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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1answer
22 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: $$\...
2
votes
2answers
51 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?
0
votes
2answers
24 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
81 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
82 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 $...
5
votes
4answers
100 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}\...
-7
votes
1answer
91 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
18 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 ...
0
votes
0answers
8 views

Can we find exact factor of inert prime ideals?

$K=\mathbb{Z}[\sqrt{m}]$ with $m$ being square free. I studied the proof of the statement that a prime ideal $\mathfrak{p}=\langle p \rangle$ of $\mathbb{Z}$ stays inert in $\mathcal{O}_K$ if $p=2$...
1
vote
0answers
37 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 ...
2
votes
3answers
47 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^{\...
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
76 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
62 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
53 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
52 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 ...
15
votes
1answer
216 views
+100

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, ...
3
votes
1answer
73 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
63 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
76 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
11 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
17 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
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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
12 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
2answers
48 views

Game of replacing number with divisors

In a game , there are N numbers and 2 player(A and B) . ...
-2
votes
1answer
21 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
150 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
57 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
57 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
36 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 $x_i \leq v$ ...
0
votes
0answers
33 views

For prime >2, is there always a power of a prime which is a primitive root? [on hold]

As the title, I'm trying to find the answer about this question. However, I can't google anything :( Thanks! EDIT: this primitive root must less than the prime
31
votes
7answers
5k views

How do you prove that a prime is the sum of two squares iff it is congruent to 1 mod 4?

It is a theorem in elementary number theory that if $p$ is a prime and congruent to 1 mod 4, then it is the sum of two squares. Apparently there is a trick involving arithmetic in the gaussian ...
5
votes
1answer
44 views

Are the extremas of $h(x)$ global?

It is well known that $li(x)$, the integral logarithm is a very good approximation of $\pi(x)$, the nunmber of primes not exceeding $x$. So, a very good approximation for the probability, that a ...
3
votes
2answers
67 views

Find the common divisors of $a_{1986}$ and $a_{6891}$

Let $(a_n)_{n \in \mathbb{N}}$ be the sequence of integers defined recursively by $a_0 = 0$, $a_1 = 1, a_{n+2} = 4a_{n+1}+a_{n}$ for $n \geq 0$. Find the common divisors of $a_{1986}$ and $a_{6891}$. ...
1
vote
0answers
55 views

Induction Method in a special case of $ n!+1 = m^2 $ (Brocard's Problem)

Context: Brocard's problem is a problem in mathematics that asks to find integer values of $n$ and $m$ for which$$ n!+1 = m^2 \tag{1}$$ Let's define, $$T=\left(\left\lfloor \frac{ (\lfloor\log(n) \...