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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2
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2answers
52 views
6
votes
4answers
127 views

Prove that $a_n$ is a perfect square

Let $\,\,\,\left(a_{n}\right)_{\ n\ \in\ \mathbb{N}\,\,\,}$ be the sequence of integers defined recursively by $$ a_{1} = a_{2} = 1\,,\qquad\quad a_{n + 2} = 7a_{n + 1} -a_{n} - 2\quad \mbox{for}\...
0
votes
0answers
82 views

Any underlying reason why these equations look similar?

Questions Is there any way to go from either of these equations to the other? Or is there any more fundamental reason for their similarities? $$ \frac{1}{\zeta(s)} = \sum_{r=1}^\infty \frac{\mu(r)}{...
2
votes
4answers
56 views

Find four positive integers having more than $100$ divisors

Find four positive integers each not exceeding $70000$ and each having more than $100$ divisors. Since we are trying to maximize divisors and minimize value, we assume that $n = 2^{\alpha_1} 3^{\...
0
votes
0answers
29 views

Can “sufficiently large” be made more concrete?

Here https://en.wikipedia.org/wiki/Prime_gap an observation of Chudakov is mentioned : For every $\theta>\frac{3}{4}$ there exists an $N$ such that $g_n<p_n^{\theta}$ for all $n\ge N$. ...
-2
votes
1answer
23 views

I need to find the least nonnegative residue

I have a hard time finding the least non negative residue of large numbers, and I'm having trouble with finding the least non negative residue of (1511)^7 (mod 3131) can someone help?
0
votes
0answers
27 views

if $a,b$ are natural numbers such that $(1+ab)|(a^2+b^2).$ then $\frac{a^2+b^2}{1+ab}$ is perfect square [duplicate]

if $a,b$ are natural numbers such that $(1+ab)|(a^2+b^2).$ then $\frac{a^2+b^2}{1+ab}$ is perfect square.please provide some hint.i have tried ,but donot know from where to start
1
vote
3answers
147 views

Proof that if $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. [duplicate]

I need help with this excercise. If $\gcd(m,n) = 1$, then $\gcd(m+n,mn ) = 1$. I don't know how to prove this, I know the definition of $\gcd$ but I can't prove it.
9
votes
6answers
222 views

Prove that $\lfloor\frac{n+1}{2}\rfloor+\lfloor\frac{n+2}{4}\rfloor+\lfloor\frac{n+4}{8}\rfloor+\lfloor\frac{n+8}{16}\rfloor+ \dots=n$

Prove $$\left[\dfrac{n+1}{2}\right]+\left[\dfrac{n+2}{4}\right]+\left[\dfrac{n+4}{8}\right]+\left[\dfrac{n+8}{16}\right] + \dots=n$$ where $[x]=\lfloor x\rfloor$ $$$$ It was suggested that ...
16
votes
1answer
235 views

Why are $L$-functions a big deal?

I've been studying modular forms this semester and we did a lot of calculations of $L$-functions, e.g. $L$-functions of Dirichlet-characters and $L$-functions of cusp-forms. But I somehow don't see, ...
2
votes
2answers
65 views

Find all triples satisfying an equation

Another question I saw recently: Find all triples of positive integers $(a,b,c)$ such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Can someone help me with it?
1
vote
1answer
37 views

Do negative binomials imply negative factorials exist?

I've seen the following identity: $$\binom{-n}{k} = (-1)^k\binom{n+k-1}{k}$$ So I tried to derive it, assuming negative factorial was a real concept, having it extend down to negative infinity: $$\...
0
votes
2answers
31 views

Why is Nim solvable with the xor operator?

In the game of Nim, played with two players, if you have $n$ stacks of stones (where you can take any number of stones from a single pile each turn), losing positions are ones where the xor of the ...
1
vote
6answers
96 views

Why is $10^k - 1$ divisible by $9$?

I know it is obvious that $10^k-1$ will always be divisible by $9$ for some integer $k$, but I am curious how to actually prove this. $$10^k - 1 \equiv 0 \bmod 9$$ $$10^k \equiv 1 \bmod 9$$ ... and ...
2
votes
2answers
83 views

How Deficient a Number is? (Finding numbers having a certain deficiency)

This question was edited, in particular equations were corrected: A number N is said to be deficient by an integer $d$ if: $\sigma(N)=2N-d$ Note that powers of 2 are deficient by 1. While a prime $...
-8
votes
1answer
129 views

is all math beyond arithmetic just advanced arithmetic? [on hold]

Is it true that at the bare-bones of all advanced math, its all just mostly arithmetic? In computer programming languages they are mostly constrained to arithmetic operators. I suppose this is ...
0
votes
0answers
19 views

New Generalized MR-test

I am conducting a new Miller Rabin (SPRP test) and editing the first step. Can someone please help me with the last step. Thanks. Original: Write $n$ $=$ $2^sd+1$ with $d$ odd. Replace: New Test: ...
11
votes
4answers
2k views

Topics on Number theory for undergraduate to do a project [closed]

Im an undergraduate in the mathematics field ..So i wanna be alittle more productive and wanted to do an essay or project mostly on number theory or Algebra(Rings or Groups) and i want to ask if you ...
1
vote
0answers
38 views

The Spacing of $e$ and $\pi$ Segments Within the Decimal Expansion of $\pi$

I discovered something seemingly very improbable today when I was searching for segments of $e$ and $\pi$ within the decimal expansion of $\pi$. I searched for $314159265$ and found it starts at the ...
0
votes
2answers
64 views

Determine all $n$-digit numbers that are divisible by the cyclic permutations of its digits

Given an integer $n \geq 2$, determine all $n$-digit numbers $M_0 = \overline{a_1a_2 \ldots a_n}$ $(a_i \neq 0, i = 1,2,\ldots,n)$ divisible by the numbers $M_1 = \overline{a_2a_3 \ldots a_na_1}$, $...
0
votes
0answers
22 views

Application of the theorem about diophantine equations having either infinite or finite solution.

How can i apply the theorem below in an equation like \begin{equation} \label{eq:(4)} 10^{n+3} a - 10^3 a + 999b = (3y)^2. \end{equation} that equation is actually from letting $m = 3$, from the ...
13
votes
2answers
2k views

Do these series converge to the von Mangoldt function?

Jeffrey Shallit formulated this recurrence for me: $\displaystyle T(n,1)=1, k>1: T(n,k) = \sum\limits_{i=1}^{k-1} T(n-i,k-1)-\sum\limits_{i=1}^{k-1} T(n-i,k)$ which is the lower triangular array ...
9
votes
1answer
80 views

A pair of sequences defined by mutual addition/multiplication

Define sequences $\{a_n\},\,\{b_n\}$ by mutual recurrence relations: $$a_0=b_0=1,\quad a_{n+1}=a_n+b_n,\quad b_{n+1}=a_n\cdot b_n.\tag1$$ The sequence $\{a_n\}$ begins: $$1,\,2,\,3,\,5,\,11,\,41,\,371,...
-3
votes
0answers
63 views

Prove that $\sqrt { 2 } +\sqrt { 3 } +\sqrt { 5 } +\sqrt { 7 } +\sqrt { 11 } +\sqrt { 13 } +\sqrt { 17 } $is irrational number? [duplicate]

I got this solution, but I didn't understand it. Assume that $b_1,b_2,b_3,\ldots, b_n$ are whole numbers (not zero). So, we have $b_1\sqrt{a_1}+b_2 \sqrt{a_2}+\ldots+b_n\sqrt{a_n}=0$. Prove it with ...
4
votes
4answers
64 views

Not understanding the wrong logic in this proof

The problem is : Suppose $a,b \in Z$. If $a^2 + b^2 $ is a perfect square, then $a$ and $b$ are not both odd. My question is why can't I answer like so: Proof by Contradiction -- Suppose $a^2 + b^2 ...
4
votes
2answers
54 views

Which primes $p$ divide $q^q-1$ for a prime divisor $q$ of $p-1$

I am looking for (a formula) for all the primes $p$ less than or equal to $X$ with the following criteria: There is at least one prime $q$ dividing $p-1$ such that $p$ divides $q^q-1$. $7$, for ...
1
vote
2answers
59 views

Subset of Coins with maximal value

Let $ n \in \mathbb{N} $ with $ n\ge 3 $ be given. Assume that you have $ k-1 $ coins of value $ 1/k $ for all $ k \in \lbrace 2,\ldots,n \rbrace $. Now you have to choose a subset of these given ...
3
votes
1answer
74 views

What is the number of Sylow p subgroups in S_p?

I am reading the Wikipedia article entitiled Sylow theorems. This short segment of the article reads: Part of Wilson's theorem states that (p-1)! is congruent to -1 (mod p) for every prime p. One ...
2
votes
2answers
75 views

Can this function be a new test for primality?

The following function returns always 0 only if a number is not prime. $$ H(x)=\prod_{i=2}^{x-1}\left\{\left[\sum_{k=1}^{x/i}(-i)\right]+x\right\} $$ what do you think? Bye!
2
votes
0answers
77 views

conjectured arithmetic properties of some continued fraction

Given the continued fraction found in this post and looking similar to the one in this post $$G(q)=\cfrac{1}{1-q+\cfrac{q(1-q)^2}{1-q^3+\cfrac{q(1-q^2)^2}{1-q^5+\cfrac{q(1-q^3)^2}{1-q^7+\ddots}}}}$$ ...
0
votes
0answers
12 views

Is the relationship between coprime integers and irreducible fractions biconditional?

Are coprime integers and irreducible fractions related biconditionally? That is, if two integers are coprime ($a$ and $b$ say) then the fractions $\frac{a}{b}$ and $\frac{b}{a}$ are both irreducible. ...
3
votes
1answer
18 views

Maximum length of a string that has no substring divisible by a prime number $p$ is $p-1$?

What is the maximum length of a string of nonzero digits that has no substring that is divisible by a given prime number? I want to find a string of length n which has no substring divisible by the ...
3
votes
1answer
93 views

multiples (of primes) coverage formula

I apologize in advance if my explanation is not clear. Please let me know if clarification is required and I will do my best to fix it! I am attempting to find an explicit formula (in terms of ...
1
vote
2answers
39 views

Getting characteristic polynomial from a small matrix

Sorry I don't know how to format matrices, but if I have this matrix $\pmatrix{1& 1& 0\\ 0& 0& 1\\ 1 &0& 1\\}$ How is the characteristic polynomial $λ^3 − 2λ^2 + λ − ...
0
votes
1answer
13 views

Matrix for a recurrence

The matrix for a recurrence of the form $a_{k+2} = ka_{k+1}+a_{k}$ where $a_0 = 0$ and $a_1 = 1$ is given by $$\begin{bmatrix}k & 1\\ 1 & 0 \end{bmatrix}^n = \begin{bmatrix} a_{k+1} & a_k \...
3
votes
0answers
49 views

little Fermat theorem generalization proof without Burnside's lemma

Burnside's Lemma Deduce That: $$\sum_{i=1}^n a^{gcd(i,n)} $$ is divisible by $n$ it's a beautiful result. but i want to prove it without any abstract algebraic tools such as Burnside's Lemma... is ...
-2
votes
1answer
23 views

Any Mersenne prime contains two consecutive 9 digits? [on hold]

The kids with me were each asked to pick a number. It crossed my mind that a smart aleck might answer with a description of some number that we have never actually computed. I remembered that a ...
68
votes
10answers
12k views

Nice proofs of $\zeta(4) = \pi^4/90$?

I know some nice ways to prove that $\zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \pi^2/6$. For example, see Robin Chapman's list or the answers to the question "Different methods to compute $\sum_{...
31
votes
3answers
2k views

Does the equation $x^4+y^4+1 = z^2$ have a non-trivial solution?

The background of this question is this: Fermat proved that the equation, $$x^4+y^4 = z^2$$ has no solution in the positive integers. If we consider the near-miss, $$x^4+y^4-1 = z^2$$ then this has ...
4
votes
1answer
45 views

Determining when ring of integers is $\mathbb{Z}[\theta]$

Something which is not difficult to prove is that if $K$ is a number field generated by an integer $\theta$, then the ring of integers $\mathfrak{O}_K$ is generated over $\mathbb{Z}$ by $\theta$ and ...
3
votes
1answer
76 views

Algorithms for finding the ring of integers

In the book's Algebraic Number theory, Ian StewarT, Third edition (page 51-52), has the following propositions: Theorem 2.20: Let $G$ be an additive subgroup of $\mathfrak{O}_K$ of rank equal to the ...
7
votes
1answer
95 views

Show that the ring of integers $A$ of the cubic field $\mathbb Q[x]$ with $x^3=2$ is principal.

Show that the ring of integers $A$ of the cubic field $K=\mathbb Q[x]$ with $x^3=2$ is principal. The hint given in the book is to majorize the discriminant of $A$ by $D(1,x,x^2)$ and then use the ...
-3
votes
1answer
158 views

Again near at Riemann hypothesis [closed]

Let $\zeta(s)$ be Riemann extended zeta function for $Re(s)>0$. Let $\eta(s)$ be Riemann alternated zeta function for $Re(s)>0$, i.e. , $$ \eta(s)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^s}=...
1
vote
2answers
51 views

Find the maximum value that the quantity $2m+7n$ can have

Find the maximum value that the quantity $2m+7n$ can have such that there exist distinct positive integers $x_i$ $(1 \leq i \leq m)$, $y_j$ $(1 \leq j \leq n)$ such that the $x_i$'s are even, the $y_j$...
4
votes
1answer
31 views

Non-negative integer solutions to $4ab-a-b=c^2$

The puzzle is as follows: Problem: Find all non-negative integer solutions to $4ab-a-b=c^2$ My Progress: There is, of course, the trivial solution of $a=b=c=0$, and I suspect there are no more (...
5
votes
1answer
58 views

Deriving sum of powers formula using generating functions

Just for fun I wanted to try to derive a formula for the sum of $p$-powers using generating functions, but without using any literature or websites for help. However I do need a small push or hint. ...
1
vote
2answers
58 views

Generating function for $\sum_{n=0}^{\infty} n^{p} x^n$

I am trying to derive the generating function for $H(x,p) = \sum_{n=0}^{\infty} n^p x^n$ I am trying to solve it with the following logic: (Edited now, trying a new framing) Base case: $$H(x,0) = \...
1
vote
2answers
37 views

Dirichlet inverse of $(-1)^n$

I was tinkering around and noticed the Dirichlet inverse of $\,f(n) = (-1)^n$ seems to be $$ f^{-1}(n) = -\mu\!\left(n\,/\,2^{\nu_2(n)}\right)\left\lceil 2^{\nu_2(n)-1} \right\rceil, $$ where $\nu_p(n)...
4
votes
2answers
63 views

Growth of $\pi(2x) - 2\pi(x)$

In Hardy & Wright's Theory of Numbers (p. 494f in 6th ed.) there's a little discussion following the proof of the prime number theorem. We have $$ \pi(2x) - \pi(x) = \frac{x}{\log x} + o\...
1
vote
3answers
38 views

The number of $n$-tuples of nonnegative integers $(x_1,\ldots,x_n)$ satisfying the equation $x_1+\cdots+x_n = r$

For given positive integers $r,v,n$ let $S(r,v,n)$ denote the number of $n$-tuples of nonnegative integers $(x_1,\ldots,x_n)$ satisfying the equation $x_1+\cdots+x_n = r$ and such that $x_i \leq v$ ...