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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3
votes
1answer
52 views

Infinitely many rationals with $|a-\frac pq|<\frac1{q^2}$

If $a$ is irrational, there are infinitely many $\frac pq$ s.t $|a-\frac pq|<\frac1{q^2}\tag1$ I have the proof but don't understand it: Take a finite set of rationals $S$ then for ...
0
votes
0answers
63 views

What is the smallest number $n$ , such that $n\uparrow^4 n>3\uparrow^5 3$ holds?

What is the smallest number $n$, such that $$n\uparrow^4 n>3\uparrow^5 3$$ holds ? $\uparrow$ stands for Knut's up-arrow-notation and is defined as follows $a\uparrow b=a^b$ $$a\uparrow ...
3
votes
3answers
86 views

The smallest number $m$, such that $m\uparrow \uparrow (n+1)>n\uparrow\uparrow n$

A natural number $n\ge 3$ is given. Denote $a\uparrow\uparrow b$ to be a power tower of $b$ $a's$. Let $m$ be the smallest natural number , such that $m\uparrow\uparrow(n+1) > n\uparrow\uparrow n$ ...
0
votes
0answers
31 views

Prove that $0_m,1_m\ldots,(n-1)_m$ are different numbers

Let $n$ be an integer that is not divisible by any square greater than $1$. Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n$. Prove that if $m$ and $x$ are ...
-1
votes
0answers
35 views

How to prove that $p$ divides $a^p -a$ for every integer $a$. [closed]

How to prove this Fermat's little theorem: $p$ divides $a^p -a$ for every integer $a$.
0
votes
0answers
16 views

Approximating the coefficients of $\prod_{i=1}^{N}\frac{1}{1-\frac{1}{2}q^i}$ for large $N$

I have $$\frac{1}{2^{N}}\prod_{i=1}^{N}\frac{1}{1-\frac{1}{2}q^i}$$ the reciprocal of the q-Pochhammer symbol $(\frac{1}{2},q)_{N+1}$ (multiplied by a power of $1/2$). Its Maclaurin series for ...
1
vote
2answers
82 views

Minimizing rational solutions of $ x^3+y^3=9$

I´m trying to solve this problem: An old alchemist had two sphercial flasks, one with a circunference of 12 inches and the other with a circunference of 24 inches. He desired to transfer their ...
0
votes
0answers
19 views

Splitting of a prime and $p$-divisibility on an elliptic curve

Let $K$ be a quadratic imaginary field and let $\lambda$ a prime of norm $l^2$, for a rational prime $l$. We consider $E$ to be an elliptic curve such that $E[p](K)$ is trivial, where $p\neq l$ is a ...
0
votes
0answers
49 views

Number of valid parenthesis

I have to find out the number of valid parenthesis.Parenthesis are of two type [] ,(). How many ways are there to construct a valid sequence using ...
2
votes
0answers
25 views

Efficient algorithm for solving a bilinear Diophantine system

I wonder if there is an efficient algorithm for finding an integer solution $(x_1, \dots, x_n)$, $(y_1, \dots, y_n)$ for the following type of system of equations $$ a_{1,1} x_1 y_1 + a_{1,2} x_1 y_2 ...
2
votes
2answers
38 views

Show that for any $a,b\in\mathbb{Z}$, $p$ prime: $(a^p+b^p)^{p^2}\equiv a+b \pmod p$

Show that for any $a,b\in\mathbb{Z}$, $p$ prime: $$(a^p+b^p)^{p^2}\equiv a+b \pmod p$$ Using the binomial expansion, I found that ...
1
vote
3answers
34 views

Arithmetic sequence whose any five consecutive elements contain a prime

Consider an arithmetic sequence $\{11 + 13k : k\in\mathbb{N}\cup\{0\} \}$ Does this sequence contain five consecutive composites? If we look at some selections of five consec. elements: $$11, 24, 37, ...
1
vote
1answer
45 views

Estimation on the accuracy of the convergents of $\sqrt{n}$

I have noticed that the accuracy of the best rational approximations to $\sqrt{n}$ given by his continued fraction expansion, when the numerator and deniminator are large numbers, is approximately ...
2
votes
1answer
1k views

How to solve difficult positive integers and co-prime word problem?

This is a very interesting word problem that I came across in an old textbook of mine. So I know its got something to do with derivative of algebra and prime numbers, which yields the shortest, ...
-2
votes
1answer
40 views

Proving a certain set is inductive?

Let $m$ be a natural number in a field $F$ and let $$ S_m= \{k:k\in N \mbox{ and } k\leq m \}\cup\{x:x\in F, m<x\} $$ Show that the set $S_m$ is inductive. Thanks in advance!
3
votes
0answers
75 views

Is there a name for these primes?

What is the name for primes $p$ where $2p-1$ is also a prime? $2p+1$ is a Sophie Germain prime. On average if $p$ is a primes how many primes of form $2p^n-1$ could we expect where $0<n<B$ ...
-1
votes
2answers
25 views

If $a,b,c \in Z$, $\gcd(a-b,b-c) = \gcd(a-b,a-c)$ [closed]

I need to prove that for every three integers $(a,b,c)$, the $\gcd(a-b,b-c) = \gcd(a-b,a-c)$. Assuming that a $a \ne b$. Having: $d_1 = \gcd(a-b,b-c)$ $d_2 = \gcd(a-b,a-c)$ How do i prove $d_1 = ...
0
votes
0answers
26 views

Calculating the period of a modular congruence

How do you in general calculate the period of a number $x^m$ modulo $n$ using the Chinese Remainder Theorem if $n$ is squarefree? We see that it is sufficient to find the residues modulo every prime ...
0
votes
0answers
30 views

Proving primality of $p$ without making any calculation involving $p$ directly

Wilson's Theorem states that a positive integer $p > 1$ is prime if and only if $(p-1)! \equiv -1 \pmod p$, showing a relationship between factorials and prime numbers. Finding it curious, today I ...
0
votes
2answers
42 views

Numbers of the form $pa+qb = n$

Is it true that numbers of the form $pa+qb = n$ where $\gcd(a,b) = 1$ and $a,b$ are positive integers and $p,q$ are nonnegative integers are unique? That is, they have unique representations ...
1
vote
1answer
69 views

Is there a way to prove that $\sqrt[7]{129}$ is irrational using the following theorem?

I want to prove that $\sqrt[7]{129}$ is irrational using the following theorem: Let $n,k$ be natural numbers. Then, $\sqrt[n]{k}$ is rational iff $k$ is the $n\text{-th}$ power of a (natural) ...
1
vote
1answer
722 views

Number of Divisors of N factorial

Say $d(N) =$ Number of factors of $N!$ Briefly: I wish to know if there is a Recurrence relation for this problem. Now I wish to Know if there is a way to calculate $d(N)$ in terms of previously ...
2
votes
5answers
110 views

Minimize $a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$ with distinct positive integers

Find the minimum value of the following: $$a^5+b^5+c^5+d^5+e^5 = p^4+q^4+r^4+s^4 = x^3+y^3+z^3 = m^2 + n^2$$ where all numbers are different/distinct positive integers. I know the answer (see ...
1
vote
1answer
45 views

Proof of Sylvester's Theorem

If $p$ and $q$ are relative primes, prove that the number of integers inferior to $pq$ which cannot be resolved into parts (zeroes admissible), nonnegative multiples of $p$ and $q$, is ...
0
votes
1answer
35 views

determine odd number pattern?

How can I determine series of such numbers which when keep dividing by 2 always produce odd quotient? For example: 15 15/2 = 7 (odd) (take only integer(floor) part) 7/2 = 3 (again odd) 3/2 = 1 (again ...
3
votes
1answer
56 views

Unramified algebraic extensions of local fields

This is a basic question from Neukirch's Algebraic Number Theory, Prop. 7.2: Fix a non-Archimedean local field $K$. Let $L/K$ and $K'/K$ be two extensions inside an algebraic closure $\bar{K}/K$ and ...
10
votes
6answers
2k views

what is Prime Gaps relationship with number 6?

Out of the 78499 prime number under 1 million. There are 32821 prime gaps (difference between two consecutive prime numbers) of a multiple 6. A bar chart of differences and frequency of occurrence ...
6
votes
2answers
93 views

Does the $5x + 1$ sequence for 7 reach a power of 2 or does it get stuck in a period?

This is much like the $3x + 1$ iteration, except that if $x$ is odd, you do $5x + 1$ [and $\frac{x}{2}$ if $x$ is even]. If $x = 7$, then we have 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, ...
1
vote
2answers
53 views

What are the applications of Sigma Function?

I read about the Sigma Function today.It tells that- The $\sigma(n)$ is the sum of all the positive divisors of $n$. But I had no idea how they can be useful.What are the practical applications ...
-1
votes
0answers
16 views

Grobner basis and number theory [closed]

Could someone help me? What is the relationship between grobner basis and number theory?
3
votes
1answer
410 views

Demystifying the asymptotic expression for the partition function

A partition of an integer $n$ is a way of writing $n$ as a sum of integers. The partition function $p(n)$ counts the number of distinct partitions of $n$. In 1918, Hardy and Ramanujan proved the ...
3
votes
1answer
60 views

How to compute $(1 \cdot 3 \cdot 5 \cdots 97)^2 \pmod {101}$ [closed]

How to compute $(1 \cdot 3 \cdot 5 \cdots 97)^2 \pmod {101}$ in easiest and fastest way?
-3
votes
0answers
36 views

Irrationality proof of the nth root of the product of two non perfect nth powers. [closed]

How to prove the following statement? Let $ a$ and $b$ be two rational numbers, except $a=b=1$ , that aren't perfect nth powers. To prove that $\sqrt[n]{ab}$ is irrational.
-2
votes
1answer
52 views

How many integer satisfy the following condition? [closed]

$\frac{n}{\lfloor{\sqrt{n}}\rfloor}\in\mathbb{Z}, 0<n<2016$, where $n$ is an integer
4
votes
1answer
29 views

On the GCD of two palindromes.

I had an observation. Which I will discuss below. My question will be Is my observation correct? If so, how can one prove it? Observation: Consider the string of palindromes below: $100...01$ and ...
0
votes
0answers
26 views

Number of solutions to equation

Let $a$ and $b$ be two nonnegative integers. Denote by $H(a,b)$ the set of numbers $n$ of the form $n = pa+qb $ where $p$ and $q$ are nonnegative integers. Determine $H(a) = H(a,a)$. Prove that if ...
0
votes
1answer
86 views

Constructing idele from a rational number.

I am a novice to concept of idele, despite the fact that I have gone through all its expositions in standard literature. Excusing my ignorance, suppose I take $q=396000$. Does it mean that the idele ...
1
vote
1answer
117 views

sum over primes involving divisor function (variation of the Titchmarsh divisor problem)

Does there exist an asymptotic estimate for the following sum over primes $$ \sum_{p\leq x} \frac{\tau(p-1)}{p}\;, $$ where $\tau(n)=\sum_{d|n}1$ is the divisor function?
0
votes
1answer
52 views

Find digit 1 in 50,000 [closed]

The following is the problem I have been working on. After spending considerable amount of time to list 1s, I got the result but it is wrong. There must be an easy way to solve the problem. Any help ...
0
votes
1answer
57 views

Prove that $n^2+11n+2$ is not divisible by $12769$ [duplicate]

My Attempt : Prime factorisation of $12769$ is $113^2$ $n^2+11n+2-113^2m=0$ The conjugate of this quadratic equation becomes: $\sqrt {113 (113m+1)} $ which can never be a rational as ...
0
votes
1answer
25 views

How to prove the expression is not a square in the following question [duplicate]

Let d be any positive integer not equal to 2,5 or 13. Show that one can find distinct a, b in the set 2,5,13,d such that ab-1 is not a perfect square. I tried it for a long time but couldn't figure ...
44
votes
2answers
791 views

Fractals using just modulo operation

Let us calculate the remainder after division of $27$ by $10$. $27 \equiv 7 \pmod{10}$ We have $7$. So let's calculate the remainder after divison of $27$ by $7$. $ 27 \equiv 6 \pmod{7}$ Ok, so ...
4
votes
1answer
43 views

Power Diophantine equation: $(a+1)^n=a^{n+2}+(2a+1)^{n-1}$

How to solve following power Diophantine equation in positive integers with $n>1$:$$(a+1)^n=a^{n+2}+(2a+1)^{n-1}$$What I have got so far: Let $p$ be a prime divisor of $2a+1$, then take modulo $p$ ...
1
vote
0answers
74 views

Is it possible to find $p-1$ natural numbers $n+1,n+2,\ldots,n+p-1$?

Let $p$ be an odd prime. Is it possible to find $p-1$ natural numbers $n+1,n+2,\ldots,n+p-1$ such that the sum of the squares of these numbers is divisible by the sum of these numbers? We can ...
0
votes
0answers
29 views

A property for a set of integers

Put $$ S=\mathbb{Z}\setminus \{ m^2-n^2: m,n\in \mathbb{Z}\setminus\{0\}\} $$ Conjecture. $S\cap(k+S)\cap(r+S)=\emptyset$, for all $k\in S$ and all $r\in S\cap(k+S)$. Is it true?
7
votes
2answers
457 views

Are there infinitely many natural numbers $n$ such that $\mu(n)=\mu(n+1)=\pm 1$?

A while ago, I answered this question here on StackExchange which asks if for any given integer $k$, whether there exists infinitely many natural numbers $n$ such that $$ ...
0
votes
0answers
19 views

Finding impact of multiple variables in a non linear equation

I am encountering a difficult(atleast to me) mathematical problem? I need to calculate the impact of a formulae on two sets of data for two different years Consider for example I have a simple ...
-4
votes
4answers
138 views

Which of these numbers is the biggest [closed]

How do I determine algebraically/without a calculator which of these numbers is the greatest? $$8^{36}, 7^{55}, 5^{72}, 2^{110}$$ Please provide the method and a little description for me to ...
-3
votes
1answer
51 views

Is the sine of a transcendental angle transcendental or algebraic? [closed]

Let $x$ be a transcendental number Algebraically Independent from $\pi$. It is known if $ \sin x $ is also transcendental or algebraic? For example, is $\sin \sqrt{2}^\sqrt{2}\pi$ algebraic or ...
1
vote
0answers
47 views

Game of Nim: Losing Positions [closed]

If you have heard of the game Nim, this is a version of the game. However, in this version, the players can only remove the amount of stones from the pile which is coprime to the current pile size. ...