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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2answers
512 views

How many prime numbers are also triangular numbers?

I've been trying to figure this out and it's been getting on me myself. I know that $3$ is not just a prime number, but also a triangular number. I'll now add a sequence: Prime numbers: $2, 3, 5, ...
0
votes
1answer
60 views

Bowers array notation : $f_{\omega^\omega}(n)\approx [n,…,n]$ ($n$ times)

I learnt at this site that $$\large f_{\omega^\omega}(n)\approx \underbrace{[n,...,n]}_{n\ n's}$$ For a simular approximation $$\large f_{\omega^2}(n)\approx \underbrace{n\rightarrow ...
13
votes
0answers
199 views

Algorithm to find primes up to $n$ in $O\left(\frac{n}{\log n}\right)$?

Consider the problem of given an integer $n$, generating a list of the primes not greater than $n$. An optimized version of the Sieve of Eratosthenes can do such task in $O(n)$, while the more modern ...
0
votes
1answer
154 views

Gradually rising or falling numbers

I'm looking for a number series I can use for gradually rising or falling numbers. The number series should not be linear and should converge to a number at some point. $\sqrt[N]{N}$ where $N > ...
0
votes
1answer
49 views

Show that there exists $s, t \in S$ such that $\gcd(s, t)$ is a prime

Let $S$ be a set containing finitely many positive integers greater than 1 with property: for all $n \in \mathbb{Z_+}$, there exist $s \in S$ such that $\gcd(s, n) = 1$ or $\gcd(s,n) = s$. Show that ...
2
votes
1answer
47 views

Numbers relative to their sum of Divisors

Define the D-Ratio as the ratio of a natural number $n$ as: the sum of $n$'s Divisors, excluding 1 and $n$ divided by $n$ itself. [Thus the D-Ratio of $24$ is $$\frac{2 + 3 + 4 + 6 + 8 + 12}{24} = ...
7
votes
0answers
125 views

Dividing the whole into a minimal amount of parts to equally distribute it between different groups.

Suppose we have a finite amount of numbers $x_1, x_2, ..., x_n$ ($x_i\in\mathbb{N}$) and an object that should be divided into parts in such a way that it can be without further dividing distributed ...
0
votes
1answer
45 views

how to solve 192-2a^2-a=m(6a+1)?

how to solve $192-2a^2-a=m(6a+1)$ ? or written as $(192-2a^2-a) \equiv 0$ (mod $6a+1$) how to calculate the integer values of $a < 41$ ? thanks to understand that serving: ...
2
votes
2answers
50 views

Is the sequence $\{0,2,6,12,20,30,…,n(n+1)\}$ admissible for every natural $n$?

Look here : https://en.wikipedia.org/wiki/Prime_k-tuple for the definition of an admissible sequence. I wonder if the sequence of differences of primes can be $\{0,2,4,6,8,...,2n\}$ for every ...
1
vote
1answer
33 views

raising elements of profinite groups to $p$-adic powers

Let $\widehat{F_2}$ be the profinite free group of rank 2, and let $\widehat{\mathbb{Z}}$ be the profinite completion of $\mathbb{Z}$, and $\widehat{\mathbb{Z}}^\times$ its group of units. For ...
58
votes
1answer
3k views

$n!$ is never a perfect square if $n\geq2$. Is there a proof of this that doesn't use Chebyshev's theorem?

If $n\geq2$, then $n!$ is not a perfect square. The proof of this follows easily from Chebyshev's theorem, which states that for any positive integer $n$ there exists a prime strictly between $n$ and ...
2
votes
2answers
65 views

How do I find(isolate) the n-th prime number?

So I wanted to solve this SPOJ problem and I did some research about finding the n-th prime number. This formula came across and it stated that the n-th prime must be in this range: $n \ln n + ...
68
votes
12answers
12k 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
4answers
74 views

Books on Prime numbers

I am a graduate student and have just finished Burton's book on number theory. Now I want to read further on prime numbers. Does anyone have any suggestion?
0
votes
2answers
59 views

Prove that $\mathbb{Z}[i]$ consists precisely of the elements of $\mathbb{Q}(i)$ which satisfy $x^2 + ax + b=0$, $a,b \in \mathbb{Z}$

I was reading Neurkich's "Algebraic Number Theory" and there was a proof in it that makes no sense. Proposition 1.5: $\mathbb{Z}[i]$ consists precisely of the elements of the extension field ...
4
votes
1answer
32 views

Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer

I'm currently reading Marcus' "Number Fields," and I'm having difficulty proving the following result: Corollary 2.2: Let $m$ be a squarefree integer. The set of algebraic integers in the quadratic ...
3
votes
2answers
64 views

Prove that for every natural number $n > 2$ there is a prime number between $n$ and $n!$

So I have already read this page with the solution: For all $n>2$ there exists a prime number between $n$ and $ n!$ Now I was able to reason that $p < n!$ Because I was given the hint that ...
-2
votes
1answer
32 views

Two Vertical Lines

What does two single vertical lines mean in math. I am thoroughly confused by this question: What describes |3/1|? Use all that apply.
1
vote
0answers
37 views

If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

(Note: This question has been cross-posted from MO.) Let $\sigma$ be the classical sum-of-divisors function. A number is said to be perfect if $\sigma(N)=2N$. If $q^k n^2$ is an odd perfect number ...
7
votes
1answer
68 views

Can the determinant of an integer matrix with a given row be any multiple of the gcd of that row?

Let $n\geq2$ be an integer and let $a_1,\ldots,a_n\in\mathbb Z$ with $\gcd(a_1,\ldots,a_n)=1$. Does the equation ...
0
votes
3answers
60 views

G is finite group. Need to proof that exists natural k that $g^k = e$ [duplicate]

How do I prove that in a finite group G, for each element in G there is natural power (say $k$) which depends on g,such that $g^k=e$ ? I need to show the existence and the dependence on which $g$ I ...
1
vote
1answer
40 views

Integral of polynomial related to prime divisors

Given the following integral $I_{m,n}=\int_{0}^{1}(1-x^n)^m \mathrm{d}x$. Prove that for any fixed $n$ and for any $m$ $I_{m,n}$ is a rational number and when written in the form $\frac{p}{q}$ with ...
5
votes
1answer
55 views

Missing values of the ratio $\frac{(x+y+z)^2}{x^2+y^2+z^2}$

Let $x,y,z$ be some positive integers. Is it true that we cannot find any positive integer $n$ for which $$ \frac{(x+y+z)^2}{x^2+y^2+z^2}=1+\frac{2}{3n}\,\,? $$
2
votes
3answers
46 views

Fastest way to perform this multiplication expansion?

Consider a product chain: $$(a_1 + x)(a_2 + x)(a_3 + x)\cdots(a_n + x)$$ Where $x$ is an unknown variable and all $a_i$ terms are known positive integers. Is there an efficient way to expand this?
3
votes
2answers
78 views

If $x^{100}$ is 31 digit number Then $x^{1000}$ contains how many digits.

If $x^{100}$ is 31 digit number Then $x^{1000}$ contains how many digits. Our Approach: $10^1$ has $2$ digits = $10$ $10^2$ has $3$ digits = $100$ $10^3$ has $4$ digits = $1000$ $10^{30}$ has ...
4
votes
1answer
60 views

Each number in $S\subseteq \{1,\ldots,2n\}$ does not divide another one, with $|S|= n$. In how many ways?

Let $f(n)$ be the number of subsets $S\subseteq \{1,2,\ldots,2n\}$ such that $|S|=n$ and $a$ does not divide $b$ whenever $a,b \in S$ are distinct. Can we evaluate $f(n)$, at least asimptotically? ...
0
votes
0answers
31 views

Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ with given conditions

If for any series $b(n)=2b(n-1)$ when $b(n)$ is odd number and $b(n)=b(n-1)$ if $b(n)$ is even number. then Find the value of $b(100)-b(97)-b(96)$ if $b(0)=1$ Our Approach: I could not ...
2
votes
0answers
26 views

What will be the last number of the set B in which a set B={$2$,$3$,$5$,$6$,$7$,$10$,_ ,_ ,_______} contain $300$ nos.

A set B={$2$,$3$,$5$,$6$,$7$,$10$,_ ,_ ,_______} contain $300$ nos. in which squares and cube of the no. are eliminated. then what will be the last number of the set B? Our Approach: As we have ...
1
vote
1answer
58 views

Smallest twin-prime-pair exceeding $10^{1000}$

I found the twin-prime-pair $$\large 10^{1000}+9705092\pm 1$$ with PARI/GP. Is this the smallest twin-prime above $10^{1000}$ ? A general question to the search of twin primes : The prime number ...
0
votes
0answers
55 views

why $\frac{a}{b}\pmod p=\frac{a\pmod p}{b\pmod p}$

It is said this following is theorem? what's this name? and How to prove it? Thanks show that $$\dfrac{a}{b}\pmod p=\dfrac{a\pmod p}{b\pmod p},a,b\in N^{+},(a,p)=1,(b,p)=1$$
0
votes
0answers
26 views

Issue with modular arithmetic problem [on hold]

So I have a problem with this question I was doing. I found that $94^6+32\cdot28^6$ is divisible by 2013, using a calculator. Since 61 divides 2013, 61 also divides $94^6+32\cdot28^6$. However, i ...
1
vote
1answer
86 views

regularization of sum $n \ln(n)$

I was testing out a few summation using my previous descriped methodes when i found an error in my reasoning. I'm really hoping someone could help me out. The function which i was evaluating was ...
3
votes
3answers
197 views

What is the Max value of n when 185! is divided by (189^n) will give an Integer Value?

What is the Max value of n when $185!$ is divided by $(189^n)$ will give an Integer Value? Options are a) $91$ b) $30$ c) $36$ d) $24$ MyApproach: $189$=$3^3$ . $7$ When $185$/$3$=$61$ ...
1
vote
0answers
26 views

Is there a tighter approximation for the least prime gap of a given length?

This link https://primes.utm.edu/notes/gaps.html gives a definition of the maximal gaps. For a number $g$ , $p(g)$ is the smallest prime $p$ followed by at least $g$ composites. The estimate is ...
1
vote
2answers
43 views

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$?

$z=100^2-x^2$. Then, how many values of $x,z$ are divisible by $6$? My approach: For $x=1$, $z$ is not divisible by $6$. For $x=2$, $z$ is divisible by $6$. For $x=3$, $z$ is not divisible by ...
0
votes
0answers
57 views

If a, b are positive integers and $(ab - 1) \mid (a² + b²)$ then prove that $q = \frac{a² + b²}{ ab - 1} = 5$. [closed]

If a, b are positive integers and $(ab - 1) \mid (a² + b²)$ then prove that $q = \frac{a² + b²}{ ab - 1} = 5$. I know it has something to do with number theory
6
votes
1answer
115 views

Ramification in $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$

Let $\pi\ne1+i$ be a prime element of $\mathbb Z[i]$. I am interested in the ramification in the extension $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$, especially over $(1+i)$. I've tried for instance to ...
-1
votes
1answer
30 views

Is there a number which describes the approaching ratio of twin primes to other primes? Or a formula for the change in density of twin primes?

Could someone shed some light on what we know about the density of twin primes? I find that it seems to be empirically true that the density of prime gaps increases as $\log(x)$ does for any gap. ...
5
votes
0answers
92 views

Diophantine equation: $13^x+3=y^2$

$$13^x+3=y^2\iff \left(4+\sqrt{3}\right)^x\left(4-\sqrt{3}\right)^x=\left(y+\sqrt3\right)\left(y-\sqrt3\right)$$ $\gcd\left(y+\sqrt3, y-\sqrt3\right)=1$, therefore ...
2
votes
3answers
108 views

Probability that the eventually a six on a dice will appear.

Dave rolls a fair six-sided die until a six appears for the first time. Independently, Linda rolls a fair six-sided die until a six appears for the first time. Let $ m$ and $ n$ be relatively prime ...
-1
votes
1answer
38 views

LCM of randomly selected integers

What is the expected LCM of 21 randomly selected positive integers under 10000000? How would someone even approach this problem? EDIT: The positive integers are chosen with replacement.
3
votes
1answer
37 views

Large regions of the plane $(x,y) \in \mathbb{Z}^2$ with no relatively prime points: $\mathrm{gcd}(x,y) > 1$

For any $\ell > 0$ can you find $M, N$ such that $ \boxed{\mathrm{gcd}(x,y) > 1}$ for all $x \in [M, M+\ell]$ and all $y \in [N, N+\ell]$ ? This is related to the statement that the set of ...
2
votes
2answers
54 views

Can this congruence be simplified?

$$p(p+1) \equiv -q(q+1) \bmod pq$$ Can this be reduced to an easier format?
13
votes
3answers
217 views

If $p$ is a prime and $p \mid ab$, then $p \mid a$ or $p \mid b$.

The proof is already given in the textbook but I tried other way around. Proof by contradiction: Let's assume that $p$ doesn't divide $a$ and $p$ doesn't divide $b$, but $p$ divides $ab$. So ...
6
votes
8answers
355 views

If a prime $p\mid ab$, then $p\mid a$ or $p\mid b$

If a prime number $p$ is a divisor of a product $ab$, $p$ has to be a divisor of $b$ or $a$. How can I demonstrate this theorem? I demonstrated this theorem on one way using Bezout's theorem in an ...
44
votes
1answer
933 views

Representing every positive rational number in the form of $(a^n+b^n)/(c^n+d^n)$

About a month ago, I got the following : For every positive rational number $r$, there exists a set of four positive integers $(a,b,c,d)$ such that ...
1
vote
0answers
50 views

Two irrational numbers are congruent iff the tails of their infinite continued fractions eventually coincide

We say that a real number $\alpha$ is $congruent$ to real number $\beta$ if there exist integers a, b, c and d with ad-bc=+1 or -1 and such that $$\alpha=\frac{a\beta +b}{c\beta+d}$$ I need to prove ...
1
vote
1answer
269 views

Unique Decomposition of Primes in Sums Of Higher Powers than $2$

Primes of the form $p=4k+1\;$ have a unique decomposition as sum of squares $p=a^2+b^2$ with $0<a<b\;$, due to Thue's Lemma. What is known about sums of $n$ higher powers resulting in ...
14
votes
0answers
697 views

Are there unique solutions for $n=\sum_{j=1}^{g(k)} a_j^k$?

Edward Waring, asks whether for every natural number $n$ there exists an associated positive integer s such that every natural number is the sum of at most $s$ $k$th powers of natural numbers ...
5
votes
0answers
50 views

Integers which are the sum of non-zero squares

Lagrange's four-square theorem states that every natural number can be written as the sum of four squares, allowing for zeros in the sum (e.g. $6=2^2+1^2+1^2+0^2$). Is there a similar result in which ...