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.

learn more… | top users | synonyms (1)

0
votes
0answers
21 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)?
0
votes
0answers
11 views

How to count the number of solutions to these matrix equations

Given: $$a_{1,1} x_1 + a_{1,2} x_2 + ... + a_{1,n} x_n = b_1 \bmod p $$ $$a_{2,1} x_1 + a_{2,2} x_2 + ... + a_{2,n} x_n = b_2 \bmod p $$ $$...$$ $$a_{m,1} x_1 + a_{m,2} x_2 + ... + a_{m,n} x_n = b_m ...
1
vote
2answers
19 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 ...
2
votes
3answers
61 views

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

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.
1
vote
0answers
16 views

Adjacent squarefree numbers of cardinality 5

The sequence of numbers 49297533, 49297534, and 49297535 is notable, because the factorizations of these numbers are each of the form $a^1 * b^1 * c^1 * d^1 * e^1$, where ${a...e}\in Primes$, and ...
4
votes
1answer
41 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 ...
1
vote
1answer
28 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
22 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
66 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 ...
0
votes
1answer
23 views

The sum of all numbers between 1 and 1000 INCLUSIVE that are divisible by 3 or 5 BUT NOT BOTH (exclusive or)

I found a problem that asked to calculate the sums of all numbers between 1 and 1000 (inclusive) that are divisible by 3 or 5 but not both. I immediately thought of Gauss which made me smile but ...
1
vote
0answers
12 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
31 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
43 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
0answers
13 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
30 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
24 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
27 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
105 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 ...
8
votes
4answers
111 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
60 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 ...
6
votes
1answer
25 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
28 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
68 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 ...
1
vote
1answer
25 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
17 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 ...
2
votes
0answers
19 views

Lehmer's totient problem generalization (adding a constant )

Lehmer's conjecture is an well-known open problem which states that the divisibility : $$\phi(n) \mid n-1$$ holds only for primes . This motivated me to ask the following : For which ...
1
vote
0answers
56 views

A diophantine problem with big numbers!

Find all pairs of positive integers $(a, b)$ such that $a^2 b^2 +300 \mid a^2(300 b^2 -a)$ and $300 b^2 -a>0$. I've tried so many different ways, I only concluded that $a<300$.
1
vote
1answer
29 views

Recurrence relation between solutions of a quadratic Diophantine equation

I have the fundamental solution of the following Diophantine equation: $$\frac{x(x-1)}{y(y-1)}=\frac{m}{n} \hspace{5 mm}, \hspace{5 mm} m \le n$$ $$nx^2-my^2-nx+my=0$$ Is it possible to derive a ...
1
vote
1answer
22 views

Largest Number that cannot be expressed as 6nm +- n +- m

I'm looking to find out if there is a largest integer that cannot be written as $6nm \pm n \pm m$ for $n,m$ elements of the natural numbers. For example, there are no values of $n,$m for which $6nm ...
1
vote
4answers
64 views

Prove that $1^2 + 2^2 + … + (n-1)^2 < \frac {n^3} { 3} < 1^2 + 2^2 + … + n^2$

I'm having trouble on starting this induction problem. The question simply reads : prove the following using induction: $$1^{2} + 2^{2} + ...... + (n-1)^{2} < \frac{n^3}{3} < 1^{2} + 2^{2} + ...
1
vote
1answer
40 views

Is this a good generating function for Sum-of-divisors function?

I have an expression for the sum-of-divisors function defined as $$\sigma(n)=\sum_{d\mid n}d.$$ However I do not know how nontrivial or practical it actually is. Let us define ...
2
votes
0answers
14 views

Wireless networks on two sequential office floors: Random partitions of a finite interval via a point process on a line

Construct a Poisson point process of density one on a line of length $L$. Allow each point in the process to "see" part of the line to their left, and part of the line the their right (such that the ...
2
votes
0answers
50 views

“Race” of the primes modulo $1,3,7,9\ \pmod {10}$

The "race" starts with the prime $11$. The number of primes $1, 3, 7, 9 \pmod {10}$ is denoted after every occurring prime. Does the lead change infinitely often? And does every "runner" have ...
6
votes
1answer
69 views

$\tau$ and grouping of prime numbers

From Prime Number Theorem and this we can state $$\frac{p_n}{\bar{p}}\sim 2$$ or $$\lim_{n\to \infty} \frac{np_n}{(p_1 + \dots +p_n)} = 2$$ If we then look at the fluctuations in the graph of $$f(n) ...
2
votes
2answers
62 views

Prove that $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \pmod p$

I'm trying to prove the statement $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} \equiv 0 \mod p$ and I don't really know where to start. Obviously $\sum_{t=1}^{p-1} \frac{t^2-1}{t^2+1} = 2\sum_{t=1}^{(p-1)/2} ...
3
votes
1answer
27 views

Prove $v,w\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and dependent when $p=3$

I need to prove that $\{v=(6,9),w=(7,8)\}\in Z_{p}\times Z_{p}$ is linearly independent when $p=2$ and linearly dependent when $p=3$. The problem is my freshman algebra course did not cover rings and ...
4
votes
2answers
149 views

a certain simple continued fraction

Given the golden ratio: $$\phi=\frac{1+\sqrt{5}}{2}$$ and the following simple continued fraction: $$G(q,k)=\cfrac{1}{1-q+\cfrac{1}{1-{q^3}^k+\cfrac{1}{1-{q^5}^k+\cfrac{1}{1-{q^7}^k+\ddots}}}}$$ For ...
3
votes
1answer
30 views

the least $m$ such that $a^m\equiv 1 \mod n $ for fixed $a,n$.

Is there any known method for calculating $\lambda_a(n)$ which returns the smallest integer $m$ such that $a^m\equiv 1 \pmod n$ where $\gcd(a,n)=1$ ? I searched but I found nothing, is there at ...
2
votes
1answer
32 views

Is there a finite initial generating set for ${\mathbb N}$ given these two operations?

This is inspired by a recent question. Suppose we have $x_j \in \mathbb{N}$, and then we are allowed to perform any sequence of the following two operations: 1) Multiply by $k$ for some fixed $k \in ...
0
votes
1answer
30 views

What should be the approach for finding the remainder

How can one approach this kind of question: Find the remainder when $\left((7!)^{6!}\right)^{17777}$ is divided by 17
6
votes
0answers
48 views

Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can $$ \zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 - p^{-s}} $$ be evaluated or written in terms of standard functions? Details: We can write the Riemann zeta function as ...
1
vote
1answer
192 views

Would this proof strategy work for proving the lonely runner conjecture?

The problem is the lonely runner conjecture. This conjecture states that if $k$ runners begin running down a circle of unit circumference with random speeds, it will always the case that all runners ...
2
votes
2answers
61 views

Solving a quadratic Diophantine equation

I want to solve the following quadratic Diophantine equation: $$\frac{x(x-1)}{y(y-1)}=\frac{p}{q} \hspace{5 mm}, \hspace{5 mm}p\le q$$ For $p=1$ and $q=2$, it is easy to solve. Let $y=x+z$. Then ...
0
votes
1answer
19 views

$(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no. Where w;x;y;z are positive odd integers?

Q $(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no where w,x,y,z are positive odd integers? Options given: 20;8;4;2 My Approach: I Choose ($9^2$+$5^2$).($7^2$+$3^2$) to get ...
-1
votes
1answer
38 views

$3x + 1$ problem: other repetitions [on hold]

The Collatz problem: Pick an integer $x > 0$ If $x$ even: $x = x / 2$, if $x$ odd: $x = 3x + 1$ repeat 2.) as long as you want This algorithm seems to always end up with the loop $4, 2, 1$ My ...