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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Solving a quadratic Diophantine equation

I want to solve the following quadratic Diophantine equation: $$\frac{x(x-1)}{y(y-1)}=\frac{p}{q} \hspace{5 mm}, \hspace{5 mm}p\le q$$ For $p=1$ and $q=2$, it is easy to solve. Let $y=x+z$. Then ...
0
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1answer
16 views

$(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no. Where w;x;y;z are positive odd integers?

Q $(w^2+x^2).(y^2+z^2)$ is always divisible by which of the max no where w,x,y,z are positive odd integers? Options given: 20;8;4;2 My Approach: I Choose ($9^2$+$5^2$).($7^2$+$3^2$) to get ...
8
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3answers
306 views

Probability that a natural number is a sum of two squares?

Some natural numbers can be expressed as a sum of two squares: $$2=1^2+1^2$$ $$25=3^2+4^2$$ $$50=7^2+1^2$$ If one chooses a random natural number, what would be the probability that that number is a ...
-2
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1answer
29 views

3x + 1 problem other repititions [on hold]

The Collatz problem: Pick an integer x > 0 if x even: $x = x / 2$, if x odd: $x = 3*x + 1$ repeat 2.) as long as you want This algorithm seems to always end up with the loop 4-2-1 My actual ...
4
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2answers
67 views
+50

Pentagonal Numbers

I recently was passing some time on Project Euler, when I came across this question. It deals with finding Pentagonal Numbers $P_j$ and $P_k$ such that $P_j+P_k$ and $P_j-P_k$ are also pentagonal ...
5
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1answer
47 views

Proving that and how $ \frac{1}{n}\sum\limits_{p\le n}\lfloor n/p \rfloor - \sum\limits_{p\le n} 1/p $ approaches $0$

Let $p$ denote a generic prime number. By Mertens' second theorem, the sequence $$\sum\limits_{\ p \le n} \frac1p - \log\log n$$ converges to the Meissel-Mertens constant $M\approx 0.2614972$. Now let ...
19
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2answers
148 views
+50

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Let $\varphi$ be the Euler's function, i.e. $\varphi(n)$ stands for the number of integers $m \in \{1,\ldots,n\}$ such that $\text{gcd}(m,n)=1$. Let $n\ge 2$ be a positive integer. Show that there ...
-3
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1answer
350 views

Is this attempted proof of ABC conjecture correct [closed]

This mathematician claims that he has tackled ABC conjecture! He uses induction and simple inequalities to achieve the result. Is this some serious stuff or is there a basic flaw in the reasoning?
2
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4answers
3k views

How many numbers below $100$ can be expressed as a difference of two perfect squares in only one way?

How many numbers below $100$ can be expressed as a difference of two perfect squares in only one way? Please explain your approach. ADDED: I can determine whether a number could be represented ...
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2answers
17 views

Simulataneous equations

Suppose you have the following system of linear congruence 2x+5y is congruent to 1 (mod6) x+y is congruent to 5 (mod6) where x,y belong to the set of Integers How would you obtain a general ...
1
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1answer
21 views

Finding Maximum in a Set of Numbers [on hold]

If I have a set of $n$ numbers: $(a_1,..., a_n)$, then how can I find the two maximum numbers in the set? Suppose that all the numbers are positive integers.
0
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0answers
35 views

Find all integers $m,n$ for which $m^2+n^2$ is a square and $\sqrt{\frac{2m^2+2}{n^2+1}}$ is rational

This is a repost of my old question here. The question is as follows: Find all integers m and n, such that $m^2 + n^2$ is a square and $\sqrt{\frac{2(m^2+1)}{n^2+1}}$ is rational. I have made no ...
2
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1answer
36 views

Can different tetrations have the same value?

Suppose, we have two numbers $a\uparrow \uparrow b$ and $c\uparrow \uparrow d$. To avoid trivial cases, suppose $a,b,c,d>2$ and $(a,b)\ne (c,d)$. Is there a quartupel $(a,b,c,d)$ with ...
6
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2answers
80 views

Does the Riemann-Hypothesis imply the Twin-Prime-Conjecture?

The Riemann hypothesis (https://en.wikipedia.org/wiki/Riemann_hypothesis) is one of the most important conjectures in number theory. I read that the Riemann hypothesis implies the Goldbach Conjecture ...
3
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0answers
37 views

Is there anything known about the zeros of $\displaystyle \sum_{n=1}^{\infty} \left(\frac{1}{\rho_n^s} +\frac{1}{\overline{\rho_n}^s}\right)$?

Assuming the RH and $\rho_n =\frac12 + \gamma_n i$ being the n-th non-trivial zero of $\zeta(s)$, then numerical evidence suggests that: $$f(s) :=\displaystyle \sum_{n=1}^{\infty} ...
0
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1answer
58 views

Roots of $\Phi_{31}(x)$ as roots of unity.

Let $\Phi_{31}(x)$ be the $31$-cyclotomic polynomial. I want to show that $\Phi_{31}(x)$ is the product of six irreducible quintic factors in $\mathbb{F}_2$. I am running into difficulties ...
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1answer
26 views
3
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0answers
33 views

Can the determinant of an integer matrix with $k$ given rows be the gcd of the determinants of the $k\times k$ minors of those rows?

I'm interested if the following is true: Let $n\geq k\geq1$ be integers, let $A\in\mathbb Z^{k\times n}$ and denote the $\binom nk$ $k\times k$ minors of $A$ by $A_1,\ldots,A_N$. Then the ...
2
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0answers
37 views

Which constellations of primes recur forever?

Having derived much joy and learning from the answers I have received to four previous questions, let me ask one more. Let a constellation of primes be a set of primes that stand in certain fixed ...
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0answers
27 views

Are there other known continued fractions that show the digits of the golden ratio?

I found a few. {16; 5, 1, 1, 5, 22} {161; 1, 4, 11, 1, 1, 3, 6, 1, 13, 8, 1, 6, 1, 4, 1, 1, 2, 1, 1, 1, 1, 13, 2, 1, 3, 8, 1, 2, 19, 1, 54, 1, 19, 2, 1, 8, 3, 1, 2, 13, 1, 1, 1, 1, 2, 1, 1, 4, 1, 6, ...
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1answer
46 views

Crude verification of Goldbach conjecture [on hold]

So the Goldbach conjecture says 'Every even integer greater than 2 can be written as sum of two primes'. Here is what I have roughly done to verify it, using probability. I don't say it is correct but ...
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1answer
22 views

Proving a simple modulo equality

I'm probably lacking some basic concept here but I'm trying to prove that $$ ((a \mod k) \cdot k + b) \mod k = (a \cdot k + b) \mod k$$ I get stuck at the passage where, applying distributive ...
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0answers
24 views

How can there be an infinite number of sequentially composite Fibonacci(p)?

I ran into this counting function a(n)>=a(k)+1 for the number of distinct prime factors of the n-th Fibonacci number, at OEIS. Thank you Robert Israel! Thank you for writing the proof there. I had ...
5
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1answer
85 views

Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $

I need help solving the Thue equation $ x^4 - 6 x^3 y - x^2 y^2 + 6 x y^3 - y^4=-1 $. It can be written as $ x(x-y)(x+y)(x-6y) = (y-1)(y+1)( y^2 +1) $. From this I found 8 solutions ...
129
votes
1answer
7k views

A variation of Fermat's little theorem

Fermat's little theorem states that for $n$ prime, $$ a^n \equiv a \pmod{n}. $$ The values of $n$ for which this holds are the primes and the Carmichael numbers. If we modify the congruence ...
0
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2answers
43 views

Can absolute value functions be moved like this?

If I have an expression that looks like $|x-a_1| + |x-a_2| + |x-a_3| + ... + |x-a_n|$ Is it the same as doing $|nx - \sum_{i=1}^{n}a_i|$
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votes
3answers
56 views

Prove that: 1. $gcd(a,b)=lcm(a,b)$ iff $|a|=|b|$ 2. $k>0\implies lcm(ka,kb)=k lcm(a,bk)$ 3. $a\mid m, b\mid m$, then $lcm(a,b)\mid m$

Let $a,b$ any non-zero integers. Prove that: $gcd(a,b)=lcm(a,b)$ If and only if $|a|=|b|$. If $k>0$, then $lcm(ka,kb)=k lcm(a,bk)$ if $m$ is multiple of $a$ and $b$, then $lcm(a,b)$ divides $m$ ...
0
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1answer
26 views

Calculate the number of zeros in square of ((4404 with base 17)) . ..?

Q Calculate the number of zeros in square of (4404 with base 17)? My approach: @Edit is it right? (4404 at base 17)*(4404 at base 17)=(10G0GF0G at base 17) So, the number of zeros will be 3.
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1answer
25 views

Show that (c,a)=(c,b)

In my book I have the implication: If $gcd(a,b)=1$ and $c|(a+b)$, then $gcd(c,a)=gcd(c,b)=1$. It gives me a hint that begins by supposing that $gcd(a,c)=gcd(b,c)=d$. But in my opinion, I do not ...
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1answer
78 views

Questions from an olympiad on number theory [closed]

The sum of the infinite series: $$ \frac{1}{2} + \frac{2}{8} + \frac{3}{16} + \frac{5}{32} + \frac{8}{64} + \frac{13}{128} + \frac{21}{256} + \frac{34}{512} +....$$ I am able to find the general term ...
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0answers
27 views

Ways Of Finding Primes and If they are efficient

I am currently in middle school and love number theory. I try and do a proof every day and today I was working on a relatively simple one involving primes. I proved that every prime above 5 can be ...
1
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1answer
59 views

What is the limit of the sum of “last half” part of harmonic series?

I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ ...
0
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0answers
46 views

Elliptic curve is self dual.

How to prove $E[p^\infty] \cong Hom ( T_E, \mathbb{Q}_p/\mathbb{Z}_p(1)) $ where $T_E$ denotes the Tate module of $E$ ?
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2answers
505 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
57 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
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0answers
198 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 ...
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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
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1answer
45 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
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1answer
44 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} = ...
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0answers
121 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
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1answer
41 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
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2answers
47 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
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1answer
26 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 ...
56
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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
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2answers
63 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
11k 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
71 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?
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2answers
56 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 ...
3
votes
1answer
27 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
60 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 ...