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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Eisenstein spectrumfor $GL(n)$

Fix a global field $F$. Does every automorphic representation of $GL(n)$ appear as an arbitrary twists in the continuous spectrum of $GL(m)$, $m>n$? What happens for the automorphic ...
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3answers
893 views

rational points of an algebraic variety

In http://en.wikipedia.org/wiki/Rational_point we read : a $K$-rational point is a point on an algebraic variety where each coordinate of the point >belongs to the field $K$. This means that, if ...
3
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2answers
354 views

Euler-Mascheroni constant expression, further simplification

The Euler-Mascheroni constant gamma is defined as: $$\gamma=\lim\limits_{n \rightarrow \infty}\left(\sum\limits_{m=1}^{n} \frac{1}{m} - \log(n)\right)$$ From this previous question Do these series ...
9
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2answers
2k views

How to check whether an ideal is a prime (or maximal) ideal?

I have a ring $R$ which is known to be a Dedekind domain, but not necessarily a Euclidian domain, and a nonzero ideal generated by one or two elements in this ring. How can I check if this ideal is a ...
11
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4answers
665 views

Is ln(n) transcendental for all integer $n > 1$?

Is $\ln(n)$ transcendental for all $n \in \mathbb{N} \setminus \{0, 1\}$? Is the answer even known?
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1answer
105 views

The discrepancy of certain sequences

To my knowledge, the best upper bound for the discrepancy of sequences of the type $(n\alpha) (\mod 1), n=1,2,...$ is $$\frac{ND_N(\alpha)}{\log N\log\log N}\to \frac{2}{\pi^2}$$ in measure. My ...
6
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1answer
591 views

Three consecutive sums of two squares

$0, 1, 2$ is an example of three consecutive non-negative integers $n, n+1, n+2$ which are each the sum of two integer squares. Using modular arithmetic you can prove that in all of these triplets $n ...
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4answers
620 views

Do these series converge to logarithms?

It is well known that $$1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}... =\log(2).$$ If we consider the array: $T(n,k) = -(n-1)\; \text{ if }\; n|k, \;\text{ else } \;1,$ Starting: ...
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3answers
447 views

Fractional part of $b \log a$

From the problem... Find the minimal positive integer $b$ such that the first digits of $2^b$ are 2011 ...I have been able to reduce the problem to the following instead: Find minimal $b$ ...
5
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1answer
184 views

Asymptotic formula for $k$-partitions of a number

Asymptotic formula for all the partitions of a number is given by $$p(n) \sim \frac{1}{4n\sqrt{3}} e^{\pi \sqrt{\frac{2n}{3}}}$$ Only fraction of those will be $k$-partitions. What is asymptotic ...
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1answer
167 views

Non-negative solutions of the equation $5^n+7^m=k^3$

How can I find all triples $(m,n,k)$ of non-negative integers such that $5^n+7^m=k^3$?
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1answer
129 views

Prove or refute that $\frac{t^a-1}{t^b-1}$ is not a integer if $a \mod b \neq 0$

Hi guys in my last question I got the wrong idea maybe because a poor problem's description or maybe because of my poor English skills. So, anyway I found out the problem requires to be a integer. ...
3
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2answers
203 views

Prove or refute that $\frac{t^a-1}{t^b-1}$ has more than 100 digits if $a \mod b \neq 0$

I'm a computer science student from Mexico and I have been training for the ICPC-ACM. So one of this problems called division sounds simple at first. The problem is straight for you have and 3 ...
4
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1answer
169 views

On a characterization of the tamely ramified coverings of the fraction field of a strict Henselian ring

Let $K$ be the field of fractions of a strictly Henselian discrete valuation ring $A$. Following Milne ("Etale Cohomology", Chapter I, Paragraph 5, example e)), $Spec(K)$ is the algebraic analogue of ...
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1answer
411 views

Algorithm for partitioning n into distinct primes

I am looking for an algorithm that will partition a positive integer into distinct primes. The number of partitions is given by this OEIS sequence: https://oeis.org/A000586 To be more specific, I am ...
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3answers
398 views

Diophantine equations: ternary forms

Thue proved that all Diophantine equations consisting of an irreducible binary form (cubic or higher) equal to a constant, i.e., $$c_nx^n+c_{n-1}x^{n-1}y+\cdots+c_oy^n=k$$ ($n,k$ fixed) have finitely ...
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1answer
132 views

Prove that a holonomic (p-recursive) difference equation returns only integral values

Consider the recurrence given by $(n+1)^2 a_{n+1} = (9n^2+9n+3)a_n-27n^2 a_{n-1}$ $a_0 = 1, a_1 = 3$. Clearly, $a_n$ is rational, but unexpectedly, the recurrence seems to output only integral ...
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4answers
401 views

Structure of $\mathbb{Z}[[x]]/(x-n)$

Is $\mathbb{Z}[[x]]/(x-6) \cong \widehat{\mathbb{Z}}_2 \times \widehat{\mathbb{Z}}_3$? It seems intuitive that $\mathbb{Z}[[x]]/(x-p)$ is the p-adic numbers, and I think this is not too hard to ...
4
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2answers
324 views

Partition function- without duplicates

Is there a function, equivalent to the partition function, that does not allow duplication? Or, alternatively, for any N, how many partitions would there be- disallowing any that have the same integer ...
6
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1answer
241 views

Modular forms database

Suppose one was given a sequence and $a_{0}, a_{1}, a_{2}, \ldots$. Is there a searchable database somewhere to see if $a_{0} + a_{1}q + a_{2}q^{2} + \cdots$ is expressible as modular form (or some ...
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1answer
174 views

The $p$-adic expansion of a function of $p$

Let $p\neq 2$ be prime. I am asked in a revision question to find the $p$-adic expansion of $(1+2p)/(p-p^3)$. The best I could do was find the $p$-adic norm, which I got as $p$ (please correct me if ...
3
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1answer
140 views

Sum of powers of divisors

How do I prove the following: $$ \sigma_k(u)\sigma_k(v) = \sum_{d|gcd(u,v)} d^k\sigma_k\left(\frac{uv}{d^2}\right) $$ when $$ \sigma_k(n) = \sum_{d|n} d^k $$ Can someone give me a clue on that one? ...
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7answers
2k views

Proof for divisibility by $7$

One very classic story about divisibility is something like this. A number is divisible by $2^n$ if the last $n$-digit of the number is divisible by $2^n$. A number is divisible by 3 (resp., by ...
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3answers
232 views

Can one show that $\sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N}$ without using the Euler-Maclaurin formula?

I would like to prove that $$ \sum_{n=1}^N\frac{1}{n} -\log N - \gamma \leqslant \frac{1}{2N} $$ without using the Euler-Maclaurin summation formula. The motivation for this is that I have come very ...
9
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2answers
679 views

Can insight be derived from direct formulae for prime number functions?

Dear StackExchange Community, I am an amateur enthusiast and was attempting to construct a formula for the n th prime using elementary functions - I didn't achieve this* but I did come up with some ...
8
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2answers
595 views

Bulgarian Solitaire: Size of root loops

I first learned of Bulgarian solitaire from one of Martin Gardner's books a while ago and have since investigated it somewhat. Google-searching has revealed that surprisingly little work has been done ...
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1answer
357 views

Are there infinitely many primes next to smooth numbers?

A side discussion over on this question has left me curious: is there any $B$ for which it's known that there are infinitely many primes adjacent to $B$-smooth numbers (i.e., for which there are ...
3
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1answer
547 views

How many solutions are there to $F(n,m)=n^2+nm+m^2 = Q$?

Let $n,m$ be two positive integers, we consider: $$F(n,m)=n^2+nm+m^2$$ Let $Q$ be one value reach by $F(n,m)$. How many different pairs $(n,m)$ verify $F(n,m)=Q$?
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4answers
3k views

Primes of the form $n^2+1$ - hard?

I met a student that is trying to prove for fun that there are infinitely many primes of the form $n^2+1$. I tried to tell him it's a hard problem, but I lack references. Is there a paper/book ...
9
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5answers
873 views

RSA: is it easy to find the public key from the secret key?

Please answer to my question about RSA, public key cryptosystem. I know that it is not easy to find the secret key from the public key . Is it relatively easy to find the public key from the secret ...
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2answers
284 views

A characterisation of tame ramification

The following is the statement from Algebraic Number Theory by Neukirch (Chapter 2 Proposition(7.7) p155) Blockquote Suppose $K$ is Henselian field, $p=char(\kappa)$ , the character of the ...
5
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3answers
306 views

Help on notation: $\mathbb{Z}/n\mathbb{Z}$ vs. $\mathbb{Z}_n$

I have difficulties understanding the difference between the following two notations: $\mathbb{Z}/n\mathbb{Z}$ (which denotes a quotient group) and $\mathbb{Z}_n$. Are they equivalent? PS1: The ...
2
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1answer
455 views

Prime power divisors of the fibonacci numbers

I came across a result that if $p^n \mid f_m$ for some $n\geq1$ then $p^{n+1} \mid f_{pm}$. I was wondering if this is true.
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3answers
2k views

Squares in arithmetic progression

It is easy to find 3 squares (of integers) in arithmetic progression. For example, $1^2,5^2,7^2$. I've been told Fermat proved that there are no progressions of length 4 in the squares. Do you know ...
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1answer
189 views

Proving $\sum\limits_{k=1}^{\pi(n)-1} [ \theta(p_k) (1/{p_k}-1/p_{k+1})] -\ln(n)$ converges

Prove the sequence $a_n$ defined by $a_n = \sum\limits_{k=1}^{\pi(n)-1} [ \theta(p_k) (1/{p_k}-1/p_{k+1})] -\ln(n)$ converges, where $p_k$ denotes the $k$-th prime and $\vartheta(x)$ is Chebyshev's ...
12
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3answers
2k views

evaluate the last digit of $7^{7^{7^{7^{7}}}}$

I found this puzzle online. Since I'm not good at number theoretic kind of problems I'm going to propose it in this form. If you have a number $x$, in this case $x=7$, how do you evaluate the last ...
7
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2answers
288 views

On natural solutions of the equation $y^{3}-3^{x}=100$

How can I solve the equation $$y^{3}-3^{x}=100$$ over positive integers?
8
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1answer
280 views

Does there exist a self-adjoint operator whose spectrum consists wholly of prime numbers?

The zeros of the canonical Riemann zeta function have been compared to the prime numbers, and they have a number of special, definite connections. The infamous zeros have also been conjectured to be ...
8
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2answers
379 views

A simple question about Iwasawa Theory

There has been a lot of talk over the decades about Iwasawa Theory being a major player in number theory, and one of the most important object in said theory is the so-called Iwasawa polynomial. I ...
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1answer
542 views

Class number Formula and Birch and Swinnerton Dyer conjecture

can anyone please explain me in simple terms ,why cant the stuff done in the case of pell conics cant be done for elliptic curves,i mean we can prove the Birch and Swinnerton Dyer in a similar way by ...
2
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2answers
201 views

If I use a result from Wikipedia, how should I cite that result?

I am writing a short article on the $\zeta$-function, and I use the "Rising factorial" representation of the function in the article. You can see that in that here. Now there is no citation in ...
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2answers
228 views

Is every even number a value of the Carmichael function?

Except for the 1, all other values of the Carmichael function $\lambda$ are even. Does $\lambda$ take all even numbers? Is there an infinite family of even numbers that are not values of $\lambda$ ? ...
89
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1answer
3k views

All polynomials with no natural roots and integer coefficients such that $\phi(n)|\phi(P(n))$

Let $P(x)$ be a polynomial with integer coefficients such that the equation $P(x)=0$ has no positive integer solutions. Find all polynomials $P(x)$ such that for all positive integers $n$ we have ...
2
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2answers
180 views

Deciding whether a given number is a totient or nontotient

The following algorithm decides if a number $n>0$ is a totient or a nontotient: ...
3
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1answer
130 views

Multiplication of coefficients in Dirichlet series

This appears to be a relationship: $\sum\limits_{p\;\text{prime}} \frac{1}{p^s} = \log\zeta (s) - \sum\limits_{n=1}^{\infty}\frac{\sqrt{a_{n}b_{n}}}{n^{s}}$ where $a_{n}$ is a sequence of fractions ...
8
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1answer
154 views

How to extend Galois character?

Let $D_p$ be a decomposition group of $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$ above $p$ (for all $p$) and $I_p$ the inertia group. Let $\chi_p$ be a character of $D_p$, such that $\chi_p$ is trivial ...
4
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8answers
599 views

Order of cyclic groups

Wikipedia says: It is known that $(\mathbb{Z}/n\mathbb{Z})^\times$ is cyclic if and only if n is 1 or 2 or 4 or $p^k$ or $2p^k$ for an odd prime number p and k ≥ 1. The statement seems provable ...
8
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1answer
334 views

what is the name of this number? is it transcendental?

Consider the number with binary or decimal expansion 0.011010100010100010100... that is, the $n$'th entry is $1$ iff $n$ is prime and zero else. This number is ...
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1answer
110 views

Efficiently finding an integer satisfying a certain congruence

I am trying to solve an exercise from a cryptography textbook and I am stuck with these specific subquestion - any kind of suggestion is welcome! Let $\alpha = 12$ be a generator of the group ...
4
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0answers
135 views

Hecke operators as endomorphism of Jacobians of modular curves

Let $p$ be a primes that does not divide $N$, then $T_p$ defined an endomorphism $J_0(N)\to J_0(N)$. what is $T_p^\vee$? In other words, we naturally have $J_0(N)^\vee \xrightarrow{T_p^\vee} ...