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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5
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1answer
52 views

Integers which are the sum of non-zero squares

Lagrange's four-square theorem states that every natural number can be written as the sum of four squares, allowing for zeros in the sum (e.g. $6=2^2+1^2+1^2+0^2$). Is there a similar result in which ...
4
votes
0answers
36 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 $$a_1^n+a_2^n+\cdots+a_m^n=b^n$$ ...
5
votes
0answers
41 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
5answers
74 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 ...
3
votes
1answer
25 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 ...
2
votes
1answer
45 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 ...
1
vote
0answers
30 views

Find the man who will survive till the last

I got the in Arthur Engel's book:Problem-Solving Strategies in the Number Theory section(p-$137$ & prob-$166$) If you are condemned to die in Sikinia, you are put into Death Row until the last ...
1
vote
0answers
17 views

Counting solutions to matrix equations

Given these 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 + a_{m,2} x_2 + ...
3
votes
4answers
112 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.
9
votes
4answers
136 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 $k=5$ ...
6
votes
1answer
76 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) ...
5
votes
1answer
51 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
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
1answer
513 views

How to determine the key-matrix of a Hill cipher where the encrypted-message-matrix is not invertible?

I am new to this subject and I have a homework problem based on Hill cipher, where encryption is done on di-graphs (a pair of alphabets and not on individuals). The alphabet domain is $\{A\dots ...
0
votes
0answers
21 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 ...
1
vote
2answers
28 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 ...
0
votes
0answers
30 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
1answer
32 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
1answer
40 views

How can there be an infinite number of sequentially composite Fibonacci(p)?

I ran into this counting function a(n)>=a(k)+1 for the number of distinct prime factors of the n-th Fibonacci number, at OEIS. Thank you Robert Israel! Thank you for writing the proof there. I had ...
1
vote
5answers
67 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 ...
4
votes
2answers
141 views

The sum of the reciprocal of primeth primes

A few days ago, a friend of mine taught me that the sum of the reciprocal of primeth primes $$\frac{1}{3}+\frac{1}{5}+\frac{1}{11}+\frac{1}{17}+\frac{1}{31}+\cdots$$ converges. Does anyone know some ...
0
votes
2answers
33 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 ...
63
votes
3answers
1k views

A real number $x$ such that $x^n$ and $(x+1)^n$ are rational is itself rational

Let $x$ be a real number and let $n$ be a positive integer. It is known that both $x^n$ and $(x+1)^n$ are rational. Prove that $x$ is rational. What I have tried: Denote $x^n=r$ and $(x+1)^n=s$ ...
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 ...
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
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
15 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 ...
2
votes
2answers
64 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} ...
-4
votes
1answer
50 views

Crude verification of Goldbach conjecture [on hold]

So the Goldbach conjecture says 'Every even integer greater than 2 can be written as sum of two primes'. Here is what I have roughly done to verify it, using probability. I don't say it is correct but ...
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: ...
2
votes
2answers
44 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) = ...
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 $\bar{\mathbb{Q}}_p$? ...
3
votes
0answers
15 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 ...
4
votes
2answers
72 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 ...
3
votes
1answer
32 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
25 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 ...
7
votes
1answer
29 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. ...
2
votes
1answer
49 views

Efficient algorithm for factorizing Mersenne numbers?

Is there an efficient algorithm (maybe Polynomial-time?) for factorizing Mersenne numbers of the form $2^p - 1$? We can assume that $p$ is a prime because if it is not, then we can reduce the problem ...
2
votes
2answers
75 views

Why are special numbers important? (Such as fermat prime, mersenne prime)

Whenever I studied topics in mathematics, I found those topics are important in purely mathematical sense and I could see some motivations. However, I cannot see neither motivation nor importance of ...
12
votes
1answer
106 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 ...
1
vote
4answers
84 views

What makes the Mersenne primes formula more special than any of these formulas?

Mersenne Primes Formula $2^n-1$ gives false results just like any of those ones: $3^n-2, 4^n-3, P_1\cdot P_2+P_1+P_2$, or $5^n-4$ and so on.. I think that each of those formulas(including ...
4
votes
3answers
120 views

Iterated square roots over finite field. When do we hit a nonresidue?

Suppose that we are working within the integers modulo $p$ where $p$ is some odd prime number. Suppose that $x_0$ is a (nonzero) quadratic residue mod $p$ then there exists some $x_1$ such that $x_1^2 ...
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
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 ...
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 ...
2
votes
2answers
191 views

Sequences containing infinitely many primes

What are some interesting sequences that contain infinitely many primes? If it takes form of a polynomial, Dirichlet's theorem answer the question completely for linear polynomial. What about ...
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 ...
7
votes
5answers
175 views

If $ a + b + c \mid a^2 + b^2 + c^2$ then $ a + b + c \mid a^n + b^n + c^n$ for infinitely many $n$

Let $ a,b,c$ positive integer such that $ a + b + c \mid a^2 + b^2 + c^2$. Show that $ a + b + c \mid a^n + b^n + c^n$ for infinitely many positive integer $ n$. (problem composed by Laurentiu ...