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
46 views

What is the Galois group of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$, where $\zeta$ is a $p^r$th root of unity?

Let $\zeta$ be a $p^r$-th root of unity and $\mathbb{Q}_p$ the p-adic numbers. I would like to know the Galois automorphisms of $\mathbb{Q}_p[\zeta] / \mathbb{Q}_p$. I already know, the degree of ...
0
votes
0answers
4 views

Dirichlet characters with values in a finite field

Although the classical Dirichlet characters are complex valued, it seems to me rather useful that the characters attain values in a finite field; thus homomorphisms from $\mathbb{Z}_N^*$ to ...
1
vote
1answer
49 views

Infinitude of the primes $p\equiv1 \operatorname{mod} n$

$\textbf{Theorem:}$ Fix $1 < n \in \mathbb{Z}$. There are infinitely many primes $p\equiv1 \operatorname{mod} n$. $\textbf{Proof}$ Recall that the $n$-th cyclotomic polynomial $\Phi_n(x)$ is ...
0
votes
2answers
29 views

Can you express the fraction 1/0 using imaginary units in any way possible?

Although in basic textbooks, 1/0 is undefined or something along those lines, can it be expressed using complex numbers (i)? One way I propose is to use a function such as 1/x and use that function in ...
0
votes
1answer
14 views

Definition of “Contractive Invariant Plane”

Can someone please explain the definition of a contractive invarient Plane found in: the paper It is nearly at the very beginning of the Introduction. By contractive do they mean a contractive map? ...
1
vote
2answers
66 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
0
votes
1answer
23 views

Elementary number theory proofs using functions

The functions $f$ and $g$ are defined by $f(x) =$ remainder when $x^2$ is divided by $7$. $g(x) =$ remainder when $x^2$ is divided by $5$. (a) Show that $f(5)=g(3)$ (b) If $n$ is an integer, ...
0
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0answers
33 views

$\eta(1) = \ln(2)$ proof using Abel's Theorem

Hi I was just wondering how does one justify $\eta(1) = \ln(2)$. Looking at the power series for $\ln(1+x)$ we have \begin{equation} \ln(1+x)= \sum_{n=1}^{\infty} \frac{(-1)^{n+1}x^{n}}{n} ...
1
vote
0answers
34 views

Does the Collatz conjecture imply the Well-Ordering Principle?

It seems to me that it does, even trivially so. My reasoning is as follows: Suppose that H is a nonempty set of positive integers. Case 1. 1 is a member of H. Then H obviously has a smallest member, ...
5
votes
1answer
63 views

Is $\pi(n)$ a Rational Function?

Are there some two-variable polynomials $P(n,\log n)$ and $Q(n,\log n)$ which we have the bellow equation for prime counting function $\pi(n)$ for $n \in \mathbb{n}$? $$\pi(n) = \Bigl{\lfloor} ...
1
vote
0answers
83 views

Prove result without using Fermat's Last Theorem

Is there a way to prove that for $x,z,n \in \mathbb{Z}$, $x > 0$, $z > 0$, $n > 2$, the equation $$ \sum_{k = 0}^{n - 1}{\binom{n}{k} x ^k} = z ^ n $$ has no solution, without using ...
4
votes
1answer
142 views
+50

What is known about these arithmetical functions?

Let $n=\prod_p p^{c_p}$, $N\in \mathbb N$ and $$ \alpha_N(n)=\prod_p p^{c_p \bmod N}. $$ The function $\alpha_N$ is multiplicative since $\alpha_N(n)\alpha_N(m)=\alpha_N(nm)$ for co-prime $n$ and $m$ ...
2
votes
2answers
71 views

How find this minimum of the $q$, if such $\frac{95}{36}>\frac{p}{q}>\frac{96}{37}$

let $p,q$ is postive integer,and such $$\dfrac{95}{36}>\dfrac{p}{q}>\dfrac{96}{37}$$ Find the minimum of the $q$ maybe can use $$95q>36p$$ and $$37p>96q$$ and then find this minimum of ...
1
vote
1answer
40 views

If $a|(p+1)$ for all but finitely many $p=3 (\text{ mod } 4)$ then $a$ divides $4$

I have the following question: Let $a$ be an integer such that $a$ divides $p+1$ for all but finitely many primes $p=3 \text{ mod } 4$ Can we conclude that $a$ must divide $4$? How we can prove ...
1
vote
1answer
45 views

How does the Riemann summation connect with prime numbers [on hold]

The Riemann sum goes as the sum of $\dfrac{1}{1^s}+\dfrac{1}{2^s}+\dfrac{1}{3^s}+\ldots$ It has been said that if $\frac{1}{2}+ix=0$ for $s$ ($x$ can be any number), then the sum will have massive ...
1
vote
1answer
115 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: ...
3
votes
1answer
38 views

Prime Zeta Function proof help: Why are these expressions not equal?

I was trying to create a formula for the Prime Zeta function and I partially succeeded except for one frustrating error. I was only able to formulate an approximation. Consider the following sum: ...
-7
votes
0answers
33 views

what is the sum of these 1230 sequences such as 1,2,1,2,2,1,2,2,2,1,2,2,2,2… A. 2411 B. 2412 C. 2413 D. 2414 [on hold]

what is the sum of these 1230 sequences such as 1,2,1,2,2,1,2,2,2,1,2,2,2,2..... A. 2411 B. 2412 C. 2413 D. 2414
2
votes
3answers
101 views

Binomial Coefficient Computation by Dividing Consecutive Terms

If I take the binomial coefficient: $$\frac{n!}{k! (n-k)!}$$ and I want to know the result of 10 choose 4 and I proceed to do the computation $$\frac{7*8*9*10}{1*2*3*4}$$ by first dividing and then ...
0
votes
0answers
26 views

Why does the Harmonic series diverge? [duplicate]

I am aware of Oresme's proof of its divergence(http://mathworld.wolfram.com/HarmonicSeries.html), but this proof could be applied to the sum of all natural numbers and it would still be valid. Yet, ...
4
votes
0answers
35 views

Substitutions that transform Fermat Equations to Elliptic Curves

I was reading Chapter 1 of Elliptic Curves - Number Theory and Cryptography by Lawrence C Washington. He was considering Fermat equations $$a^4+b^4=c^4\text{ and }a^3+b^3=c^3.$$ For the 1st equation, ...
1
vote
0answers
29 views

Notation for indexing the factorizations of a number?

Background Given any $n \in \mathbb{N}$, the ordered factorization count of $n$ can be computed and is traditionally written $H(n)$. This is, essentially, the number of unique decompositions of $n$ ...
2
votes
2answers
48 views

Number Theory- Mathematical Proof

I am trying to prove a theorem in my textbook using another theorem. What I need to show that if a,b, and c are positive integers, show that the least positive integer linear combination of a and b ...
2
votes
4answers
2k views

Floor Function Proof

Let $m \in \mathbb{R}$ and $n \in \mathbb{N}$. Prove the following facts: $\lfloor \hspace2mm\rfloor$ Means Floor function: And $\lfloor \lfloor m\rfloor/n\rfloor$ mean the floor of $m$ and then the ...
2
votes
1answer
60 views

prove that this number contains two equal digits

We delete the first digit from the number $7^{1996}$ and then we add it to the remaining number, repeat this until we get a number consisting of $10$ digits, prove that, this number contains two equal ...
126
votes
1answer
7k views

A variation of Fermat's little theorem

Fermat's little theorem states that for $n$ prime, $$ a^n \equiv a \pmod{n}. $$ The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence ...
6
votes
1answer
36 views

counting function of system of equations and Circle method

I came up with the follwing question while looking on Davenport's book: Analytical Methods for Diophantine equations and Inequalities. When introducing the Circle method gives an example on how to ...
6
votes
1answer
40 views

Cyclotomic polynomials, properties.

Let $F$ be a field of characteristic prime to $n$, and let $F^a$ be an algebraic closure of $F$. Let $\zeta$ be a primitive $n$th root of unity in $F^a$. I know that the monic polynomial $\Phi_n(X)$ ...
13
votes
1answer
158 views

Sum of Reciprocals of Primes in Imaginary Quadratic Field Diverges (2014 Miklós Schweitzer)

Problem 5 of the 2014 Miklós Schweitzer states: Let $\alpha$ be a non-real algebraic integer of degree two, and let $P$ be the set of irreducible elements of the ring $\mathbb{Z}[\alpha]$. Prove that ...
5
votes
1answer
61 views

How to extract solutions to a Pell's equation satisfying certain congruences?

I'm trying to solve $y^2=3x^2+3x+1$ for integers, which transforms into $(2y)^2-3(2x+1)^2=1$. I know how to solve pell's equation, but how can we extract only (odd,even) pair from the solutions of the ...
6
votes
1answer
124 views

To solve $n(n+1)(n+2)=6m^3$ in positive integers $m,n$

How to find all positive integers $m,n$ such that $n(n+1)(n+2)=6m^3$ ? I can see that $m=n=1$ is a solution , but is it the only solution ?
0
votes
2answers
62 views

Is $n^2+1$ always a product of Pythagorean primes?

Is it true that every positive integer $1$ bigger than a square is a Pythagorean prime (one which is the sum of 2 squares, and I'm including $2 = 1^2+1^2$) or a product of same? If so, is there an ...
0
votes
0answers
7 views

Comparing conway chains

See https://en.wikipedia.org/wiki/Conway_chained_arrow_notation for the details how conway chained arrow notation works. I want to calculate the approximate value $n$ such that $$n\rightarrow ...
3
votes
1answer
86 views

To find positive integers $n$ such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square

How many positive integers $n$ are there such that $\dfrac {n(n+1)(n+2)}6$ is a perfect square ? I know $n=1 , 2$ works ; are there any more ? Are there only finitely many such $n$ ?
2
votes
0answers
15 views

Show that the equation has a natural solution [duplicate]

let $n$ be a natural number and $r$ , $s$ be rational such that $n=s^2+r^2$ show that there are natural numbers a,b such that $n=a^2+b^2$
2
votes
1answer
29 views

$\operatorname{lcm}(2x, 3y-x, 3y+x)$

Problem Find $\textrm{lcm} (2x, 3y-x, 3y+x)$, where $y > x > 0$ and $\gcd(x,y) = 1$. Attempt I noticed after some numerical calculation that the answer seems to depend on the parity of $x$ ...
65
votes
11answers
11k views

$\sqrt a$ is either an integer or an irrational number.

I got this interesting question in my mind: How do we prove that if $a \in \mathbb N$, then $\sqrt a$ is an integer or an irrational number? Can we extend this result? That is, can it be shown ...
-4
votes
2answers
68 views

Can I divide $50$ cars on$ 5$ days? any trick? [on hold]

Can I divide $50$ cars on $5$ days, on condition that the numbers should be prime numbers? is there any trick? I'm asking for $5$ prime numbers whose sum is $50$
0
votes
1answer
55 views

Does $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ hold for infinitely many values of $x$ and $y$?

The problem is (assume $\pi(x)$ to be the prime-counting function), Does there exist infinitely many solutions to the equality $\pi \left(\dfrac{x+y}{2}\right)=\pi\left(\sqrt{xy}\right)$ with ...
2
votes
1answer
49 views

p-th root does not become a p-th power when adjoined?

Suppose $k$ is a number field of characteristic zero, and $u$ is a unit of infinite order, which is not a $p$-th power in $k$. Show $\sqrt[p]{u}$ is not a $p$-th power in $k(\sqrt[p]{u})$. (You can ...
0
votes
1answer
67 views

Is the Riemann zeta function $\zeta(s)$ exactly $\pi(x)$?

Let $\pi(x)$ denote the number of primes less than or equal to a certain x value. The prime number theorem says that $x/\log x$ (or more accurately $x/(\log x-1)$) has been the most popular method ...
0
votes
1answer
40 views

Let p be an odd prime number. Then show the following:

Let $p$ be prime, $p \geq 3$. Then show that $K_p$ is the union of $\frac{1}{2}(p-1)C_p$. I am once again at a loss for a starting point. Maybe just a small hint so I can work through this myself ...
6
votes
1answer
83 views

Prove that $2AB$ is square [duplicate]

Let $$A= 1! \cdot 2! \cdot 3! \cdots 1002!$$ $$B= 1004!\cdot 1005! \cdots 2006!$$ Prove that $2AB$ is square. Help guys, I tried, I really did but I couldn't.
0
votes
1answer
32 views

Factorials and trailing zeroes: more methods

I have following problem: how many trailing zeros are in 50! I know that there is a method dividing the number by next powers of 5, for example: 50/5 + 50/25 + 50/125 + ... = 10 + 2 + 0 + ... = 12. ...
4
votes
1answer
52 views

For any $n$ positive integers ($n\geq 5$) exactly 3 or 4 of them are equal to each other modulo $2^m$ for some $m$

How can one prove that for any $n$ distinct positive integers, $n\geq 5$, there exists $m$ such that exactly 3 or 4 of them are equal to each other modulo $2^m$? I tried to prove it for small $n$. ...
1
vote
0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
5
votes
3answers
77 views

Can a square be in the form $2x + 1$, when $x$ is odd?

I was given this question, and I think I have solved it, but I'm not sure it is correct because this differs from how the answer is given. What is the most common way to solve this problem? Let's ...
2
votes
2answers
650 views

True or false identity?

I found the logo from The Eighth Congress of Romanian Mathematicians. I think this is the von Mangoldt summatory function and with a simple computation, using this definition, I obtained $83$. Am I ...
4
votes
0answers
47 views

A sequence avoiding 3-term power progressions

Rankin1 studied sequences of integers that avoid 3-term geometric progressions, $(a, a c, a c^2)$, e.g., $$\{1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, \ldots \} \;$$ So, $18$ is excluded ...
3
votes
1answer
34 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...