GRE practice exam and I was sure I had cracked the twist in this one question... only to find in that there was no twist (apparently). Here is what I know: $$ \sqrt a = \pm b ...
1
vote
1answer
17 views

Proof of a complete residue system without using congruences

I am taking a elementary class on number theory this semester, and among the exercises from the third lecture there is this one: For $m > 1$ and $gcd(m, a) = 1$, show that the remainders from the ...
0
votes
1answer
16 views

Count of solutions to matrix equations

Given these modular equations: $$a_{1,1} x_1 + a_{1,2} x_2 + \cdots + a_{1,n} x_n = b_1 \bmod p $$ $$a_{2,1} x_1 + a_{2,2} x_2 + \cdots + a_{2,n} x_n = b_2 \bmod p $$ $$\vdots$$ $$a_{m,1} x_1 + ...
15
votes
2answers
807 views

Who is the “father of number theory”?

I noticed that some sources state Fermat as the father of modern number theory while others say Gauss. I am trying to start a paper on the history of number theory for a presentation, but I cannot ...
1
vote
1answer
18 views

Proof Verification - $\exists a \in S (a\ge S_a)$

I wanted to prove that $\exists a \in S (a\ge S_a)$ where $S$ is an finite set of real numbers with order $n$ and $S_a$ is the average of the set. This is my proof so far: Assume $a_i = a_k, i,k ...
0
votes
2answers
48 views

Fermat's Last Theorem - Variation with arithmetically descending exponents

Are there solution(s) to the following variant of Fermat's Last Theorem in the positive integers? $$ a^n + b^{n-i} = c^{n-2i} $$ I haven't been able to identify any trivial solutions. To my ...
1
vote
3answers
55 views

Proof that if $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. [duplicate]

I need help with this excercise. If $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. I don't know how to prove this, I know the definition of $\gcd$ but I can't prove it.
0
votes
1answer
31 views

Is $p_n \sim \frac{5}{4}n\log(n) + \frac{1}{2}n + \frac{(p_1+\ldots+p_{n-1})}{n-1}$ a good approximation for the $n^\text{th}$ prime?

If you plot the following function $$f(n) = \left|\frac{(p_1+\ldots+p_{n})}{n} - \frac{(p_1+\ldots+p_{n-1})}{n-1}\right|$$ you get a graph that is similar to $$f(x) = \frac{5}{4}\log(x) + ...
0
votes
0answers
20 views

Implementing FizzBuzz game

I need to build an electrical-circuit for the FizzBuzz game. There's a signal, called next which increment the current number by one. The rules are simple - You ...
3
votes
0answers
38 views

Trisectible Angle

How do we prove that a triangle with sides $(one, x, y)$, where $x$ is any constructible length from one to three at the elliptic curve $$y^2 = x^3 -x^2 -x +1$$then the triangle possess at least ...
6
votes
0answers
60 views

Fermat's Last Theorem: A natural extension

It is well known that there are no solutions to $$a_1^n+a_2^n=b^n$$ for $a_1,a_2,b\in\mathbb{Z}^+$ and $n>2$. Is it then true that there are no solutions to $$a_1^n+a_2^n+\cdots+a_m^n=b^n$$ ...
8
votes
0answers
64 views

Can anyone improve on this work and find a closed form of $\zeta(3)$?

This was something I and another user came across independently, although he decided to post it on reddit. So while its already online, let me reproduce it here with the hope that someone will be able ...
3
votes
1answer
27 views

Was this arithmetic Möbius/Mangoldt function ever used for something?

Let $n=\prod_k p_k^{c_k}$, with $p_k \in \mathbb P$ and $$ A(n)=\sum_{d|n} \mu(d)\Lambda(d), $$ with the $\mu$ Möbius function, which has values in {−1, 0, 1} depending on the factorization of n ...
4
votes
5answers
104 views

What is $\limsup_{n\to\infty} \frac{p_{n+1}}{p_n}$?

Let $(p_n)_{n\in\mathbb N}$ be the strictly increasing sequence of all primes. I'm wondering what $$S:=\limsup_{n\to\infty} \frac{p_{n+1}}{p_n}$$ is. Is the result already known? By Bertrand's ...
1
vote
2answers
26 views

Proof - Uniqueness part of unique factorization theorem

The uniqueness part of the unique factorization theorem for integers says that given any integer $n$, if $n=p_1p_2 \ldots p_r=q_1q_2 \ldots q_s$ for some positive integers $r$ and $s$ and prime ...
0
votes
0answers
24 views

Limit of an euler product

Before I can ask my question, I have to state a couple of definitions. Let $f$ be a multiplicative function and let $$ D_f(s) = \sum_1^{\infty} \frac{f(n)}{n^s}, $$ and define $\Lambda_f(n)$ as ...
0
votes
0answers
35 views

Does this set contain these numbers?

How would I go about proving whether or not every number $n=k^8$ is included in the set of all numbers $m=k^4$ ($n$ and $k$ are integers in both cases)?
1
vote
2answers
32 views

Lucas's proof of a special case of Beal's conjecture

While studying the properties of a certain elliptic curve, I came across the equation $x^4+y^4=z^3$. There is no solution of this equation in relatively prime integers, and this is a special case of ...
3
votes
4answers
121 views

How to show that $2\times 10^{18}<20!<3 \times 10^{18}$ without calculator? [on hold]

I want to find the first digit of $20!$ By calculator $20! = 2.43290200817664 \times 10^{18}$. So I want to show that $2\times 10^{18}<20!<3 \times 10^{18}$ Thank you.
3
votes
2answers
80 views

Consecutive squarefree numbers of 5 prime factors each, mostly small

The sequence of numbers 49297533, 49297534, and 49297535 is notable, because the factorizations of these numbers are each of the form $a^1 \cdot b^1 \cdot c^1 \cdot d^1 \cdot e^1$, where $\{a\ldots ...
5
votes
1answer
55 views

How to prove$\displaystyle \sum_{i=0}^{k}(-1)^i\binom{n}{k-i}\binom{n+i-1}{i}=0$

I saw a combinatorial identity when i study linear-algebra, But the author didn't explain how to get it. $\displaystyle \sum_{i=0}^{k}(-1)^i\binom{n}{k-i}\binom{n+i-1}{i}=0$ I tried $n=10$ or ...
2
votes
1answer
36 views

Likelihood at least 2 out of $n$ numbers are visible to each other in $\mathbb{Z}^n$

Two points in $ \mathbb{Z}^n $ are said to be visible to each other, if they can be connected by a straight line, which doesn't intersect any points of $ \mathbb{Z}^n $ In Apostol's book "An ...
0
votes
0answers
24 views

On some factorial inequalities

Denote $P_n$ to be product of primes at most $n$. What is the minimum value of $m$ such that $P_m\geq P_n^2$? What is the minimum value of $m$ such that $m!\geq n!^2$? What is the minimum value of ...
1
vote
5answers
69 views

there does not exist a perfect square of the form $7\ell+3$

I have been trying to prove that there does not exist a perfect square of the form $7\ell+3$. I've tried using $n$ as even or odd, and I'm getting stuck. Can someone put me on the path? Is this an ...
1
vote
0answers
15 views

Why does the uniqueness theorem for Dirichlet series hold for the infinite sums, while obviously not for partial sums?

I asked in a previous question whether a function, $a_n$, is unique to $F(s)$ for any Dirichlet function defined by the following $$F(s)=\sum_{n=1}^\infty{\frac{a_n}{n^s}}.$$ Its uniqueness property ...
1
vote
0answers
17 views

What are the asymptotic considerations in the following?

The following is from this paper that discusses polynomials and classic number theory functions. The proof of theorem 1.3 has a final statement saying that $R$ must be null because we arrive at ...
0
votes
2answers
38 views

Number of divisors $d$ of $n^2$ so that $d\nmid n$ and $d>n$

I just wanted to share this nutshell with you guys, it is a little harder in this particular case of the problem: Find the number of divisors $d$ of $a^2=(2^{31}3^{17})^2$ so that $d$ does not ...
2
votes
2answers
45 views

Existence of integer solution of $a^2 -17b^2 = $ any constant

When checking whether if $9-\sqrt{17}$ in the ring $\{a+b\sqrt17: a,b \in \mathbb{Z}\}$ is a prime. Suppose $$\alpha\cdot \beta = 9-\sqrt{17},$$ using norm argument $$N(\alpha)N(\beta) = ...
0
votes
1answer
20 views

Integer solution to the following root

If $a,b\in\Bbb{N}$, then what is the smallest non-trivial solution to this equation? $$\frac{\lfloor100{\sqrt[3]{a}}\rfloor}{100}+\frac{1}{100}=b$$ So I want the answer like this: ...
3
votes
1answer
26 views

Simplifying a Taylor polynomial that involves Stirling numbers of the second kind

I am currently trying to evaluate a complicated function $f$ at a point $x+a$ using a high order Taylor polynomial about the point $a$. The polynomial has the standard form: $$\sum_{n=1}^k ...
3
votes
1answer
33 views

Largest Triangular Number less than a Given Natural Number

I want to determine the closest Triangular number a particular natural number is. For example, the first 10 triangular numbers are $1,3,6,10,15,21,28,36,45,55$ and thus, the number $57$ can be ...
1
vote
0answers
26 views

Proof of the binomial theorem through Dirichlet convolution?

Here I gave a proof for $\sum_{k=0}^n\binom nk(-1)^k=0$ based on the fact that $\mu*1=\varepsilon$ (the Dirichlet identity). I am wondering if using a similar technique we can prove that ...
0
votes
0answers
28 views

Moebius Identity

Is there alternative proof of Moebius identity i.e. sum of moebius function over divisor of n is zero than as suggested n page: ...
0
votes
0answers
13 views

Dirichlet product is associative

Is there alternative proof of fact: Dirichlet product on arithmetic function is associative than given in Dirichlet's product with number theoretic functions
12
votes
1answer
107 views

When is $\sum_{n=0}^{\infty}\frac{n^k}{3^n}$ an integer?

The motivation for this was to find some nice expression for the sum $$\sum_{n=0}^\infty\frac{P(n)}{b^n}$$ where $P(n)$ is a polynomial and $b$ is a positive integer greater than 1. Clearly, it ...
9
votes
4answers
152 views

Diophantine equation $(x+y)(x+y+1) - kxy = 0$

The following came up in my solution to this question, but buried in the comments, so maybe it's worth a question of its own. Consider the Diophantine equation $$ (x+y)(x+y+1) - kxy = 0$$ For $k=5$ ...
2
votes
1answer
37 views

How to prove the sum of squares larger than 1/n without induction? [duplicate]

known that: $1\geq R_1 \geq R_2 \geq \dots \geq R_n \geq 0$ and $\sum_{i=1}^n R_i=1$ To prove: $\sum_{i=1}^n R_i^2 \geq \frac{1}{n}$ Using induction, the problem can be easily proved. I'd like to ...
3
votes
1answer
61 views

$\left[\frac n1\right]+ \left[\frac n2\right] + \cdots+\left[\frac nn\right]+\left[\sqrt n\right]$ is even [duplicate]

Let $n$ be any natural number. Prove that $\left[\dfrac n1\right]+ \left[\dfrac n2\right] + \left[\dfrac n3\right]+\cdots+\left[\dfrac nn\right]+\left[\sqrt n\right]$ is even. I tried this by ...
7
votes
1answer
31 views

Smallest witness for checking the primality of a number

In this link https://primes.utm.edu/prove/prove2_3.html it is stated that the smallest witness for a composite number is always less than $2ln(n)^2$ , assuming the extended Riemann-hypothesis. ...
6
votes
0answers
29 views

For all Dirichlet series, is $a_n$ unique to $f(s)$?

For any Dirichlet series, $$f(s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ is the function, $a_n$, always unique to $f(s)$? In other words, is it possible to show that $a_n$ is the only function that will ...
7
votes
0answers
34 views

Does $S(n)$ contain infinite many primes? [duplicate]

Denote $p_j := j\text{th prime}$ and $S(n)\:=\sum_{j=1}^n p_j$ (The sum of the first $n$ primes). Is it known whether $S(n)$ is prime for infinite many $n$? OEIS gives the sum of the prime ...
4
votes
2answers
80 views

Rational solutions to $x^4+y^4=cz^2$

Suppose $c\neq 1$ is a squarefree number, and consider the curve $x^4+y^4=cz^2$. How can I find rational points on this curve? What I really want to know is how to transform this into an elliptic ...
2
votes
1answer
29 views

zeros of p-adic power series

Suppose I have $f(x) \neq 0 \in \mathbb{Z}_p[[x]].$ If it can help it is of the form $g(x^p)+ph(x)$. Under which conditions can I say that $f(x)$ has finitely many zeros in $\bar{\mathbb{Q}}_p$? ...
0
votes
0answers
18 views

Quadratic Diophantine equation

How to find the integral solutions of $$ \ x^2 + y^2 =2z^2 $$ such that x,y are distinct and $$ z^2 < 2(min(x,y))^2 $$ This can be reduced to $$ a^2 + b^2 = 2 $$ such that a,b are rational ...