**2**

votes

**2**answers

175 views

### A new result on the Diophantine equation $x^3 + y^3 +z^3 = 3$

The above Diophantine equation is unknown to have any further integer solutions other than $(x, y, z) = (1, 1, 1)$ and $(4, 4, -5)$.
I am a prospective undergraduate mathematics student in Zimbabwe ...

**0**

votes

**0**answers

76 views

### Are there unconditional results for boundedness of finitely many rational points on $f(x,y)=n$ for all $n$?

Major rewrite due to comments.
Let $f(x,y) \in \mathbb{Q}[x,y]$ and $f$ depends on both $x,y$.
Q1 Is it possible the number of rational solutions to $f(x,y)=n$
to be uniformly bounded for all ...

**12**

votes

**2**answers

625 views

### Number theory and physics

I was following some lectures by Edward Frenkel about Langlands correspondence. He was describing some analogies between number theory and theoretical physics (Mirror symmetry). At some point ( my ...

**17**

votes

**2**answers

226 views

### Deep/precise relationship between two approaches to FLT for polynomials, $n = 3$

David Speyer commented the following here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials ...

**3**

votes

**2**answers

141 views

### family of polynomials with square discriminant

The title pretty much sums it up: do people know of nice parametrized families of polynomials (with integer coefficients) with square discriminant. I should say that one such family consists of ...

**0**

votes

**1**answer

107 views

### On the number of divisors in a given range

Given $\alpha\in\Bbb N$, can there be more than $(\log N)^4$ divisors (composites allowed) of $N$ in $\big[\frac\alpha2,\alpha\big]$ when $\sqrt N\in\big[\frac\alpha2,\alpha\big]$?
What is the ...

**10**

votes

**7**answers

899 views

### Open problems in continued fractions theory

I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.

**17**

votes

**3**answers

754 views

### Is there any pattern to the continued fraction of $\sqrt[3]{2}$?

Is there any pattern to the continued fraction of $\sqrt[3]{2}$ ? Wolfram Alpha returns for cube root of 2:
$\sqrt[3]{2}=$ [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1, 3, ...

**10**

votes

**1**answer

660 views

### Estimate on radical of $2^n \pm 1$

Not sure if this belongs to MO or not.
Are there any lower bound on radical of $2^n \pm 1$ (Recall that radical of integer $rad(k)$ is a product of primes which divide integer $k$)?
As an example If ...

**6**

votes

**1**answer

252 views

### Analytic continuation for $L$-functions of elliptic curves

Let $E$ be an elliptic curve over a number field.
When $E$ has no CM and is a $\mathbb Q$-curve (i.e. it is $\overline{\mathbb Q}$-isogenous to all of its conjugates), it is nowadays known that $E$ ...

**9**

votes

**0**answers

329 views

### Quaternions: ellipse effect

I would be interested in an explanation of the "six ellipse effect" produced by the pseudocode below (I also wonder how close are these to being actual ellipses). Note the code is somewhat similar to ...

**4**

votes

**1**answer

160 views

### Goldbach for certain classes of $n$

Asked on MSE without response here.
$\#$ of ways even $n$ can be represented by prime additions is heareafter denoted $G(n)$.
The Wiki article on the Goldbach conjecture states that
In 1975, ...

**1**

vote

**1**answer

209 views

### rational numbers and triangular numbers

This question is an offshoot of Ratio of triangular numbers. Suppose $ka(a+1)=nb(b+1)$, where $k,n >1$ are relative prime integers, and $a,b \geq 0$ are integers. Which $k,n$ pairs have no solution ...

**0**

votes

**0**answers

59 views

### Euler Characteristic of simple sheaves

Let $X$ be a projective curve over a field $K$ (any characteristic). Let $\mathcal{F}$ be a coherent simple sheaf
(In the sense, that $\mathcal{F}$ doesn't have non-trivial subsheaves). What is the ...

**0**

votes

**2**answers

175 views

### Minimal solution of simultaneous congruences

I would to determine the set of values $\lbrace a_1,a_2,a_3,\ldots,a_n \rbrace$ that minimizes the value of $x$ such that:
$$x\equiv a_1\mod p_1$$
$$\vdots$$
$$x\equiv a_n\mod p_n$$
where every ...

**2**

votes

**0**answers

286 views

### How big can a set of integers be if all pairs have bounded gcd

In this recent MO question, it was shown that the maximal cardinality of a subset $A(M,N)$ of $[1,N]$ where the pairwise GCD's of all set elements are upper bounded by $M,$ with $M^2\leq N$ has size ...

**2**

votes

**0**answers

62 views

### Number of multipartite partitions with odd components

For some positive integer $r$, by an $r$-vector I will mean an $r$-tuple $(a_1,a_2,\dots,a_r)$ with $a_1,\dots,a_r$ nonnegative integers not all zero, and I will call it odd if $a_1,\dots,a_r$ are all ...

**18**

votes

**3**answers

946 views

### Is the sequence of partition numbers log-concave?

Let $p(n)$ denote the number of partitions of a positive integer $n$. It seems to me that we have for all $n>25$
$$
p(n)^2>p(n-1)p(n+1).
$$
In other words, the sequence $(p(n))_{n\in ...

**5**

votes

**0**answers

173 views

+150

### A characterization of quadratics similar to an inverse sieve problem

Suppose $\mathscr{A} \subset \mathbb{N}$ enjoys for all large enough cutoffs $X$ the following properties:
$|\mathscr{A} \cap [1,X]| > \sqrt{X}/10$; and
the discriminant $\prod_{\alpha \neq ...

**0**

votes

**1**answer

78 views

### For a given $n$, under what condition(s) there exists (at least) two different $c$ and $c′$ such that $X_n^c=X_n^{c'}$

Let $X_n^c=\{\cos\left((4k-c)\frac{\pi}{2n}+\frac{\pi}{4}\right): k=0, 1, \dots, n-1\}$ where $c\in\{0, 1, \ldots, \lfloor\frac{n}{2}\rfloor\}$ and $n$ is any positive integer greater than 3. I want ...

**17**

votes

**3**answers

739 views

### Ratio of triangular numbers

Which natural numbers can be written as the ratio of two triangular numbers?
That is, which natural numbers can be written as
$$\frac{n(n+1)}{m(m+1)}$$ for natural numbers $m,n$.

**35**

votes

**5**answers

3k views

### Has decidability got something to do with primes?

Note: I have modified the question to make it clearer and more relevant. That makes some of references to the old version no longer hold. I hope the victims won't be furious over this.
Motivation:
...

**4**

votes

**1**answer

194 views

### Unit in cyclotomic field

Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...

**3**

votes

**1**answer

234 views

### Reducible polynomials

Say one only seeks to identify whether a given polynomial over $\mathbb{Z}[x]$ is reducible, then what are the best ways known to solve this?
$(1)$ If the polynomial is reducible, the algorithm ...

**3**

votes

**1**answer

208 views

### Index of the Hecke algebra with operators omitted

This is a spin-off to the question Omitting primes from a Hecke algebra by David Loeffler.
Let $N$ be a positive integer. For a finite set of primes $\Sigma$, let $\mathbb T^{\Sigma}$ be the $\mathbb ...

**13**

votes

**1**answer

449 views

+100

### Small quotients of smooth numbers

Assume that $N=2^k$, and let $\{n_1, \dots, n_N\}$ denote the set of square-free positive integers which are generated by the first $k$ primes, sorted in increasing order. Question: what is a good ...

**2**

votes

**1**answer

117 views

### Gradual monotonic morphing between two natural numbers

Let $a < b$ be two natural numbers. I will use these as an example:
\begin{align*}
a & = 2^5 \cdot 3^2 \cdot 5^2 = 7200\\\
b & = 2^3 \cdot 3^5 \cdot 7^1 = 13608
\end{align*}
I seek to ...

**7**

votes

**2**answers

716 views

### What is the lower bound for highly composite numbers?

if $x=d(n)$ is the number of divisors of $n$, what is the tightest lower-bound for $n$ only given $x$?
http://en.wikipedia.org/wiki/Highly_composite_number

**1**

vote

**0**answers

91 views

### Zeros of a nearly holomorphic form

Let $f$ be a nearly holomorphic modular form on a Hilbert modular variety $Sh$. Suppose that $f$ vanishes on a Zariski dense subset of CM points on $Sh$. How to show that $f$ is identically zero?

**11**

votes

**2**answers

418 views

### No nonconstant coprime polynomials $a(t)$, $b(t)$, $c(t) \in \mathbb{C}[t]$ where $a(t)^3 + b(t)^3 = c(t)^3$

See David Speyer's answer here.
I saw Brian Conrad give an excellent one hour talk to undergraduates where he proved that there do not exist nonconstant, relatively prime, polynomials $a(t)$, ...

**0**

votes

**0**answers

28 views

### Voronoi-type summation formula for coefficients of symmetric square $L$-functions

given a primitive form $f$ for the full modular group $SL_2(Z)$ and $\lambda_f(n)$ be the $n$th Hecke eigenvalue. Various Voronoi-type formulas are fulfilled by these coefficients and there are ...

**18**

votes

**2**answers

503 views

### Rational points on the “quintic circle” $x^5 + y^5 = 7$

I suspect that the curve $x^5 + y^5=7$ has no $\mathbb Q$ points, and a brief computer search verifies this hypothesis for denominators up to $10^4$. What techniques can be used to show that there are ...

**1**

vote

**0**answers

123 views

### Does data suggest $| \pi_2 (n) - 2\Pi \int_2^n \frac{dx}{\ln(x)^2} | < \ln(n+2)^2 \sqrt (n+2) $?

Let $\Pi$ be the twin prime constant and $\pi_2(n)$ the twin prime counting function.
Define
$$ t(n) = \left| \pi_2(n) - 2 \Pi \int_2^n \frac{dx}{\ln(x)^2} \right| $$
Is it consistent with current ...

**2**

votes

**1**answer

202 views

### Central binomials and irrationals

I suppose this is une cause perdue, but it would be nice if the following held.
Let $\theta$ be an irrational, and let $c_m = {{2m}\choose m}$, the central binomial. For a real number $r$, let $d(r)$ ...

**5**

votes

**1**answer

127 views

### Upper bound on level of a congruence subgroup of the modular group

Let $\Gamma = PSL(2,\mathbb{Z}) = \langle S,T \ | \ S^2=(ST)^3=1 \rangle$. Let $G$ be some mystery normal subgroup of $\Gamma$ that we happen to think may be congruence. Recall that a subgroup of ...

**10**

votes

**2**answers

350 views

### BSD and congruent numbers

Let $n$ be a positive integer, and let $E_n$ denote the elliptic curve $y^2=x^3-n^2x$. By work of Tunnell, it's known that if $E_n$ satisfies the BSD conjecture, then there is an algorithm to decide ...

**4**

votes

**1**answer

433 views

### A general question about strictly non-palindromic numbers

For a definition, see the wikipedia page: http://en.wikipedia.org/wiki/Strictly_non-palindromic_number
So according to the wikipedia page, under properties, all strictly non-palindromic numbers with ...

**1**

vote

**0**answers

58 views

### How much extra ramification in a residual representation

Suppose $\rho:G _{\mathbb{Q}} \rightarrow GL_n(\mathbb{Q}_p)$ is a Galois rep. It has a uniquely defined (up to semisimplification residual rep $\bar{\rho}$. $\bar{\rho}$ is unramified where $\rho$ ...

**-5**

votes

**3**answers

131 views

### Collecting sufficient conditions for Sorli's conjecture on odd perfect numbers [on hold]

Sorli's conjecture predicts that, for an odd perfect number $N$ given in the Eulerian form $N = {q^k}{n^2}$ (where $q$ is prime with $\gcd(q, n) = 1$ and $q \equiv k \equiv 1 \pmod 4$), the condition ...

**-6**

votes

**0**answers

64 views

### Continuity of Real line [on hold]

f:R to R such that f attains every value exactly twice i.e. for all a in
R, {x in R|f(x)=a} is either empty or doubleton set.prove that,f is
discontinuous at infinitely many points .

**6**

votes

**1**answer

291 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 $a_i = \lfloor \alpha_i \rfloor$, ...

**5**

votes

**5**answers

822 views

### Fermat over Number Fields

If $n>2$, does the impossibility of solving $x^n +y^n=z^n$ with $x, y, z$ rational integers imply the same with $x, y, z$ algebraic integers?
Rather, If insolvability in algebraic integers does ...

**4**

votes

**0**answers

158 views

### Is this a viable public key cryptosystem?

Would this, or a variant, work as a public key cryptosystem?
Alice takes a (computable) function p of several variables, let us say $p(x,y)$. We can think of p as a polynomial and x and y as ...

**0**

votes

**0**answers

57 views

### How is the p-adic norm calculated when using universal witt vectors?

How is the p-adic norm calculated when using UNIVERSAL WITT VECTORS?
Is the p-adic norm calculated in the familiar way, in the sense that we look to the last digit to the right, and the prime number ...

**3**

votes

**1**answer

163 views

### Polynomials of even degree with solvable Galois group

Let $f(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \cdots + a_0$ be a polynomial with integer coefficients and irreducible over $\mathbb{Q}$. For $n \geq 3$, $f$ is generically unsolvable by radicals. ...

**-7**

votes

**0**answers

181 views

### Prove of “sane” [closed]

An integer is called “sane” if 3|(n^2 + 2n). (That is, if (n^2 + 2n) mod 3 = 0.)
(a) Prove or disprove that all odd integers are sane.
(b) Prove or disprove that, if 3| n, then n is sane.

**1**

vote

**0**answers

104 views

### Relations between Mirimanoff polynomials

Let p be an odd prime number, let $f_i(X)$ be the i-th Mirimanoff polynomial (with respect to p) :
$f_i(X) = X + 2^{i-1}X^{2} + ... + (p-1)^{i-1}X^{p-1}.$
Mirimanoff noted that the three polynomials ...

**1**

vote

**0**answers

74 views

### Bounds on the number of zeros of a quadratic form

Let $Q(x_1, \dots, x_n)$ be a non-degenerate indefinite quadratic form with integer coefficients. Let $N(Q,T)$ be the set of vectors $x=(x_1, \dots, x_n) \in {\mathbb Z}^n$ such that $|x|<T$ and ...

**34**

votes

**1**answer

3k views

### Infinitely many solutions of a diophantine equation

If $P(x,y,...,z)$ is a polynomial with integer coefficients then every integer solution of $P=0$ corresponds to a homomorphism from $\mathbb{Z}[x,y,...,z]/(P)$ to $\mathbb{Z}$. So there are infinitely ...

**6**

votes

**1**answer

262 views

### When do two lattices have the same stabilizer in the diagonal torus?

This is moved from MSE, where I asked and didn't receive an answer (see http://math.stackexchange.com/questions/1145151/lattices-in-mathbbq-pn-with-the-same-stabilizer)
Let $T$ be the diagonal torus ...