Questions tagged [number-theory]
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.
28,762 questions
0
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0answers
17 views
Detect if an element is in an orbit.
Suppose $n = pq$ is a product of two distinct primes. We also know that integer $r$ divides $\varphi(n) = (p-1)(q-1)$ and $r^2$ doesn't divide it. Fix element $x$ of $\mathbb{Z}_{n}$. Can we tell ...
1
vote
1answer
23 views
Number of Collatz steps for Mersenne numbers
I noticed that for all $k \in \mathbb{N} \geq 1$ the following is true (I tested up to $2^{5000}$):
$\text{Collatz_Steps}(2^{2k+1} - 1) + 1 = \text{Collatz_Steps}(2^{2k+2} - 1)$
Where $\text{...
-2
votes
1answer
31 views
Help in answering the below mathematical trivia question
can someone help me answer the below question?
Five plus zero is five, and fifty is divisible by five. what is this mathematical trait called
0
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0answers
47 views
Is there interest in this elementary equivalent of the Riemann Hypothesis
It can be shown that the Riemann Hypothesis is equivalent to the following:
Let $B$ be the Meissel-Mertens constant, $p$ prime, and $\theta(x)=\sum_{p\le x}\log p$. Then, $\sum_{p\le x}1/p-\log\log ...
2
votes
3answers
48 views
If $p$ is prime then $p^{n-1}\mid \binom{p^n}{p}$.
I am stuck with this problem:
If $p$ is prime then $p^{n-1}\mid \binom{p^n}{p}$.
The thing is that I don't know much properties of binomial coefficients and I'd accepts hints.
0
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0answers
15 views
Books on polynomial Pell equations
I am interested in polynomial Pell equations and their links with polynomial continued fractions, Padè approximants and the conditions under which they are solvable.
Could you suggest me some books ...
-1
votes
0answers
22 views
Counting without counting [on hold]
In an organization, there are 15 employees, 9 men and 6 women. In how many distinct ways a president, vice-president and a secretary can be selected, providing that the secretary is a woman?
2
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0answers
49 views
5 digits numbers such that when the sum of digits divided by 4 leaves remainder 2.
How many 5 digits numbers such that when the sum of digit divided by 4 leaves remainder 2.
Example:-
Consider a 5 digit number-
(x1,x2,x3,x4,x5)
Then (x1+x2+x3+x4+x5)= must be of form(4n+2)
I tried ...
1
vote
0answers
17 views
A funny inequality problem of Logarithmic integral function and prime number
Let
$E(x)=\left(5.5 \times 10^{9}+2.3 \times 10^{-8} \operatorname{li}(x)+10^{-11} x\right) \log x$,
$E_{1}(x)=\left(5.5 \times 10^{9}+2.3 \times 10^{-8} \operatorname{li}(x)\right) \log x$,
$\...
4
votes
0answers
26 views
Pairs of palindromic primes without $1$ and have a palindromic product
While discussing about prime numbers with other users, I noticed that:
$(1)$ There are very few pairs of palindromic prime numbers that have products which are palindromes.
Ex : $[2, 3], [2, 30203],...
7
votes
0answers
41 views
if $(\{n\alpha\}-c)(\{n\beta\}-c)\ge 0,\forall n\in N^{+}$ then have $\{\alpha\}=\{\beta\}$?
The following question was asked by one of my students.He didn't know the conclusion was correct. I thought for a long time and felt right.Because we seem to be using Kronecker theorem
It follows ...
1
vote
3answers
25 views
Congruence Class of mod p where p is a prime number
If $n =p$ is a prime number and if $[0]\neq[a]$ is in $J_p$ ,
then there is an element $[b]$ in $J_p$ such that $[a][b] =
[1]$. ($J_p$ being the set of congruence classes mod $p$)
This was a remark ...
1
vote
1answer
57 views
Find the smallest $n$ such that the $n$-th prime $p_n \equiv 330 \mod n $.
Find the smallest $n > 1$ such that the $n$-th prime $p_n \equiv 330 \mod n $.
I was investigating the remainders when the $n$-th prime is divided by $n$. For every positive integer $a < 330$, ...
0
votes
0answers
13 views
Binary Quadratic Forms of Discriminant -3
The following is a question in my textbook:
Show that any positive definite binary quadratic form of discriminant $-3$ is equivalent to $f(x, y) = x^2 + xy + y^2$.
Show that a positive integer $n$ ...
1
vote
1answer
25 views
Show that $\operatorname{Cl}(\mathbb{Z}[\sqrt{-5}])$ is of order 2 using a specific lemma.
Show that $\operatorname{Cl}(\mathbb{Z}[\sqrt{-5}])$ is of order 2 using a specific lemma.
$\textbf{Lemma:}$ For a ring of integers $O_k$, $\exists$ positive integer $M$ only depending on $K$ with ...
1
vote
1answer
25 views
Proper Ideals with Norm Relatively Prime to Conductor
Let $K$ be an imaginary quadratic number field, and $\mathcal{O}_K$ the ring of integers. Let $\mathcal{O}$ be an order. Call the $\textit{conductor}$ $f = [\mathcal{O}_K:\mathcal{O}]$.
Given some $\...
0
votes
4answers
59 views
Number Theory theorem regarding ring of integers in $\mathbb{Q}[\sqrt{D}]$
Here is the theorem that I need to prove
For $K = \mathbb{Q}[\sqrt{D}]$ we have
$$\begin{align}O_K = \begin{cases}
\mathbb{Z}[\sqrt{D}] & D \equiv 2, 3 \mod 4\\
\mathbb{Z}\...
2
votes
1answer
32 views
Pell like equation solution
If I am looking for solutions of $x^2-2y^2= 24n+23,$
for $ n \in \mathbb N$ and I want to know if there are patterns to the solutions i.e. perhaps are the $x,y$ dependent on $24n+23$ in any ...
1
vote
0answers
19 views
Meaning of absolute constant
In my analytic number theory course, we discussed Dirichlet Series of Dirichlet Characters. A question I was asked was to show that:
If $\chi_1, \chi_2$ are two Dirichlet Characters of modulo $q_1, ...
3
votes
2answers
31 views
Proof that every common divisor divides GCD (solve only by Bézout's identity)
As part of the course's assignments, we received a task to prove the following sentence using only Bézout identity:
Every common divisor of $a, b$ divides the gcd $(a, b)$.
I tried the following ...
1
vote
1answer
37 views
RSA - statements re $\phi(n)$, $\lambda(n)$ and $n$, given $n$ and $e=65537$
Assume $n=pq$, with $p,q$ primes, $e=65537$, and length of $n$, $|n|=N=1024$ bits = 309 decimal digits. $p,q$ are unknown.
I am trying to understand the information sourced from Wikipedia page on RSA ...
0
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0answers
42 views
How can I calculate $x=y^{\frac{1}z} \pmod n$?
How can I calculate $$x=y^{\frac{1}z} \pmod n$$
I have $y, z$ and $n$, but I really don't have any good ideas on how to get x without brute-forcing it. Is there any good way to calculate it?
-2
votes
3answers
38 views
If $a \equiv b \mod n$ then is $a \equiv b\mod mn$ for any integer $m$? [on hold]
I somehow cannot prove this. Does this hold?
1
vote
1answer
25 views
Solution to the Pell Equation and Pell like Equation
I have read through a lot of similar posts so I am not trying to re ask a question but just seeking some clarity.
I am looking at the Pell and Pell like equations:
$x^2-2y^2=1$ and $x^2-2y^2=k$ ...
0
votes
0answers
8 views
Looking for the explicit formula of a possible bijection from natural numbers to integers. [duplicate]
As a gratuitous exercice, I'm trying to find a way to establish a 1-1 correspondance between the set of natural numbers and the set of integers.
Can it be done in the following way?
My idea would ...
0
votes
0answers
40 views
Computing Integers' Prime Factorization Using the General Number Field Sieve
Recently, I have taken upon myself the task of writing an algorithm to compute the prime factorization of an integer. I am neither a mathematician nor a programmer/computers' engineer as an occupation,...
1
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0answers
25 views
A problem of sum set and difference set
Given a set of $N$ distinct positive integers $ {A}=\left\{a_1,a_2,\cdots,a_N\right\}$. Denote the sum set of elements in $A$ as $B=\left\{a_m+a_n|a_m,a_n\in A \right\}$. Then, denote the difference ...
2
votes
1answer
55 views
Are there integer solutions to the equation ${^n}a+{^n}b={^n}c$?
A couple days ago, someone posted a question about using integer solution to the equation $a^a+b^b=c^c$ to disprove Fermat's last theorem. The question has since been deleted but I was curious as to ...
4
votes
0answers
58 views
Are 2, 3 the only prime numbers that don't have the digit 1 and are palindromes whose squares are also palindromes?
While thinking about prime numbers, I noticed that:
$(1)$ Very few prime numbers have squares that are palindromes.
Ex: $2$, $3$, $11$, $101$, $307$
$(2)$ Even rarer are prime numbers that are ...
1
vote
1answer
64 views
How many positive integer solutions?
I'm interested in ${\bf integer}$ solutions of
$$abcd+1=(ecd-c-d)(fab-a-b)$$
subject to ${\bf a,b,c,d \geq 2}$, and ${\bf e,f \geq 1}$.
A few comments:
(1) If $e,f \geq 2$ or $e=f=1$ then there ...
3
votes
2answers
79 views
Distribution of prime numbers modulo $4$
Are primes equally likely to be equivalent to $1$ or $3$ modulo $4,$ or is there a skew in one direction?
That is my specific question, but I would be interested to know if there exists a trend more ...
-2
votes
2answers
36 views
Show that the five digit number abcde is congruent (mod11) to $(a + c + e) - (b + d)$ [on hold]
Show that the five digit number $abcde$ is congruent (mod $11$) to $(a + c + e) - (b + d)$
8
votes
0answers
53 views
Are there infinitely many solutions such that the digit sum of a prime power is a smaller power of the same prime?
While discussing prime powers and divisors, I came up with the following problem.
Examples
$\to$ prime $p=3$
digit sum (in base ten) of $p^3=27$ is $p^2=9$, a power of $p$,.
$\to$ prime $p=7$
...
-1
votes
3answers
47 views
Prove: $\gcd(b,a) = \gcd(b,c)$ if $c \equiv a \pmod{b}$ [duplicate]
$b$ is an integer where $b > 1$ and $a, c$ are integers.
Prove: $\gcd(b,a) = \gcd(b,c)$ if $c \equiv a \pmod{b}$
I am completely stumped on where to start. Any help is appreciated.
0
votes
1answer
25 views
degree extension over filed of $p$-adic numbers
Let $K = \mathbb{Q}(\theta)$ is a numberfiled and $[K:\mathbb{Q}]=n$. When $\mathbb{Q}_p$ is the field of $p$-adic numbers and $K_p=\mathbb{Q}_p(\theta)$, what about $[K_p : \mathbb{Q}_p]$?
-1
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0answers
33 views
Let m and n be coprime integers. Prove that of the m+n−2 fractions below, precisely one lies inside each of the open intervals [on hold]
Let m and n be coprime integers. Prove that of the m+n−2 fractions below, precisely one lies inside each of the open intervals (1, 2), (2, 3), . . . , (m + n − 2,m+n−1):
m+n/m, 2(m+n)/m, ..., (m−1)(m+...
1
vote
0answers
17 views
Is there a closed formula for $b$ in $2 \sum_{n=1}^{x-1} \sum_{i=n+1}^x (-1)^{n+i} \frac{\cos(b \ln(\frac{n}{i}))}{\sqrt{n}}$?
I was wondering if there was a closed formula for b such that $$\lim_{x \to \infty} 2 \sum_{n=1}^{x-1} \sum_{i=n+1}^x (-1)^{n+i} \frac{\cos(b \ln(\frac{n}{i}))}{\sqrt{n}} = - \sum_{n=1}^x \frac{1}{n}$$...
5
votes
1answer
93 views
Show that $x^2 - 79y^2 = \pm 3$ has no solutions in $\Bbb Z$
I am trying to compute the ideal class group of $\Bbb Q(\sqrt{79})$ and this came up:
"Show that $x^2 - 79y^2 = \pm 3$ has no solutions in $\Bbb Z$ using congruences."
I've tried using quadratic ...
1
vote
0answers
48 views
Does this explicit formula for the prime-counting function $\pi(x)$ converge?
This question is related to an answer I posted earlier at the following link.
Explicit Formula for $\pi(x)$
A potential explicit formula for the fundamental prime-counting function $\pi(x)$ is ...
0
votes
0answers
33 views
Advice on Henri Cohen's Number Theory books
I am a graduate student willing to learn Number Theory. I have come across Henri Cohen's "Number Theory" (Vol. I and II) and I would like to hear from someone who has read these books before: are they ...
0
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0answers
21 views
Question about primitive roots and multiplicative groups
Let $p$ be an odd prime and let $k \in \mathbb{N}$. We know that $U(\mathbb{Z}_p) = \{\overline{1},\overline{2},\dots,\overline{p-1}\}$. Can it happen and when that $U(\mathbb{Z}_p) = \{\overline{1}^k,...
5
votes
1answer
72 views
Request for info about Real Analysis to a beginner [on hold]
So I'm a junior HS student very interested in mathematics and I think I have pretty good chances to get admitted into Stanford online HS and take a university level real analysis course that's worth 5 ...
1
vote
0answers
36 views
Find all triplets of integers $(a, b, c)$ such that $3^a + 3^b + 3^c$ is a square. [on hold]
Find all triplets of integers $(a, b, c)$ such that $3^a + 3^b + 3^c$ is a square. How can you find them?
0
votes
1answer
35 views
$\lfloor \frac{n}{27} \rfloor=\lfloor \frac{n}{28} \rfloor$ [duplicate]
If $n$ be a non-negative integer then solve for $n$:$$\left\lfloor \frac{n}{27} \right\rfloor=\left\lfloor \frac{n}{28} \right\rfloor$$
My Attempt:
$$\left\lfloor \frac{n}{27} \right\rfloor=\left\...
1
vote
3answers
42 views
Show $2 = (2, 1+\sqrt{-5})^2$ in $\mathbb{Z}[\sqrt{-5}]$ [duplicate]
Show $2 = (2, 1+\sqrt{-5})^2$ in $\mathbb{Z}[\sqrt{-5}]$.
Apologies if this may seem a trivial question, however, I am some difficulty showing this.
My Attempt:
$(2, 1+\sqrt{-5}) * (2, 1+\sqrt{-5})...
7
votes
3answers
101 views
Solve for x: $x^3-\lfloor x\rfloor=5$
Solve for x:$$x^3-\lfloor x\rfloor=5$$
My Attempt:
$x^3-5=\lfloor x\rfloor$
Now, $x-1<\lfloor x\rfloor\leq x$
$x-1<x^3-5\leq x$
Not able to proceed from here
0
votes
0answers
12 views
Determine the set of integers that are represented by the binary quadratic form (1,0,-1) [duplicate]
I need help with finding the set of integers represented by the form (1,0,-1).
This is essentially f(x,y) = x^2 - y^2 which can be factorised into (x + y)(x - y) and the determinant is d = 4 > 0.
0
votes
1answer
53 views
Is $2(2^{p} − 1)$ a divisor of $n$? How about $2^2(2^p − 1)$? Finish the proof. [on hold]
A positive integer is called perfect if it equals the sum of its positive divisors.
Example: $6$ is perfect because the divisors of $6$ are $1,2$ and $3$ and $6=1+2+3$.
Show that $28$ is perfect.
...
-1
votes
1answer
29 views
Prove that the binomial coefficient is composite [on hold]
Let $n > 3$ and $k$ be integers, with $1 < k < n-1$. Prove that the binomial coefficient is composite.
1
vote
1answer
24 views
Analogue of normality for continued fractions
For our traditional representation of numbers we can study properties of the digits such as normality: Normal number (Wikipedia).
This is easily extended to other bases but how about the rather ...
