**16**

votes

**5**answers

917 views

### What arrangement of unit cubes minimizes surface area?

For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below).
Question A. How does one arrange $n$ unit cubes ...

**16**

votes

**4**answers

893 views

### Given a prime $p$ how many primes $\ell<p$ of a given quadratic character mod $p$?

There was this question for which my response was unusally popular, so I dare to ask the following:
(1) Given a prime $p>2$, how many primes $\ell < p$ there exist which are quadratic residues ...

**2**

votes

**1**answer

60 views

### Compactness of adelic quotients for unipotent groups over global fields

Let $K$ be a global field, $\mathbb{A}_K$ the ring of adeles, and $U$ a unipotent algebraic group over $K$. Why is $U(\mathbb{A}_K)/U(K)$, when endowed with the quotient topology, compact?

**48**

votes

**4**answers

2k views

### What is the limit of gcd(1! + 2! + … + (n-1)! , n!) ?

Let $s_n = \sum_{i=1}^{n-1} i!$ and let $g_n = \gcd (s_n, n!)$. Then it is easy to see that $g_n$ divides $g_{n+1}$. The first few values of $g_n$, starting at $n=2$ are $1, 3, 3, 3, 9, 9, 9, 9, 9, ...

**3**

votes

**2**answers

248 views

### Are there any new results on approximating Riemann $\Xi$ function by Polya-like Fourier transforms?

I posted [this question][1] at math.stackexchange.com and was told that it is more appropriate to post this research related question here at mathoverflow.
So I re-post it below.
Riemann $\Xi(z)$ ...

**49**

votes

**4**answers

4k views

### Strange (or stupid) arithmetic derivation

Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...

**12**

votes

**4**answers

459 views

### Is the sequence of Apéry numbers a Stieltjes moment sequence?

Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...

**4**

votes

**1**answer

107 views

### $L^2$ discrepancy bound for sequences in $[0,1)$

Given a sequence $x_1,x_2,\dots$, let $D_n$ be the $L^2$-norm of the function $f_n$ whose value at $t \in [0,1)$ is $nt$ minus the number of $1 \leq i \leq n$ with $x_i \leq t$. What can be said ...

**1**

vote

**1**answer

195 views

### Asymptotic formula for the number of ways to write a number as the sum of $k$ triangular numbers

How would one derive an asymptotic formula for the number of representations of a number $n$ as the sum of $k$ numbers of the form $\frac{m(m + 1)}{2}$
I think that one could use the circle method, ...

**1**

vote

**0**answers

109 views

### Approximation of the form $\frac{1}{u}\pm\frac{1}{v}$

Given positive integers $m<n\in\mathbb{N}$ is there an algorithm to find integers $z_1, z_2\in \mathbb{Z}\setminus\{0\}$ such that $\frac{m}{n}$ is best approximated by ...

**0**

votes

**1**answer

186 views

### A question on degree 4 binary forms

Suppose that we have a binary form $f(x,y) \in \mathbb{Z}[x,y]$ of degree 4, and that we explicitly have
$$\displaystyle f(x,y) = a_0 x^4 + a_1 x^3 y + a_2 x^2 y^2 + a_3 xy^3 + y^4,$$
so that $(0,1)$ ...

**1**

vote

**0**answers

211 views

### How to find generators to Mordell weil groups of elliptic curves?

I am new to branch of elliptic curve and algebraic number theory .I want to find generators to Mordell Weil group of the Elliptic Curve $y^2=x^3-6321363052$ and class number of $\mathbb ...

**5**

votes

**3**answers

369 views

### Bounds on horizontal minima of the Riemann zeta function

It is known that $\zeta(s)$ has an infinity of zeros in the strip $0<\sigma<1$ and that those zeros become closer together as $t\rightarrow\infty$. More precisely, Littlewood showed that there ...

**-7**

votes

**0**answers

131 views

### Theory of mnemonics [on hold]

Even for the typical most skilled (human) number theorist it is hard to reproduce only the first 10 digits of $\pi$ in moderate speed (without physically reading them off).
On the other hand there ...

**7**

votes

**1**answer

926 views

### An interesting double coset in the theory of automorphic forms

Does anyone have some idea to describe the double coset $P(F)\backslash G(F)/H(F)$ , say using Weyl group elements ? Here $G=GL_n\times GL_{n-1}$ is defined over a number field $F$ , $H=GL_{n-1}$ ...

**0**

votes

**0**answers

104 views

### q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity.
Show that the polynomial ...

**1**

vote

**0**answers

121 views

### Rational map and diophantine sets [on hold]

A subset $A$ of $\mathbb{Q}^m$ is a diophantine set over $\mathbb{Q}$ if there is $P(\vec{a},\vec{x}) \in \mathbb{Q}[a_1,...,a_m,x_1,...,x_n]$ such that $\forall \vec{a} \in \mathbb{Q}^{m}$,
...

**13**

votes

**3**answers

406 views

### Probing the generalization of the abc conjecture to more than 3 variables

Browkin and Brzezinski, in "Some remarks on the $abc$-conjecture", Math. Comp. 62 (1994), no. 206, 931–939, state the following generalization of the $abc$ conjecture to more than three variables:
...

**3**

votes

**1**answer

127 views

### Optimal lower bounds for the sum of digits in base $b$

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$
(e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...

**5**

votes

**2**answers

189 views

### Estimates on derivatives of Bessel function

In Duke, Friedlander and Iwaniec's Erratum on "Bounds for automorphic L–functions. II"
They have the following estimates for derivatives of Bessel functions: For $k \geq 2$
\begin{align}
& ...

**11**

votes

**1**answer

641 views

### The “Level N modular equation for delta” in characteristics 3, 5, 7 and 13

When $N > 1$, the modular forms $\Delta(z)$ and $\Delta(Nz)$ are algebraically independent over the complexes, and the same then is true of their expansions at infinity. But using the fact that
the ...

**1**

vote

**0**answers

63 views

### Are all complex zeros of $Li_s(i)\, + \, Li_{1-\overline{s}}\,(-i)$ equal to the $\rho$'s?

Take the well known square relationship for polylogarithms:
$$Li_s(z)\, + \, Li_{s}(-z)=2^{1-s}Li_s(z^2)$$
Assume $z=i$:
$$Li_s(i)\, + \, Li_{s}(-i)=2^{1-s}Li_s(-1)=-2^{1-s}\,\eta(s)$$
with ...

**11**

votes

**1**answer

608 views

### Tight prime bounds

This is a cross-post of this question on MSE. I would not usually do this, but have decided to in this case since it has had no responses having been posted as a bounty question. I did not delete the ...

**2**

votes

**1**answer

142 views

### Quotient of Projective line over rationals with an infinite subgroup of PGL(2,Q)

I am looking for references for the following; how to calculate quotient of the projective line over the field of rationals with an infinite subgroup of PGL(2,Q), e.g, of the form
$
\left(
...

**7**

votes

**1**answer

335 views

### A question on BSD conjecture

If $E$ is an elliptic curve over $\mathbb{Q}$, and $K$ is an imaginary quadratic field. If $rank E(K)\leq 1$ and both $E$ and the quadratic twist of $E$ by $K$ satisfy the full BSD conjecture, does ...

**-1**

votes

**0**answers

64 views

### Are (odd) perfect numbers divisible by a repdigit (in another base)? How about by a repunit?

[This question was cross-posted from MSE.]
A positive integer $N$ is said to be a perfect number if
$$\sigma(N) = 2N,$$
where $\sigma(x)$ is the sum of the divisors of $x$. For example, $6$ is ...

**0**

votes

**0**answers

52 views

### odd squarefree and squareful neighbors [migrated]

There are squarefree numbers $n=\prod_{i=1}^{k}p_i$ so that $n+2$ is not squarefree (e.g. $115+2=3^2.13$).
Are there infinite many such $n$?
Are there numbers n with arbitrarely many prime-factors?
...

**6**

votes

**0**answers

184 views

### Invariant obstructions to gluing Galois representations on elliptic curves

Let $E$ be an elliptic curve over $\overline{\mathbb F_p}$, or another separably closed field of characteristic $p$. Let $K$ be the function field of $E$, and let $K_p$ be the local field at a point.
...

**9**

votes

**1**answer

480 views

### Why is the section conjecture important?

As in the title, I want to know the reason for importance of the section conjecture. Of course, the statement of conjecture is important as itself, even I cannot fully grasp the soul of it. However, ...

**9**

votes

**2**answers

302 views

### Ideal classes fixed by the Galois group

Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...

**9**

votes

**2**answers

351 views

### Elliptic curves with trace of Frobenius values always congruent to 0 modulo 2

Let $E/\mathbb{Q}$ be an elliptic curve and suppose that the trace of Frobenius values are such that $a_{p}(E) \equiv 0 \pmod{2}$ for all odd primes avoiding the conductor. Is it the case that $E$ ...

**0**

votes

**0**answers

93 views

### Any result, example or conjecture about the computational complexity of transcendental number that can not be computed by linear time Turing Machine [on hold]

The Hartmanis-Stearns Conjecture is restated by Prof.Lipton as:
Suppose that a linear time Turing Machine computes the first $n$ digits of the real number $r$ in base ten. Then, the number is either a ...

**0**

votes

**0**answers

72 views

### Are there any relevance between coefficients of simple continued fraction of quadradic algebraic number and algebraic number with degree $2^n$ [on hold]

Let $\sqrt{c}$ be quadratic algebraic number, We know that $[a_0;a_1,a_2,\dots ]$ the coefficients of simple continued fraction of $\sqrt{c}$ the quadratic algebraic number is periodical. ...

**9**

votes

**0**answers

139 views

### Cubic fields correspond to $3$-torsion ideals in quadratic fields, or to order $3$ characters of quadratic class groups?

I was watching Dick Gross's laudation for Manjul Bhargava, followed up by one of Bhargava's talks, and I realized I was confused about something.
Bhargava says (around 21 minutes) that the orbits of ...

**2**

votes

**0**answers

44 views

### Image of the typenorm contains the squares

I am having a look at the paper Explicit CM-theory for level 2-structures on abelian surfaces by Bröker, Gruenewald and Lauter, and there is an argument which I don't understand. The main reason being ...

**12**

votes

**0**answers

218 views

### Artin L-function and Zeta function of twisted Dirac operator

If one thinks of a Frobenius as an element in the fundamental group of an arithmetic curve and of a Galois representation $\sigma$ as a flat connection on the curve, then the definition of the Artin ...

**0**

votes

**0**answers

36 views

### Stable analytic manifold under simple action

For an integer $m > 1$, let us define the action
$$
f: X_i \to (1+X_i)^{m - 1 }
$$
on variables $X_1,\dots,X_N$. Consider the analytic manifold $V(I)$ defined by the ideal $I$ in
...

**2**

votes

**0**answers

149 views

### Metric on the set of subsets of the rational primes

Note: this is a revision of an earlier post. It was kindly pointed out that my initial proposed metric was in fact not a metric, so this is a revised version.
I was thinking how to say that two sets ...

**2**

votes

**2**answers

362 views

### Exponential Sum Bound

In
http://131.220.77.52/files/preprints/diophantine/bruedern/wpminicubefinal.pdf, Bruedern and Wooley mention the following fact on the bottom of page 6:
Let
...

**2**

votes

**2**answers

210 views

### Reference for Skinner-Urban on the Iwasawa main conjecture for $GL_2$

Does anyone know the existence of an expository paper or a report discussing the work of Skinner-Urban
"The Iwasawa main conjecture for $GL_2$"?
I am interested in partucular in the case of elliptic ...

**1**

vote

**1**answer

149 views

### Quadratic Gauss sums: Explicit determinations?

Can anyone please tell me (give me a reference, preferably) if there is any explicit determination of sums of the form $g(n,\chi):=\sum_{r=1}^{q}\chi(r)e(\frac{rn+r^2}{q})$ where $\chi$ is a Dirichlet ...

**3**

votes

**1**answer

137 views

### Existence of real modular function with specific behavior as $q\to 0$

I am looking for a real modular function $F(q,\bar{q})$ such that in the limit of small $q,\bar{q}$ it behaves as:
$F(q,\bar{q})=(a_0 + a_1 (q + \bar q)+...)\log q \bar q+ (b_0 + b_1 (q + \bar ...

**2**

votes

**1**answer

243 views

### Does the set of automorphisms of a cyclic group exhibit some sense of randomness?

I prefer to proceed with a concrete example if I may. I appreciate that the answer might well be better explained with group theory, geometry and/or notions from probability theory, which I welcome.
...

**2**

votes

**0**answers

49 views

### On the uncountability of a subset of U-numbers of type $\leq m$

We say that $\xi\in \mathbb{R}$ is an $m$-ultra number if there exists a sequence $(\alpha_n)_n$ of $m$-degree real algebraic numbers, such that
$$
|\xi-\alpha_n|<(\exp^{[3]}(H(\alpha_n)))^{-n},\ ...

**1**

vote

**1**answer

211 views

### Is it possible to sum the divergent series with prime coefficients?

It is known that the series $$ P := \sum_{n=1}^{\infty} p_{n} \qquad \text{where } p_{n} \text{ is the n'th prime} $$ cannot be summed by means of (prime) zeta function regularization. (The result was ...

**0**

votes

**1**answer

206 views

### A Problem Concerning Odd Perfect Number

Briefly, prove that every odd number having only three distinct prime factors cannot be a perfect number.
I know there are results much stronger than the one above, but I am looking for an answer ...

**10**

votes

**3**answers

1k views

### Finding integer points on elliptic curves via divisibility conditions like $(a+b)^2 \mid (2b^3+6ab^2-1)$

Is the following conjecture correct?
Conjecture. The divisibility condition $(\alpha+\beta)^2 \mid (2\beta^3+6\alpha\beta^2-1)$ has no solutions in positive integers $1 \le \beta < \alpha < ...

**4**

votes

**1**answer

134 views

### Constructing a family of convergents from continued fractions formed by a set of prime partial quotients

For a given real number $x$, the continued fraction representation $x = [a_0; a_1, a_2, \cdots]$ where $(a_n)_{n \geq 0}$is defined by setting $x = \alpha_0$, then $a_i = \lfloor \alpha_i \rfloor$, ...

**2**

votes

**0**answers

83 views

### What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers $\sqrt{A}$ and the integer $A$

What is the relation between the length of period of simple continued fraction expansion of quadratic algebraic numbers and the integer
As we know,$\sqrt{2} = [1;2,2,2,2,…]$; while $\sqrt{14}= ...

**3**

votes

**1**answer

209 views

### Dynamics in the integers - Floor function

Let $\alpha$ be an irrational with $0<\alpha<1$. Consider the function given by \begin{align*}
f: &\mathbb{N}\longrightarrow \mathbb{N}\\ &x\longmapsto [ \alpha\cdot x]\end{align*} where ...