Questions on more advanced topics of number theory. Consider first if (elementary-number-theory) might be a more appropriate tag before adding this tag.

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Sum of product partitions of divisors

Let $M(n)$ be the the set of the multiplicative partitions of $n$, and let $D(n)$ be the set of the sum of the multiplicative partitions of the divisors of $n$. eg $M(30)=\{\{30\},\{2,15\},\{3, ...
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Sum of Gauss sum

Let $p$ be an odd prime, $v \in \mathbb{N}$ be a positive integer, and $c\in \mathbb{Z}$. Set \begin{align} G(c,p^v):=\sum_{\substack{d \bmod p^v \\ (d,p^v)=1}}{ \left(\frac{d}{p^v}\right) {e}^{ { ...
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The n-th k-gonal number

I was doing some school work and got bored so I started messing with k-gonal numbers. I started with the triangular numbers, square numbers and looked for patterns. I noticed something. Let ...
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Conjecture on OEIS A167055

OEIS A167055 Numbers n such that $12n + 5$ is prime. $0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS A167055. I conjecture that the set of the sum of every two items of this ...
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Is there an irrational number which satisfies this condition?

In fact, every irrationals $x$ can be uniquely expressed as an infinite simple continued fraction: $$x=a_0(x)+\cfrac{1}{a_1(x)+\cfrac{1}{a_2(x)+\cfrac{1}{a_3(x)+\ddots}}},$$ where ...
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Major arcs in the proof that every odd number is the sum of at most 5 primes

In his proof that all odd numbers greater than 1 are the sum of at most 5 primes, Terence Tao uses one large major arc around 0 rather than small ones around the rationals, which I am more accustomed ...
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Updated:Sum of entries in continued fraction of $\sqrt d$ and $\sqrt{d}-\lfloor \sqrt{d}\rfloor$ equals (divides) $d$.

(1)I noted as a joke in class, for $\sqrt{13}$ which has continued fraction expansion $[3;\overline{1,1,1,1,6}]$ that $3+1+1+1+1+6=13$. Another eg. $\sqrt{22}=[4;\overline{{1,2,4,2,1,8}}]$, as ...
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Elements of a Dedekind domain can be chosen to have valuation $1$ with respect to one prime, $0$ everywhere else

I noticed this is true for $\mathbb{Z}$, but I was wondering whether it was true in general. Let $R$ be a Dedekind domain and $P_1, ... , P_s$ maximal ideals. The localized ring $R_{P_i}$ is a ...
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Is there a known method for finding extremely huge squarefree numbers?

People often compete to beat the record for largest known prime (it is currently $2^{57,885,161}-1$). There are also big money prizes for finding explicit prime numbers exceeding specific magnitudes. ...
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Formula to round up to the next multiple not divisible by $2$ or $3$?

I want a formula that rounds up any integer to the next multiple of a given prime, which is not divisible by $2$ or $3$, so it is either $p$ or $5p \pmod{6p}$. The simplest formula is preferred. I've ...
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The zeros of $2\,\xi(s)-1$. Is there anything known about the curves they lie on?

Take the well known integral: $$\displaystyle \pi^\frac{-s}{2}\,\Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\int_1^{\infty} \left({x}^{\frac{s}{2}-1} + {x}^{\frac{-s}{2}-\frac12}\right)\,\psi(x)\, ...
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Find extension of $\mathbb{Q}$ containing components of eigenvectors of a matrix

Given a matrix $\mathbf{A} \in \mathbb{Z}^{d \times d}$ I need to find an algebraic number $a$ of minimal degree, such that all eigenvalues and eigenvector's coordinates of $\mathbf{A}$ belong to the ...
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Problem on the digits of $n!$

let $m$ be a natural number, is it always possible to find an $N\in \mathbb{N}$ such that $m$ or more "$0$" digits (excluding the terminal ones) appears amongs the decimal digits of $n!$ if $n\ge N$
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Number of Solutions to a Diophantine Equation

I am asked the following: 1) Solve the diophantine equation $y^3=x^2+2$. 2) Show that the number of integer solutions to $y^p=x^2+2$ for any odd prime $p$ is at most $p-1$. The first part is easily ...
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find all coprime sets with size $m \leq n$

I am trying to find all sets of size $m$ with elements in $\{1, 2, 3, \dots n\}$ such that all of the elements in the set are pairwise coprime. I would like to find the number of sets as well as the ...
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Generating all lesser numbers of two coprime numbers

Let's say I have two coprime positive integers, $a$ and $b$. How would you go about proving that it is possible to make all integers between 1 and $max(a,b)$ by subtracting them from each other? For ...
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Number of points over elliptic curve is p+1 given…

Suppose that -1 is not a square in $\mathbb{Z_p}$. Show that the number of points on the elliptic curve $y^2=x^3+ax$ over $\mathbb{Z_p}$ is $p+1$. Hint: Use the fact that $x^3+ax$ is an odd function. ...
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Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
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Problem following detail of Szemeredi's original proof that sets of positive upper density have arbitrarily long arithmetic progressions

Here is a link to Szemeredi's original proof of Szemeredi's theorem: http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27132.pdf I am stuck on Fact 6 (page 212) because I don't follow the "induction" ...
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$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let ...
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Ratio of maximal to minimal jump in the set of angle multiples (corrected)

(This is the corrected version of the question I asked here: Ratio of maximal to minimal jump in the set of angle multiples.) Let $S^1$ be the unit circle in the complex plain. Let $d:S^1\times ...
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News on SG values of Grundy's Game?

Is there any recent research into the Sprague-Grundy values of Grundy's game? It was calculated to $2^{35}$ integers but with no sight of recurrence. Has anyone come up with anything new to compute ...
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Useful restrictions for prime factors of a sum of two powers with coprime exponents

Every prime factor of a number of the form $$a^2+b^2$$ with gcd(a,b)=1 has -1 as a quadratic residue. Does this work only for exponents with a common factor, or are there useful restrictions also ...
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polynomials and functions on $\mathbb{Z}/n\mathbb{Z}$

My general question is How is the set of all polynomial functions on $\mathbb{Z}/n\mathbb{Z}$ structured? What is the number of such functions? How, given a function, one can recognize that it is ...
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An application of Hensel's lemma to some different fields

I would like to prove the following. Let $p>2$ be a prime number, $\mathbb{Q}_{p}$ the field of p-adic numbers Let $u\in \mathbb{Z}_{p}^{\times}$ be a unit. (1)Prove that the following are ...
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Are the Fourier coefficients of a new form real?

Let $f\in S^\text{new}_k(Γ_0(N))$ be a $\text{newform}$ . Are all its Fourier coefficients real? Of course the Hecke operators $T_n$ are selfadjoint for $(n,N)=1$, but is it also true for all $n$?
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$\frac{p^p-1}{p-1}, \frac{p^p+1}{p+1} $ cannot be prime power at the same time

$p\gt3$ is a prime, then the two numbers $$\frac{p^p-1}{p-1}, \; \frac{p^p+1}{p+1} $$ cannot be prime power at the same time I have no clue about it. Could anyone help me? Thanks a lot.
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Prime factor of $14 \uparrow \uparrow 3 + 15\uparrow \uparrow 3$ wanted

Checking the prime factors of the numbers $$z(a,b) \ := a \uparrow \uparrow 3 \ +\ b \uparrow \uparrow 3 \ ,$$ a,b positive integers with a < b the number $$z(14,15) = 14 \uparrow \uparrow ...
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Transcendental numbers involving primes?

Is the prime zeta function value $$ P(2)=\sum_{p \in \mathrm{primes}} \frac{1}{p^2} = 0.452247420041065498506543364832247934173231343\ldots $$ a transcendental number ? What about the following sum ...
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How generalize the alternating Möbius function?

Here is what I want to do, I have this matrix: $$\displaystyle T = \begin{bmatrix} +1&+1&+1&+1&+1&+1&+1 \\ +1&-1&+1&-1&+1&-1&+1 \\ ...
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Showing that a number is part of sequence A000275 in OEIS

Consider the sequence of integers defined recursively by $c_0 = 1$ and $$ c_p = \sum_{l = 0}^{p-1} (-1)^{p+l+1} \binom{p}{l}^2 c_l $$ for $p \geq 1$. This is sequence A000275 in the online ...
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Squarefree numbers

From the Wolfram MathWorld page on squarefree numbers, "There is no known polynomial time algorithm for recognizing squarefree integers or for computing the squarefree part of an integer." ...
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$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n)$?

Let $K$ be a totally ramified extension of $\mathbb Q_p$ of degree $n$. Then $$ \mathcal O_K/(p) \cong \mathbb F_p[T]/(T^n) .$$ What is this isomorphism?
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Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
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How prove this $\frac{a_{1}+a_{2}+a_{3}+\cdots+a_{p^2-q^2}}{p^2-q^2}=p^2-q^2+1$

Nice Question: let $p,q\in N^{+}$,and $p>\sqrt{2}q\ge \sqrt{2}$ show that: there are exsit $a_{1},a_{2},a_{3},\cdots,a_{p^2-q^2}$ such $a_{i},(i=1,2,\cdots,p^2-q^2)$ are all positive square ...
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special values of zeta function and L-functions

I was reading in some lectures notes about the Riemann zeta-function which takes on special values: $$\zeta(2) = \sum \frac{1}{n^2} = \frac{\pi^2}{6}$$ In fact, we can compute even values of the ...
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Equations with operations as variables

There are number puzzles which go like this: given 2_2_1=5. Insert operations (addition and multiplication) to make the equation valid. Solution: (+,+) or ($\cdot$,+). My question is: does anybody ...
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Totient summatory function

Let $\Phi(n) = \sum_{k=1}^n \phi(k)$ be the totient summatory function. Here is an interesting conjecture I've made: The ratio $\Phi(n^2)/\Phi(n)$ is an integer only for $n=1,2,3,5$ and $6$. I made a ...
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Can one define informational content of a mathematical expression?

At least in physicist's thinking, information, vaguely, is something that allows one to select a subset from a set. Say, a system can be in states A and B, we have done a measurement on it ...
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Remark in “Multiplicative Number Theory” by Davenport

On page 9 in "Multiplicative Number Theory" by Davenport, he remarks that "It is a remarkable fact that no one has yet given a simple and direct proof that the value of the finite sum in (7) is ...
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Conditions for polynomial $f$ such that $f(n) \in \mathbb{N}$ for enough $n \in \mathbb{N}^+$ implies $f$ has rational coefficients

This question is suggested by this one: prove: coefficients of $f(x)$ are rational numbers What are the weakest sufficient conditions and strongest necessary conditions on a set of positive integers ...
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How to solve $a^x=x\,(\text{mod } 10^{12})$

For given $1< a< 50001$, I want to find $x$ for which $a^x=x ~(\text{mod }10^{12})$. My idea is to find $x_1$: $a^{x_1} = x_1~ (\text{mod } 2^{12})$ and $x_2$: $a^{x_2} = x_2 ~(\text{mod } ...
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Solving $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$

If $m$ and $n$ are distinct positive integers then does the equation $(x^m+y^m-z^m)^n=(x^n+y^n-z^n)^m$ $\space$has any solution , for $x,y,z$ , in positive integers with $x,y,z$ all not equal ?
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Symmetry in the Sum of Factors

The factors of 72 are 1,2,3,4,6,8,9,12,18,24,36,72. The factors can be organized as such $$m=1,2,3,4,6,8$$ $$n=72,36,24,18,12,9$$ Knowing the symmetric properties and that the 12 factors exist. Pair ...
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A computation modulo 1223

I want to compute $48^{306}$ modulo $1223$. I can't use a calculator, hence I tried to simplify something. I have $306=2^8+2^5+2^4+2$, thus $48^{306}=48^{2^8+2^5+2^4+2}=48^{2^8}\cdot 48^{2^5}\cdot ...
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Proving that the finite sum of the each reciprocal of any sequence of integers with common difference is not an integer.

Question : Could you show me how to prove that $\sum_{j=1}^{n}\frac{1}{a+jd}$ is not an integer for any integers $a\gt1, d\gt0$. A week ago, I found the following question in a book: Prove that ...
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Borel lemma on rationality

I am looking for an enlightening proof of the following fact: Let $F(t)\in \mathbb{Z}[[t]]$ and suppose $S$ is a finite set of places on $\mathbb Q$ containing $\infty$. If for every $v\in S$ $F(t)$ ...
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$\pi$, disjunctive numbers, and finite sequences of given length

It is an open problem whether the number $\pi$ is disjunctive in base $10$, i.e., whether every finite sequence appears (at least once) in the base $10$ expansion of $\pi$. Of course, every sequence ...
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Why does there exist a totally positive element?

Given a totally real number field $K_0$ and its totally imaginary quadratic extension $K$. Does there exist an element $\psi\in O_K$ such that $-\psi^2$ is totally positive in $K_0$? Why?
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How does Hildebrands proof of the prime number theorem via large sieve work?

How does the sieve inequality (I may not know the most general form) lead to the distribution of primes? To me, these concepts do not seem to be related. Can their connection be described in a ...