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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0answers
17 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 ...
22
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 ...
2
votes
1answer
31 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
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0answers
31 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
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1answer
32 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 ...
2
votes
2answers
161 views

Infinite irrational number sequences?

Is an irrational number, such as $\pi$ or $\sqrt2$, guaranteed to contain every possible digit sequence somewhere within it? Is there no proof for this? Is there any clue as to whether this is so? It ...
4
votes
1answer
48 views
+300

How to see that the prime gaps functions isn't monotonic?

Let $g(n)$ be the distance between the $n$th prime and the next. By elementary means we can see that $g(n)$ is not eventually constant and that $g(n)$ is not strictly monotonic. Further we know that ...
0
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0answers
13 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
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0answers
15 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
40 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
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2answers
101 views

solutions for Diophantine equation for ${k_1} + 2{k_2} + 3{k_3} + … + n{k_n} = n$

Consider the Diophantine equation of the form ${k_1} + 2{k_2} + 3{k_3} + .... + n{k_n} = n$, where ${k_1},{k_2},...{k_n} \in Z^+$ . For a given $n$, how can I obtain the solutions of a given equation? ...
1
vote
1answer
40 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 ...
1
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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: $a^2$ ...
2
votes
1answer
49 views

proof of chinese remainder theorem $x=a_1M_1y_1+…+a_nM_ny_n$?

I can't understand the proof of Chinese Remainder Theorem let $x ≡ a_1 (\text{mod }m_1 ),$ $x ≡ a_2 (\text{mod }m_2 ),$ · · · $x ≡ a_n (\text{mod }m_n )$ such that $m_1,m_2,...,m_n$ are relatively ...
10
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0answers
54 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 ...
3
votes
2answers
42 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
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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
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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 ...
12
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1answer
111 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
15 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
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3answers
229 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
12 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 ...
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 ...
1
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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
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 ...
15
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2answers
153 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
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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. ...
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 ...
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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
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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 ...
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
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1answer
81 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
73 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
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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
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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! $ ?
-1
votes
0answers
29 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
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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 ...
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 ...
-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 ...
1
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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
vote
1answer
39 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
82 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
102 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
733 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
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: $0^2+1^2=1^2$ $3^2+4^2=5^2$ $20^2+21^2=29^2$ $119^2+120^2=169^2$ ...
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) = ...