**0**

votes

**0**answers

45 views

### Estimating solutions to a binary form congruence with small moduli and prime inputs

Currently I am dealing with the following problem. Suppose that $F(x,y) \in \mathbb{Z}[x,y]$ is a binary form of degree $D \geq 2$ and $k \geq 2$ is an integer such that for all primes $p$, there ...

**1**

vote

**0**answers

429 views

### Is $ a^2b^2-ab-ap$ a perfect square for suitable $a,b\in \mathbb{Z}^+$?

Consider the expression
$$
a^2b^2-ab-ap\qquad (a,b\in \mathbb{Z}^+),
$$
where $p\equiv1\pmod{4}$.
Question. For every prime $p\equiv1\pmod{4}$ do there exist $a,b\in\mathbb{Z}^+$ such that ...

**9**

votes

**0**answers

234 views

### Do all complex zeros in the strip of $\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$ lie on the critical line?

Numerical evidence suggests that the complex zeros of:
$$f(s):=\frac{\zeta(s)}{\Gamma(s)} - \frac{\Gamma(1-s)}{\zeta(1-s)}$$
all reside on the line $\Re(s)=\frac12$, except for a finite few outside ...

**22**

votes

**0**answers

643 views

### On certain representations of algebraic numbers in terms of trigonometric functions

Let's say that a real number has a simple trigonometric representation, if it can be represented as a product of zero or more rational powers of positive integers and zero or more (positive or ...

**4**

votes

**0**answers

199 views

### A relation between the Gamma function and the Mobius function?

It is well known how altering the integral for the Gamma function:
$$\displaystyle \Gamma(s) = \int_0^\infty t^{s-1} e^{-t}\,dt$$
through substituting $t=nx$,
$$\displaystyle \Gamma(s)\frac{1}{n^s} ...

**2**

votes

**1**answer

237 views

### Prime Number Theorem on APs under various conjectures

I'm trying to find the best asymptotic expansions for $\pi(x; a, q)$ in various states:
Unconditionally we have
\begin{equation}
\pi(x; a, q) = \frac{\operatorname{li(x)}}{\phi(q)} + O\left(x ...

**6**

votes

**1**answer

366 views

### Some questions about the ring Z((x))

$\newcommand{\ZZ}{\mathbb{Z}}$
$\newcommand{\dim}{\text{dim }}$
Let me begin by apologizing for the length of this question, but I thought this might be interesting to some of you. This ring isn't ...

**5**

votes

**0**answers

404 views

### For which rational values of $c$ and $d$ are the numbers $\sin{(\pi\cdot c)}$, $\sin{(\pi\cdot d)}$ and $1$ linearly dependent over $\mathbb{Q}$?

A year ago, I posted this problem on [MSE]. After a number of edits, I have arrived at the following more general problem (suggested by Hjalmar Rosengren; see the comments below).
For which ...

**35**

votes

**2**answers

1k views

### Why does this sequence converges to $\pi$?

One of my daughters was having a small programming exercise.
Let's consider following algorithm:
Take a list of length $n$: $\ (1\,\ 2\,\ \ldots\,\ n)$.
Remove every $2$nd number.
From the ...

**14**

votes

**3**answers

2k views

### How did Ramanujan discover this identity?

Let $$\small F_n=(a+b+c)^n+(b+c+d)^n-(c+d+a)^n-(d+a+b)^n+(a-d)^n-(b-c)^n$$ and
$ad=bc$, then
$$64*F_6*F_{10}=45*F_8^2$$
This fascinating identity is due to Ramanujan and can be found in ...

**5**

votes

**1**answer

518 views

### The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$

I. If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\tag3$$
where $u=a+b+c,\; v = ab+ac+bc,\;w = abc$, then,
$$(a + u)^k + (b + ...

**6**

votes

**1**answer

135 views

### Buildings associated to generalized $BN$ pairs

I'll begin by asking a general question, and then specializing to the situation I really care about.
Let $G$ be a group and let $(B, N)$ be a $BN$-pair in $G$ (see, for instance, page 39 of Tits' ...

**0**

votes

**0**answers

138 views

### Does the proportion of non trivial zeroes of given real part increase on [0,1/2] for all L-functions?

Here what I call an L-function is either an element of the Selberg class or an automorphic L-function. For such a function $F$, $x\in [0,1/2]$ and $T\gt 0$, let's define $\delta_{F,T}(x)$ such that ...

**4**

votes

**0**answers

149 views

### Automorphisms of k((X))

I'm looking for a good reference for the following fact:
Let $k$ be a perfect field of characteristic $p$ and let $K=k((X))$.
Then every $k$-linear automorphism of $K$ is continuous with respect
to ...

**2**

votes

**1**answer

942 views

### Why Riemann Hypothesis so important [closed]

I am often hearing people emphasized how important the RH is,one of them said that it should lead to an efficient way of determining whether a given large number is prime,and the other said,RH would ...

**5**

votes

**1**answer

316 views

### An old conjecture of M.Newman

M.Newman raised several questions in his 1957 paper on modular forms.
Definition: $H_n$ is the subclass of all zero-free weakly modular forms of weight 0 on $\Gamma_0(n)$, where $n$ is a composite ...

**3**

votes

**0**answers

147 views

### The Bilu-Linial conjecture and Ramanujan graphs

The Bilu-Linial conjecture claims that every $d-$regular graph has a $2-$lift such that for the signing matrix has its eigenvalues between $[-2\sqrt{d-1},2\sqrt{d-1}]$ (the ``signing matrix" is the ...

**3**

votes

**0**answers

155 views

### On quintic roots $x_1^{1/5}+x_2^{1/5}+\dots+x_5^{1/5}$

I. Given the roots $x_i$ of the general cubic,
$$x^3+c_2x^2+c_1x+c_0=0\tag1$$
with $c_i \in \mathbb Q$, it is easy to show that the expression,
$$F_3 = (x_1^{1/3}+x_2^{1/3}+x_3^{1/3})^3$$
is an ...

**0**

votes

**0**answers

118 views

### Opportunistic idea about constructing curves with many rational points

This is just an opportunistic idea about constructing curves
with many rational points using rational surfaces.
Working over the rationals.
Take a rational surface given by parametrization
...

**2**

votes

**0**answers

366 views

### A “Take a Square Root When You Can” conjecture related to the prime factorization

I would tend to think that the following has already been investigated.
But as implied from the title, I have no idea how to even start looking for it.
Let $P_n$ denote the sum of the squares of ...

**1**

vote

**0**answers

96 views

### Is the stationary distribution of this Markov chain uniform?

First, a little bit of background: Since 2012, Canada has decided to phase out the penny for its coinage system. Product prices may still use arbitrary cents, especially since prices do not typically ...

**6**

votes

**1**answer

187 views

### Extensions of an abelian variety by a torus vs. extensions of their $\ell$-adic Tate modules

Let $K$ be a number field, let $A$ be an abelian variety over $K$, and let $H$ be a torus over $K$. For a prime $l$, we have the natural map
$$\mathrm{Ext}^1(A, H) \otimes_{\mathbb{Z}} \mathbb{Z}_l ...

**1**

vote

**1**answer

223 views

### Nepero game (by Yacov Perelman)

I have already posted this question time before on stackexchange, but didn't receive a definitive solution.
So this is the game: consider a positive integer number $n$ and divide it in a finite ...

**6**

votes

**0**answers

316 views

### Large sets not containing arithmetic progressions of length 3 in intervals

Given a large enough natural number $N$, let $\Delta_N=\{A \subseteq [N, 2N]: A$ contains no arithmetic progressions of length $3 \},$ where for natural numbers $N<M$ we have $[N, M]=\{N, N+1, ..., ...

**2**

votes

**1**answer

509 views

### Conjecture on the square root of the sum of the squares of the prime factors of a number

Let $A_{n}$ denote the square root of the sum of the squares of the prime factors of $n$.
For example, $A_{60}=\sqrt{2^2+2^2+3^2+5^2}\approx6.48$.
I have recently made the following observations:
...

**7**

votes

**0**answers

136 views

### Is there an infinite increasing sequence of naturals for which Landau's function can be efficiently computed?

Landau's function
$g(n)$ is the largest order of an element of the symmetric group $S_n$.
Equivalently, $g(n)$ is the largest least common multiple (lcm) of any partition of $n$.
In general $g(n)$ is ...

**3**

votes

**2**answers

166 views

### Examples of component crossing between families of modular forms

Is there a reference that contains explicit examples of component crossing of Hida families at height one primes? The paper of Emerton, Pollack, and Weston addresses component crossing obtained ...

**12**

votes

**3**answers

1k views

### On $e^{\pi\sqrt{4\cdot163}}$ and unusual connections

We are familiar with the expansion of the j-function,
$$j(\tau) = \tfrac{1}{q}+744+ 196884{q} + 21493760{q}^2 + \dots\tag1$$
and maybe with the approximation,
$$e^{\pi\sqrt{652}} = ...

**1**

vote

**0**answers

194 views

### Ramification: Riemann surfaces vs Number fields

I am trying to understand the connection between Riemann surfaces and number fields. I am wondering if there an inconsistency in the definition of ramification in terms of Riemann surfaces vs number ...

**1**

vote

**1**answer

213 views

### Error term for prime harmonic

What is known about the asymptotic behavior of
$$
f(x)=\sum_{p\le x}\frac1p-\log\log x-B_1?
$$
Of course by Mertens we know that $f(x)=o(1),$ but has more been proved (in terms of $O$ or ...

**3**

votes

**1**answer

220 views

### Is this property of the Bell's number evident?

Let $B(n)$ denote the Bell's number which is the number of the equivalence relation which can be defined on a set of cardinality $n$.
While I was trying to solve a problem, I reached another result;
...

**1**

vote

**2**answers

93 views

### Transforming a recurrence to the product of two other recurrences

This question deals with sequence:
$$a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6$$
The question is about establishing that $a_n$ is a composite number (except some finite cases).
In one ...

**1**

vote

**0**answers

91 views

### Arithmetic Progression [duplicate]

Show that for all $a,d,n\in\mathbb{N}$ :
$$\sum_{i=0}^{n}\frac{1}{a+id}\notin\mathbb{N}$$
It means that a finite sum of reciprocal of arithmetic progression of natural numbers with at lest two ...

**4**

votes

**0**answers

187 views

### Telescoping series for $\zeta(s)$, question about the basic ideas and a specific series

There are many known telescoping series that enable analytic continuation of $\sum _n \frac {1}{n^{s}}$ into a variety of domains, however they seem to all be derived from two basic ideas:
1) The ...

**2**

votes

**2**answers

521 views

### Is it possible on an elliptic curve both $x,y$ to be arbitrary large powers infinitely often?

Let $f(x,y)=0$ be irreducible elliptic curve over the rationals.
Are there $f$ for which:
Both $x,y$ are arbitrary large powers infinitely often,
i.e. infinitely many rational points $(u^k,v^m)$ ...

**1**

vote

**1**answer

117 views

### basis of the lattice generated by the integer points inside a subspace of R^L

Consider $K$ linearly independent vectors $\mathbf{a}_1, \mathbf{a}_2, ..., \mathbf{a}_K \in \mathbb{Z}^L$, where $1 \leq K<L $. Hence, the span of $\lbrace\mathbf{a}_1, \mathbf{a}_2, ..., ...

**5**

votes

**2**answers

224 views

### Density of multi-grade solutions to $x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k$ for $k = 5$ or $6$?

Given the Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
there is the rather curious observation that the smallest positive solutions for $k=5$ or $6$ is multi-grade.
...

**1**

vote

**1**answer

99 views

### asymptotic for restricted partitions

Let $m$ and $n$ be two positive integers and denote by $P(n,m)$ the number of partitions of $n$ into $m$ non-negative integers.
Is there an asymptotic formula for $P(n,m)$ ?? Any reference is ...

**3**

votes

**0**answers

220 views

### On 7th and 8th powers for $x_1^k+x_2^k+x_3^k+x_4^k = y_1^k+y_2^k+y_3^k+y_4^k$

The Diophantine equation,
$$x_1^k+x_2^k+x_3^k = y_1^k+y_2^k+y_3^k\tag1$$
for either $k=5$ or $6$ is quite well explored, and it has long been known that it has an infinite number of primitive ...

**10**

votes

**1**answer

615 views

### Divergence of a series similar to $\sum\frac{1}{p}$

Suppose we start with $k$ primes $p_1,p_2,\ldots ,p_k$ (not necessarily consecutive) and a residue class for each prime $r_1,r_2,\ldots ,r_k$.
We denote the least integer not covered by the arithmetic ...

**1**

vote

**1**answer

179 views

### automorphism group of a given period

Maxim Kontsevich and Don Zagier defined the algebra of periods and conjectured that one can pass from a representation of a given period to another one using only three rules. Assuming this ...

**3**

votes

**0**answers

184 views

### Possible weakening of Robins criterion for RH

I hope 2015 doesn't start with gross nonsense question for me.
Robins equivalence for RH is
$$ \frac{\sigma(n)}{n \log\log n} < e^{\gamma}\qquad(1)$$
for $n \ge 5041$. We have
$ \limsup ...

**24**

votes

**1**answer

933 views

### $x^4+y^4$ powerful for relatively prime $x,y$

I asked this question on the NMBRTHRY mailing list on
17 February 2014, but it remains unsolved as far as I know.
Recall that a "powerful
number" is a positive integer whose prime factorizations
$m = ...

**0**

votes

**0**answers

122 views

### Number of lattice points in a given triangle

Given a triangle with real coordinates, does anybody know how to find the number of lattice points contained within it? What if the points are only rational? I know Pick's formula can be used for the ...

**1**

vote

**0**answers

57 views

### Showing that a crypto hash function is not permutation, possibly conditionally?

Let $f$ be some crypto hash function, say MD5 with output $n$ bits. Restrict the input to $n$ bits.
Cryptographer told me it is open problem if such restricted collision
exists, i.e. $f(x)=f(y),x \ne ...

**3**

votes

**0**answers

217 views

### What are the values of this sequence?

Let $F_n$ denote the $n$th Fibonacci number.
Then $\prod\limits_{i=1}^{\infty}(1-x^{F_i})$ is a series all of whose coefficients are either $-1$, $0$ or $+1$.
The sequence of the coefficients in ...

**3**

votes

**0**answers

110 views

### On the computational complexity of the Hilbert polynomial of numerical semigroup rings

Let $(R, \mathfrak{m}) = k[[X^a, X^b, X^c]]$, $a<b<c$, $gcd(a, b, c) = 1$, be a semigroup ring. We have $R$ is a Cohen-Macaulay local ring of dimension one. It is well known that ...

**7**

votes

**3**answers

535 views

### Is $n = p-q$ equivalent to Goldbach's Conjecture?

One open conjecture is that every even integer greater than two is the difference of two primes. (Some superficial discussion here.)
Goldbach's conjecture states that every even integer greater than ...

**19**

votes

**1**answer

826 views

### When is “independence of l” known?

My question is for which varieties over local fields is "independence of l" known for
etale cohomology. Say $X/{\mathbb Q}_p$ is a complete non-singular variety and $W_l$ is the (complex) ...

**0**

votes

**0**answers

85 views

### Generalized Dedekind Sum Reciprocity Law

Is there a reciprocity law for generalized Dedekind sums of the form:
$$S(a,b;x,y;c)=\sum_{k \mod c}\tilde{B}_1\left(\frac{ak+x}{c}\right)\tilde{B}_1\left(\frac{bk+y}{c}\right)$$
such that the other ...