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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3
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0answers
174 views

Hauptmoduls for modular curves

If I have a modular curve, how does one in general find a Hauptmodul for this curve?
15
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1answer
1k views

Using floor, ceiling, square root, and factorial functions to get integers

So, several days ago, I was introduced to the "Four game" again. The object of the game is to use four 4's to produce as many integers as you can. You are allowed to use addition, subtraction, ...
6
votes
2answers
158 views

Does $\log p$ make sense in a finite extension of $\mathbb{Q}_q$?

Can we make sense of the logarithm of prime in some algebraic extension of $\mathbb{Q}_q$, where either $q \neq p$ or $p = q$ and both prime numbers? Some reflections: A naive starting point is ...
7
votes
2answers
309 views

Goldbach's conjecture and difference of squares

Someone came to me with the following observation: If $2n=p+q$ then $pq=n^2-m^2$ for some value of $0<m<n$ (namely, $m=n-p$ given $p\le q$). Now he claims that this is actually equivalent: that ...
3
votes
1answer
99 views

Equidistribution results vs transcendence degree

Consider $\alpha =(\alpha_1, \dots, \alpha_n) \in \mathbb{R}^n$ linearly independent over $\mathbb{Q}$, then the map $q \mapsto q \alpha:= ( q \alpha_1, \dots, q \alpha_n)$ gives a dense embedding ...
5
votes
1answer
133 views

In what locally compact abelian groups does $\mathbb{Q}$ embed densely?

I know that there is classification of local fields, but here is a closely related question: Can the additive group of $\mathbb{Q}$ be a proper dense subgroup of a locally compact abelian group, whose ...
1
vote
1answer
108 views

Something basic in “l-adic properties of the partition function” paper

I am trying to understand the basic result in this paper: http://www.aimath.org/news/partition/folsom-kent-ono.pdf My problem is with the example at the end of page 2. I understand it's supposed to ...
7
votes
4answers
798 views

Find all prime $p$ such that $x^2=-1$ has a solution in $\mathbb{Z}/p\mathbb{Z}$

I have found by a numerical experiment that first such primes are: $2,5,13,17,29,37,41$. But I cannot work out the general formula for it. Please share any your ideas on the subject.
3
votes
1answer
89 views

Where can I find rigorous statements about the spectral decomposition of reductive groups?

Given a global field $F$ and a reductive group $G$, where can I find the spectral decomposition of $$ L^2( Z(\mathbb{A}) G(F) \backslash G( \mathbb{A})).$$ I will need the result in this generality, ...
10
votes
2answers
891 views

What is the Pontryagin dual of the rationals?

Endow the rational numbers (or any global field) with the discrete topology, what will be the (compact) Pontryagin dual of the additive group and of the multiplicative group? I am suprised nobody ...
4
votes
1answer
138 views

Nature of zeros of $4\cdot2^{2p}\cdot3^p-4\cdot3^{2p}-7\cdot2^{3p}+8\cdot2^{2p}+8\cdot3^p-4=0$

Recently while coming up with an example for a paper I'm writing I find myself wanting something to say about how 'awful' the first positive root of the equation $$ ...
9
votes
2answers
384 views

Closed-form Expression of the Partition Function $p(n)$

I feel like I have seen news that a paper was recently published, at most a few months ago, that solved the well-known problem of finding a closed-form expression for the partition function $p(n)$ ...
5
votes
2answers
645 views

Find primitive element such that conductor is relatively prime to an ideal (exercise from Neukirch)

This is an exercise from Neukirch, "Algebraic Number Theory", Ch I, Sec 8, Exercise 2, pg 52. It really has me stumped. Suppose $A$ is a Dedekind domain, $K$ its field of fractions, $L$ a finite, ...
8
votes
2answers
161 views

Given $N$, count $\{(m,n) \mid 0\leq m<N, 0\leq n<N, m\text{ and } n \text{ relatively prime}\}$

I'm confused at exercise 4.49 on page 149 from the book "Concrete Mathematics: A Foundation for Computer Science": Let $R(N)$ be the number of pairs of integers $(m,n)$ such that $0\leq m < N$, ...
11
votes
2answers
678 views

Fermat's Last Theorem: implications (there is no new proof)

I am not experienced in Number Theory but what I know is that some results of this filed are applicable in other areas, e.g. algebra. For sure FLT made (and makes) people be interested in Number ...
5
votes
1answer
142 views

What is a good number theoretic interpretation of primitive geodesics on the modular surface?

Given $SL_2( \mathbb{Z})$, what interpretation is available for the hyperbolic elements? What is true, if we consider a congruence subgroup? I heard that there is a connection with certain class ...
2
votes
1answer
39 views

Search limit for even nontotients

n is an even nontotient if n is even and Euler phi(m) = n has no solution. I am looking for a search limit L(n) depending on n. By this I mean: If for all k, 1 <= k <= L(n) phi(k) = n has no ...
4
votes
1answer
99 views

A sine product with (almost) integer values

Let n, k be integers, $n>1$ and $k \perp n$ denote that k, n are coprime and let $S_n = \{1 \le k \le \lfloor n / 2 \rfloor : k \perp n \}.$ Then $$ n \left( \prod_{k \in S_{n}} \sin \left( k ...
2
votes
1answer
121 views

Alternatives to valuations

All valuations of $\mathbb{Q}$ are associated to either a p-adic field or the field of real numbers. Now, the classification of fields gives that every local field in characteristic $0$ is a finite ...
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vote
2answers
79 views

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 ...
9
votes
3answers
640 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 ...
2
votes
1answer
301 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
votes
2answers
1k 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 ...
10
votes
4answers
534 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?
0
votes
1answer
98 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
votes
1answer
479 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 ...
12
votes
3answers
522 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: ...
5
votes
3answers
434 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
votes
1answer
173 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 ...
2
votes
1answer
164 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 $5^n+7^m=k^3$?
3
votes
1answer
123 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
votes
2answers
193 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 ...
3
votes
1answer
150 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 ...
1
vote
1answer
359 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 ...
1
vote
3answers
343 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 ...
8
votes
1answer
127 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 ...
10
votes
4answers
382 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 ...
0
votes
1answer
264 views

Sum of two squares [duplicate]

Possible Duplicate: Prove that $n$ is a sum of two squares? I was reading this and began wondering if there is a general theorem that a number of a given form is the sum of two squares. I ...
3
votes
2answers
247 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
votes
1answer
218 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 ...
3
votes
1answer
159 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 ...
2
votes
1answer
130 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? ...
8
votes
7answers
1k 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 ...
8
votes
3answers
221 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 ...
8
votes
2answers
576 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 ...
7
votes
1answer
470 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 ...
5
votes
1answer
284 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
votes
1answer
504 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 $F(n,m)=Q$?
19
votes
4answers
2k 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
votes
5answers
766 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 ...