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)

6
votes
3answers
79 views

Is there any integral for the Golden Ratio?

This is a curiosity. I was wondering about math important/famous constants, like $e$, $\pi$, $\gamma$ and obviously $\phi$. The first three ones are really well known, and there are lots of integrals ...
3
votes
2answers
36 views

Let $C$ be the set of all complex numbers of the form $a+ b \sqrt {5}i$, where $a$ and $b$ are integers…

Let $C$ be the set of all complex numbers of the form $a+ b \sqrt {5}i$, where $a$ and $b$ are integers. Prove that $7$, $1 + 2\sqrt {5}i$, and $1 - 2\sqrt {5} i$ are all prime in $C$. -I am really ...
1
vote
1answer
67 views

A problem of decimals..

The exact problem: For any natural number n>1, write the infinite decimal expansion of $\frac{1}{n}$ (for example, we write 1/2 = $0.4\overline9$ as it's infinite decimal expansion, not 0.5). ...
1
vote
1answer
34 views

number pair's in the self-root function $f(x) = x^{1/x}$

in the self-root function $f(x) = x^{1/x}$ the output is in pairs of numbers i.e. $f(2) = f(4)$ , the inputs are 2 apart producing the same output , the square root 2 is equal to the 4th root of 4 ...
11
votes
1answer
102 views
+100

Is there a polynomial such that $F(p)$ is always divisible by a prime greater than $p$?

Is there an integer-valued polynomial $F$ such that for all prime $p$, $F(p)$ is divisible by a prime greater than $p$? For example, $n^2+1$ doesn't work, since $7^2+1 = 2 \cdot 5^2$. I can see that ...
0
votes
1answer
24 views

What are the invariants of a number field? [on hold]

How is defined an invariant of a field? Given a certain field extension $L/K$, is it related with the Galois group ${\rm Gal}(L/K)$? In the case of number fields, which are the invariants associated ...
0
votes
1answer
13 views

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…

$a =\prod_{i = 1}^{r} p_{i}^{ai}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$, deduce a formula for the sum of the squares of the positive divisors of $a$. -The section we ...
1
vote
3answers
222 views

What is wrong with this infinite sum [on hold]

We know that: https://www.youtube.com/watch?v=w-I6XTVZXww $$S=1+2+3+4+\cdots = -\frac{1}{12}$$ So multiplying each terms in the left hand side by $2$ gives: $$2S =2+4+6+8+\cdots = -\frac{1}{6}$$ This ...
0
votes
2answers
10 views

$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$…

$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ with $a_{i} > 0$ for each $i$ is the canonical representation of $a$,Prove that $a$ is the square of an integer if and only if $a_i$ is even for each $i$. -The ...
2
votes
1answer
54 views

Norm restricted to $\mathbb Q$

Consider an algebraic extension $K$ of $\mathbb Q$ and suppose that on $K$ we have a non trivial norm $||\cdot||$. Is it possible to have that the restriction $||\cdot||_{\mathbb Q}$ is the trivial ...
0
votes
0answers
47 views

Any good math books? [on hold]

I was wondering if there are any books about a bunch of random math theories, areas and topics. For example topics like group theory, randomness. klein bottles and other interesting things
3
votes
3answers
43 views

Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$

Show that we cannot have a prime triplet of the form $p$, $p + 2$, $p + 4$ for $p >3$ I was a bit lost with this proof until I found a similar looking proof-based question from a previous ...
14
votes
2answers
141 views
+300

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 ...
1
vote
3answers
39 views

What is the significance of using prime numbers in proving: $x$ is a multiply of $y$?

I came to a problem where it asks me to prove, for example, $n^4-n^2$ is a multiple of $12$. Now, factorize the multiple: $n\times n\times (n-1)\times (n+1)$. Here we have $3$ consecutive integers. ...
18
votes
1answer
122 views

Is there any palindromic power of $2$?

My question is in the title: Is it possible to find $n≥4$ such that $2^n$ is a palindromic number (in base $10$)? A palindromic number is a number which is the same, independently from which ...
-1
votes
0answers
19 views

$k$ be a positive integer ; $\{a_n\}$ be a sequence of positive integers such that $\gcd(m,n)=1 \implies a_m^k+a_n^k|m^k+n^k$ ; then $a_n=n$?

Let $k$ be a positive integer ; $\{a_n\}$ be a sequence of positive integers such that g.c.d.$(m,n)=1 \implies a_m^k+a_n^k|m^k+n^k$ ; then is it true that $a_n=n , \forall n \in \mathbb N$ ?
1
vote
0answers
11 views

Is there K and an infinite amount of different primes $a_i,b_i$ so that min|$a_i^y-b_i^x$| <K on natural x,y for all i?

First of all I know that it was proved recently that prime gaps are less than like 7 million for an infinite amount of primes, but I'm not smart enough to follow the proof. I am looking for a ...
0
votes
4answers
39 views

No solutions to diophantine equation

I am trying to deduce that $x^2-5y^2=0$ having shown that $x^2 \equiv 5y^2 (mod 7)$ has no integer solutions (not 0). How do I go about this?
0
votes
1answer
79 views

Solving Quadratic Diophantine Equation with initial solutions.

I have read here about a method to generate integer solutions for a Diophantine equations like the following: \begin{align*} an^2+bn+c = d^2 \end{align*} By knowing an initial integer solution for it. ...
0
votes
2answers
70 views

Range of inverse harmonic mean of two integers

Today I was solving an exercise and one of the things I tried (which later turned out to be useless) involved considering the following: Is there a simple way to describe in terms of $n$ the range of ...
0
votes
1answer
39 views

Is the Euler prime of an odd perfect number a palindrome (in base $10$), or otherwise?

Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $\gcd(q,n)=1$ and $q \equiv k \equiv 1 \pmod 4$). (That is, $2N=\sigma(N)$ where $\sigma$ is the ...
1
vote
0answers
41 views

Is it true that for every positive integer $m$ , there exist a positive integer $n$ such that $\phi(n)=m! $ ?

Is it true that for every positive integer $m$ , there exist a positive integer $n$ such that $\phi(n)=m! $ ?
-1
votes
0answers
27 views

Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers upto $\frac{1}{2}p^2_x -2$ exvert all non-trivial $\Xi_x$-classes?

This question follows from the information provided below. Are there any ways we can determine whether the $\Xi_x$-classes of natural numbers up to $\frac{1}{2}p^2_x -2$ exvert all non-trivial ...
0
votes
1answer
41 views

Does the product of two numbers with a primitive representation have a primitive representation?

I know the theorem that $n = x^2 + y^2, \, \textrm{gcd}(x, y) = 1 \iff p | n \implies p \equiv 1 \bmod 4$. We call an expression of $n$ in this form primitive. I'm trying to prove the statement. I've ...
0
votes
2answers
47 views

Prove Expression cannot be factored

I am currently working on primes which can be expressed in form of a polynomial. For eg, Find all primes which can be expressed in form $n^4-52n^2+595$ It is very essential to tell whether a ...
-3
votes
2answers
81 views

Investigating Nicolas' criterion for the Riemann Hypothesis. [on hold]

Throughout this note, $N_k$ denotes the $k$-th primorial number (the product of the first $k$ primes), $\varphi(n)$ the Euler totient function, and $\gamma$ is the Euler-Mascheroni constant. By the ...
1
vote
0answers
30 views

Find the minimum number of tickets to guarantee the win of a n-bit binary lottery?

Here's the problem. I just don't know how to approach it. If the 'one error tolerance' were removed, then this would be a simple binomial distribution problem. But now I can't figure it out. In ...
1
vote
1answer
38 views

Prove that $\operatorname{lcm}(ak, bk) = k \cdot \operatorname{lcm}(a, b) $.

Prove that $\operatorname{lcm}(ak, bk) = k \cdot \operatorname{lcm}(a, b) $. How to prove the above statement? I have tried writing out the lcm relationships as a series of 'divides' ...
2
votes
2answers
779 views

question about prime numbers [on hold]

Prove that for all odd prime numbers $p_1$ and $p_2$, there exist prime numbers(exclude 2) $p_3$ and $p_4$ such that $$p_3 + p_4 = p_1 + p_2 + 2.$$ Hints would be appreciated.
0
votes
1answer
25 views

Incipit of chapter VI of Neukirch's ANT book.

The title of the chapter VI of the neukirch's ANT is "Global class field theory", and the first few lines are the following: the author doesn't explain what is $K$ here, but from the previous ...
3
votes
2answers
80 views

Kummer Theory - Example of Subgroup of $K^{*}$ containing $K^{*m}$ for global fields.

I am trying to understand Kummer theory and I wish to apply it to global fields, so our field $K$ containing $\mu_m$ should be $\mathbb{Q}(\zeta_m)$. Let $B$ be a subgroup of $K^{*}$ containing ...
0
votes
0answers
97 views

How to use an inverse Mellin transform to get to the $\mathrm{core}(n)$?

From this question here: Moreover, if multiplicative function $\mathrm{core}(n)$ is defined to map positive integers "$n$" to square-free numbers by reducing the exponents in the prime power ...
22
votes
2answers
731 views

Seeking proof for the formula relating Pi with its convergents

Could anyone try to prove that the below conjectured formula is valid for relating $\pi$ with ALL of its convergents - those, which are described in OEIS via $\mathrm{A002485}(n)/\mathrm{A002486}(n)$ ...
13
votes
1answer
80 views

Pythagorean triplets of the form $a^2+(a+1)^2=c^2$ and the space between them

I was searching for pythagorean triples where $b=a+1$, and I found using a python program I made the first 10 integer solutions: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
1
vote
2answers
23 views

Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$

Find all $7$-digit numbers which use only the digits $5$ and $7$ and are divisible by $35$. Attempt It is easy to see that all numbers of this form must be of the form _ _ _ _ _ _ 5. Working ...
0
votes
3answers
31 views

Positive integer not a power of 2

It's given that if a positive integer $n$ is Not a power of two, then $n$ must have an odd prime factor, meaning $$n = pr, p>2, 1\leq r< n $$ Is it really this trivial? There's a proof that ...
-7
votes
0answers
54 views

Has Brochard's Problem been solved? I'm sure it has… [on hold]

I had a conversation with him, and Wan Chan solved it. Proof: Let $n! +1 = m^2$; $n! = m^2 -1$; $n! = (m +1)(m -1)$. Let $m +1 = k$, and $n! = k*(k -2)$. Thus, for $n = 4$, $4! = 1*2*3*4 = 6*(6-2) = ...
0
votes
1answer
62 views
0
votes
1answer
39 views

If $p = a^2 + b^2$, prove that $(ab^{-1})^2 \equiv -1 \pmod{p}$

Let $p \equiv 1 \pmod{4}$ be a prime, where $p = a^2 + b^2$. Show that $(ab^{-1})^2 \equiv -1 \pmod{p}$ I'm having trouble with this question. Any help is appreciated.
1
vote
0answers
30 views

How to show that a set of elements is a basis for the ring of integers of a number field?

Let $K$ be a number field of degree $n$ (that is $[K:\mathbb{Q}]=n$) with ring of integers $\mathcal{O}_K$. I know that there exists algorithms to find $\mathcal{O}_K$ and hence determine a ...
0
votes
0answers
28 views

How common are diophantine equations for which the local global principle is invalid?

The local global principle says that in some families of diophantine equations the solvability over the rationals is equivalent to solvability over the reals and in p-adic fields $Q_p$ for each prime ...
4
votes
0answers
39 views

AMM 2488: Primitive Root Relatively Prime to p-1

(from American Mathematical Monthly, problem 2488. I hope this hasn't been posted before but I'm new and maybe not very good at using the search function effectively..) Let $p>3$ be a prime. Show ...
1
vote
1answer
68 views

Given M, can we find $2$ primes $a,b$ so that for all naturals $x,y$, $|a^x-b^y|>M$?

For example, if $M = 2$, one can show that $3,17$ satisfy the above: For any naturals $x,y$, $|3^x-17^y|>2$.
1
vote
0answers
22 views

On an inequality involving primorial numbers.

Let $N_k$ denote the $k-th$ primorial number. That is, the product of the first $k$ primes and $\phi(n)$ be the Euler totient function. How can one show that there exists a constant $\theta>1$ ...
2
votes
4answers
41 views

Showing that Harmonic numbers are $\Theta(\log n)$, intuitively

I wish to verify that Harmonic numbers $H_n = \sum_{k=1}^{n} \frac{1}{k}$ are $\Theta(\log n)$. One idea I have is to approximate the sum with an integral: $$\int_{1}^{n} \frac{1}{k} ~dk = \log(n) - ...
7
votes
2answers
131 views

Is $k+p$ prime infinitely many times?

I have the following conjecture: Let $k\in\mathbb{N}$ be even. Now $k+p$ is prime for infinitely many primes $p$. I couldn't find anything on this topic, but I'm sure this has been thought of ...
1
vote
1answer
23 views

Is solvability of diophantine equations over a p-adic field decidable?

As far as I understand, the decidability of solvability of diophantine equations over the rationals is an open problem. What about the decidability of solvability over a given p-adic field?
0
votes
2answers
27 views

Congruence problem $12x\equiv3\pmod{45}$ [on hold]

$$12x\equiv3\pmod{45}$$ Find all possible solutions to above congruence and show procedure in detail.
1
vote
1answer
44 views

Show that $st$, $(s^2-t^2)/2$ and $(s^2+t^2)/2$ are relatively prime.

Let $s$ and $t$ be odd integers. Show that $st$, $(s^2-t^2)/2$ and $(s^2+t^2)/2$ are relatively prime. I've seen this question on here, but unfortunately some of the cases were not covered, and I ...
39
votes
8answers
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 ...