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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4
votes
0answers
14 views

Decomposition of an algebraic number into a sum or product of algebraic numbers with smaller degree

An algebraic number can be identified by its minimal polynomial together with isolating intervals with rational bounds for its real and imaginary parts. The degree of an algebraic number is the degree ...
0
votes
1answer
38 views

Question about 'Knuth's up-arrow notation'

Am I right? $$x\uparrow x=x^x$$ $$x\uparrow\uparrow x=x^{x^x}$$ $$x\uparrow\uparrow\uparrow x=x^{x^{x^x}}$$ $$x\uparrow\uparrow\uparrow\uparrow x=x^{x^{x^{x^x}}}$$ ...
6
votes
1answer
51 views

Given $k$, are there infinitely many $n$ so that $w(n) = w(n+k)$?

$w(n)$ denotes the number of distinct prime factors of $n$. I am wondering if any such result is known.
2
votes
0answers
18 views

Question on a proof of the euler product of the zeta function

Let $\zeta(s)$ be the Riemann zeta function, then we know it satisfies the Euler product for Re$(s) > 1$, $$ \zeta(s) = \prod_{p} (1 - p^{-s})^{-1}. $$ The proof I read, if I recall correctly, was ...
1
vote
2answers
27 views

Generalizing Legendre's Theorem on prime factorization of factorials

From Legendre's Theorem we know $$n!=\prod_{p}p^{\lceil \frac{n}{p}\rceil +\lceil \frac{n}{p^{2}}\rceil +.. }$$. Since $\Gamma (n+1)=n!$, i wonder if there is a generalization of this formula for ...
4
votes
2answers
43 views

Roots of $x^p + x + [\alpha]_p \in \mathbb{Z}_p[x]$

Let $$g(x) = x^p + x + [\alpha]_p \in \mathbb{Z}_p[x],$$ where $p$ is prime. For which $\alpha, p \in \mathbb{Z}$ does $g(x)$ have at least one root? And for which $\alpha, p \in \mathbb{Z}$ ...
-4
votes
1answer
33 views

Find a formula for this

I need help. I don't know if it is possible. Example formula that uses English instead of math! $f(x) = 3x$ + all previous values of $f(i)$ with $i$ from $0$ to $x-1$, where $x$ is a positive ...
3
votes
1answer
81 views

Is $\arctan2$ irrational?

Is $\tan^{-1}2$ an irrational number or a rational number? How to show that? Or generally how to show $\tan^{-1}3, \tan^{-1}4, \tan^{-1}5...$ is irrational or rational?
-1
votes
1answer
32 views

How would you prove $ab|c$ knowing that $a|c$ and $b|c$ and $\gcd(a,b) = 1$? [on hold]

If $a|c$ and $b|c$ and $\gcd(a,b) = 1$, prove that $ab|c$.
2
votes
2answers
15 views

Square Free congruence modulo n

I am trying to show that if $a^n\equiv a\pmod n$ for all integers $a$ that $n$ is square free. I have an idea to start with the contradiction that suppose $n=p^2m$ for some prime $p$, then n does not ...
-3
votes
2answers
30 views

Finding Square Roots

Well, i have a method to find square roots to any number. for Eg To find $\sqrt{58}$ or any number. We have to find which perfect square is place before it. 49 is placed before 58. What we have to do ...
1
vote
1answer
39 views

How I can prove that for any natural number $n$ such that $30<n$, $\pi(4n-3)<n$?

I need to proove that for any natural number $n>30$: $$\pi(4n-3)<n.$$ In this inequality, $\pi(x):\mathbb{N}\to \mathbb{N}$ is the defined as follows: $$\pi(x):=Card(\lbrace p \ | \ p\leq x\ \ ...
0
votes
1answer
7 views

Injectivity of idele norm map

Let $K/F$ be an extension of global fields (I'm considering number fields, but my question may be also considered in function fields). We may define a norm map on the idele groups $$N_{K/F}:\Bbb ...
1
vote
3answers
56 views

For which $n \in \mathbb{N}$ does $x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$ have at least $7$ distinct solutions?

I have to find one $n \in \mathbb{N}$ such that $$x^8 + [2]_nx^7+[6]_nx^6-x^2-[2]_nx - [6]_n = [0]_n$$ has at least $7$ distinct solutions in $\mathbb{Z}_n$ (or, equivalently, $f(x) = x^8 + ...
1
vote
1answer
40 views

Find all $n$ such that $3^{2n+1}-4^{n+1}+6^n$ is prime

I'll try to format my question in a manner such that you can skip (irrelevant) parts. Exercise: Find all natural $n$ such that $3^{2n+1}-4^{n+1}+6^n$ is prime. Motivation: I'm trying to ...
0
votes
0answers
35 views

How to square a number that got more digits than search results “digits” on Google.

I am implementing the quadratic sieve algorithm. And I got run in unexpected problem. Take a look at those two final steps of the algorithm as described in wiki. Use linear algebra to find a ...
3
votes
2answers
77 views

Product of Primes

Let $\mathbb{P}$ denote the set of prime numbers. How would one evaluate $$\prod_{p\in \mathbb{P}}\frac{p-1}{p}$$ I do not think that the fact that ...
1
vote
1answer
62 views

Show by mathematical induction that the gcd(n,n+1) = 1 for every integer n.

By mathematical induction, how would you show $\gcd(n,n+1)= 1$ for every integer $n$? I'm thinking you would start by knowing that some integer, d, would divide both n and n+1. That's all I have so ...
0
votes
1answer
19 views

Checking if a number is Polygonal

Polygonal numbers are of the form $\cfrac{n^2(s-2)-n(s-4)}{2}$, where $s$ is the number of sides of the polygon and $n$ is to say which one it is (the $n^{th}$ $s$-gonal number) So my question is, ...
5
votes
1answer
85 views

How many solutions are there of the equation $(\cos a)^x+(\sin a)^x=1$, $x \in \mathbb{R}$.

Is there any solution of the equation other than $x=2$? Please help me. Thank you in advance.
2
votes
1answer
34 views

How are the nontrivial zeros of the Riemann zeta function calculated?

The Riemann zeta function, is the function of the complex variable $s$, defined in the half plane $\Re(s)>1$ by the absolutely convergent series $\zeta(s) = \sum_{n} n^{-s}$ and extends to the ...
0
votes
0answers
46 views

Find the Summation of Summation [on hold]

An Array A consisting of N integers .We perform the following operation M times: for i = 2 to N: Ai = Ai + Ai-1 We have to find xth element of the array ...
1
vote
1answer
16 views

Connecting homomorphism in Galois homology using the standard resolution

Let $G$ be a finite group, although this may not be necessary for almost everything that follows. One of the ways of defining Galois homology groups is using the standard resolution for the ...
2
votes
1answer
46 views

Gaussian elimination algorithm performance

I am developing the quadratic sieve algorithm and I reached a new bottle neck: The matrix processing. I been reading quit a lot about this topic and I found many solutions Gaussian elimination: ...
1
vote
2answers
41 views

Uniqueness of log function with relaxed conditions?

Question If: $$f(a) + f(b) = f(ab)$$ $$ f(1) = 0 $$ $$ a<b \implies f(a) < f(b) \forall a,b \in N $$ where $N$ is the set of natural numbers. Prove or disprove $f$ must be the $\log$ ...
-1
votes
5answers
75 views

Is it accurate to say that multiplication of two integers yields an integer?

I am reading a book in discrete mathematics and it assumes that a multiplication of two integers yields an integer. Although that this book's saying is justifiable since the book is making an ...
0
votes
2answers
17 views

Showing Modulo Congruence Amongst Prime Divisors (Number Theory)

I'm having trouble figuring out how to show the general existence part of the following problem. Suppose $n\in\{1,2,3...\}$ and $n\equiv 7\mod{10}$. Show that $\exists$ a prime divisor $p$ of $n$ ...
1
vote
0answers
38 views

Important numerator and denominators in the evaluation of the integral: $\int_0^\infty x^t \operatorname{csch} x\text{ d}x$

$$\int_0^\infty x^t\operatorname{csch}x\text{ d}x=\frac{a\zeta(t+1)}{b}$$ for $t\in\Bbb{N}$ How might one represent $a,b$ in terms of $t$? (Note that $a,b\in \Bbb{N}$) If possible, could one also ...
2
votes
1answer
22 views

Modular Arithmetic and Greatest Common Divisor.

In my algebraic structures textbook I have come across a tricky question that I am trying to solve which goes as follows: suppose that $d|(a^n-1) $ and $d|(a^m-1)$ where $m,n$ are natural numbers ...
0
votes
1answer
34 views

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$

Find all the numbers $a$ such that the number $an(n+2)(n+4)$ is an integer for all $n \in \mathbb{N}$ It's trivial to see that if $a$ is irrational, we get no solution. Thus $a \in \mathbb{Q} ...
29
votes
5answers
2k views

$-1$ as the only negative prime.

I was recently thinking about prime numbers, and at the time I didn't know that they had to be greater than $1$. This got me thinking about negative prime numbers though, and I soon realized that, for ...
0
votes
0answers
38 views

on the least primitive root of a prime

There is an article on the least of primitive root of a prime in this link On the second page you will see ...
2
votes
1answer
52 views

Homeomorphism between $\mathbb{R}$ and $\mathbb{Q}$ - why does cardinality matter?

When I look up why $\mathbb{R}$ and $\mathbb{Q}$ are not homeomorphic, almost all the answers just say something along the line of "Because, Cardinality" and then ends there. Can someone provides ...
0
votes
2answers
37 views

Intersection of dense sets in $\mathbb{N}$

Let's call $A\subseteq\mathbb{N}$ dense if $$\text{lim inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n} = 1.$$ Is the intersection of two dense sets dense again? Or does the intersection of two dense ...
2
votes
1answer
35 views

What is the significance of Coleman maps arising in Iwasawa thoery?

I have come across two instances of "Coleman map" Let $E$ be an elliptic curve defined over $\mathbb{Q}_p$. Let $k_\infty$ be the unique $\mathbb{Z}_p$ extension of $\mathbb{Q}_p$ contained in ...
0
votes
0answers
29 views

Is there an arbitrarily large set of naturals so that the sum of each two has exactly $n$ prime divisors? What about an infinite set? [on hold]

Is there an arbitrarily large set of naturals so that the sum of each $2$ has exactly $n$ prime divisors where $n$ is fixed? What about an infinite set? For $n=1$ this is clearly false, what ...
1
vote
0answers
12 views

Best pattern of cinnamonbuns on a baking tray?

Imagine that i have a 50 x 100 cm baking tray, and i have a load of cinnamonbuns, shaped like a circle with a diameter of 10cm. How do i calculate the best place to place my cinnamonbuns, as the ...
1
vote
0answers
19 views

Stern-Brocot Tree and sum of coefficients of continued fraction

Suppose we are given a continued fraction $$\frac{p}{q}=a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\cdots}}}$$ I am trying to find an expression, possibly asymptotic, for the sum of the ...
0
votes
0answers
25 views

Sampling efficiently conditioned on linear constraints modulo both $\mathbb{F}_p$ and $\mathbb{F}_2$.

Given a prime $p$ and positive integer $t \ll \log p$ (say $t = \sqrt{\log p}$), is there an algorithm that is polynomial time in $\log p$ to sample uniform $X, Y \in \mathbb{F}_p$ conditioned on the ...
2
votes
2answers
41 views

Given integers $x,\,y$ s.t. $x^2-16=y^3$, show that $x+4$ and $x-4$ are perfect cubes

Suppose $x$ and $y$ are some integers satisfying $$x^2-16=y^3.$$ I'm trying to show that $x+4$ and $x-4$ are both perfect cubes. I know that the greatest common divisor of $x+4$ and $x-4$ must divide ...
1
vote
0answers
21 views

How can we prove a statement is provable?

Given a concrete mathematical statement, such as BSD conjecture(https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture), do we know if it is provable?
1
vote
1answer
56 views

An odd property of Egyptian fractions

This question arose through a response to [this post] Is there any partial sums of harmonic series that is integer? For which integers $N>1$ does the fraction $\frac 1N$ appear in the Egyptian ...
1
vote
0answers
25 views

square monotonic numbers

A monotonic number is a number in which the digits are in non-decreasing order. I found by computer that most of these numbers are squares of these numbers $$3 \ldots 34,3 \ldots 35,3 \ldots 37,3 ...
-7
votes
0answers
45 views

Why are there twin primes? [on hold]

Speculation encouraged. Isn't it strange that there are probably infinitely many despite the size of the numbers? Why is that?
1
vote
2answers
64 views

How can Mersenne Prime rule be valid if $2047$ isn't prime?

The rule of Mersenne Prime says that $2^p - 1$ is prime if $p$ is prime. $2^{11} - 1 = 2047$ satisfies the condition, but it's not a prime as it can be divided by two prime numbers $23$ and $89$. ...
3
votes
4answers
91 views

Remainder when ${45^{17}}^{17}$ is divided by 204

Find the remainder when ${45^{17^{17}}}$ is divided by 204 I tried using congruence modulo. But I am not able to express it in the form of $a\equiv b\pmod{204}$. $204=2^2\cdot 3\cdot 17$
0
votes
1answer
21 views

Connected Components of p-adic rationals

Notation: $p$ - a prime integer, $\Bbb{Z}_p$ - set of $p$-adic integers, $\Bbb{Q}_p$ - set of $p$-adic rationals, $\Bbb{Q}$ - set of rationals, $\Bbb{R}$ - set of reals. While reading up on ...
0
votes
0answers
21 views

Prime division algebra level 5

Let $P$ be the number of integers $n$  for which $n^4-52n^2+595$ is prime, and let $D$ be the number of distinct primes that can be represented in this form. Find $P+D$.
2
votes
1answer
65 views
+300

$n^a$ integral for all integer $n$ implies $a$ integral

Let $a>0$ be a real number, such that for all integers $n\geq 1$: $n^a \in \mathbb N$ Show that $a$ must be an integer. It's not difficult to show this when $a$ is a rational number: ...
2
votes
0answers
19 views

Finite amount of consecutive smooth numbers

is there a short proof of the fact that there is a finite amount of consecutive smooth numbers (meaning Given a finite set B, there is a finite amount of pairs $n,n+1$ so that both can be expressed as ...