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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6
votes
2answers
39 views

How many integer-sided right triangles are there whose sides are combinations?

How many integer-sided right triangles exist whose sides are combinations of the form $\displaystyle \binom{x}{2},\displaystyle \binom{y}{2},\displaystyle \binom{z}{2}$? Attempt: This seems like ...
1
vote
0answers
37 views

Integer Factorization problem - New Idea

I been thinking about slightly different approach of solving the problem, and I want you to tell me if my idea is reasonable and if it's original(If someone already thought about this, I would be ...
0
votes
1answer
33 views

Number theory and primitive roots

I wanted to find the primitive roots of 35 . What i know is that we find the euler totient of 35 which is 24 and we check the divisors of 24 to be the powers but it was a great time consuming ... how ...
0
votes
0answers
14 views

Is there any significance in such Heegner numbers (or class number 1) representation symmetry?

$\mathrm{A003173}(n) = 1+((1 + \sqrt{3})^{n-1} - (1 - \sqrt{3})^{n-1})/(2\sqrt{3})$ for n = 1,2,3,4. $\mathrm{A003173}(n) = 19+24((1 + \sqrt{3})^{n-6} - (1 - \sqrt{3})^{n-6)})/(2\sqrt{3})$ for n = ...
0
votes
0answers
16 views

SAGE function for calculating mod p reductions of a modular form [on hold]

In Magma there exists a Reductions(f,p) command, which for any modular form $f$ defined over a number field $K$ and a prime $p\in\mathbb{Z}$, outputs all the $``f \mod \mathfrak{p}''$ reductions for ...
4
votes
0answers
44 views

if $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$?

is it true that for any $k,l\in\{2,3,4,\dots\}$, $k\neq l$, if $x\in\mathbb{R}$ satisfies $x^k-x\in\mathbb{Z}$ and $x^l-x\in\mathbb{Z}$, then $x\in\mathbb{Z}$? This is a generalisation of if ...
2
votes
1answer
32 views

arbitrarily long sequences without perfect powers

The fact that there are arbitrarily long sequences of consecutive numbers without prime number is well known, the proof is easy and goes like this: let $n\ge 2$, then the number $n!+k$ is greater ...
1
vote
1answer
19 views

Product of Quadratic Residues

If a is a quadratic residue, and ab is a quadratic residue, how can I show that b is also a quadratic residue? Would appreciate a hint. So far I thought about the problem a little and I have: $a^2$ ...
10
votes
0answers
56 views

Numbers that are clearly NOT a Square

Although I have never studied math very seriously, I have heard of Brocard's Problem, which asks for integer solutions for the following Diophantine Equation:$$n!+1=m^2$$ The only solutions are ...
24
votes
9answers
1k 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 ...
1
vote
1answer
41 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 ...
1
vote
1answer
41 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
2answers
13 views

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

$a =\prod_{i = 1}^{r} p_{i}^{a_i}$ 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 ...
0
votes
1answer
17 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
0answers
52 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
2answers
43 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 ...
3
votes
3answers
53 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 ...
1
vote
1answer
70 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
3answers
230 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 ...
1
vote
0answers
13 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 ...
1
vote
3answers
43 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. ...
-1
votes
0answers
20 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
42 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! $ ?
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?
18
votes
1answer
127 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 ...
0
votes
2answers
51 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 ...
1
vote
0answers
34 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
votes
0answers
30 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 ...
-3
votes
2answers
84 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 ...
2
votes
1answer
58 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
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 ...
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
56 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
63 views

I need a best proof that e is a transcendental? [on hold]

Where can I find the best proof that $e$ is transcendental?
0
votes
1answer
40 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.
0
votes
1answer
44 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 ...
4
votes
0answers
40 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
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 ...
1
vote
0answers
23 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 $\theta>1$ ...
13
votes
1answer
83 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: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
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) - ...
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
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?
1
vote
1answer
46 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 ...
0
votes
0answers
28 views

speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$

I would like to find the speed of divergence of $\prod_{m=1}^n (\frac{z}{m} + 1)^m$ for any $z$. For example, if $|z| < 1$, doing taylor expansion we know that it is roughly $e^{zn}.$ But I need it ...
1
vote
1answer
71 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 $x,y$, $|3^x-17^y|>2$.
0
votes
1answer
33 views

Is/when is this property about the totative subset of $(\mathbb{Z}_n , +_n)$ true?

Is it possible to show (or when is it true); that for the group $\mathbb{Z}^+_n :=(\mathbb{Z}_n , +_n) $, there exists an $a \in \mathbb{Z}^+_n$ for each $z \in \mathbb{Z}^+_n$, where both $z+_n a$ ...
-1
votes
0answers
45 views

Can this number be rational?

Let $K = e^{-\gamma}\log\log n$, where $\gamma$ is the Euler-Mascheroni constant and $n\geq 2$ is a positive integer. Can $K$ be rational for any integer $n\geq 2$ ? I seem not to find any argument ...
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 ...