**3**

votes

**0**answers

62 views

### realizing uniform boundedness of Galois representations associated to elliptic curves

This is less of a question and more of an argument that I've been worried about for a while and want to check (apologies for the length and if my writing is unclear).
Suppose I have an elliptic curve ...

**0**

votes

**0**answers

26 views

### Weighted maximal number of disjoint chains in the integer divisibility poset for $\{1,2,\ldots,n\}$

In the mathoverflow question here the asymptotic growth of antichains in the divisibility poset ${\cal P}_n$ of the set of natural numbers $\{1,\ldots,n\}$ is considered. I have a somewhat dual ...

**0**

votes

**1**answer

83 views

### Linear forms that avoid numbers with lot of factors

Is following true?
For every given $c>0$ there is an $n_c>0$ such that for every $n>n_c$ there are integers $n<a,b<2n$ such that there are two positive integers $\frac{n}{2(\log ...

**-5**

votes

**0**answers

99 views

### Can the following expressions be regarded as general formula of prime numbers? [on hold]

Commonly accepted opinion is that there is no general formula for prime numbers. But we propose expressions for two pairs of 2-dimensional arrays which contain indexes $p$ in the sequences $6p + 5= 5, ...

**8**

votes

**2**answers

560 views

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

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 ...

**5**

votes

**3**answers

221 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 ...

**18**

votes

**2**answers

340 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 ...

**1**

vote

**1**answer

220 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

73 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 ...

**4**

votes

**1**answer

173 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, ...

**0**

votes

**1**answer

124 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 ...

**9**

votes

**0**answers

382 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 ...

**17**

votes

**3**answers

791 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, ...

**4**

votes

**1**answer

209 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 ...

**2**

votes

**1**answer

120 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 ...

**6**

votes

**1**answer

260 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 $E$ ...

**0**

votes

**0**answers

30 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 ...

**14**

votes

**3**answers

838 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 ...

**1**

vote

**0**answers

126 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

**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 ...

**1**

vote

**0**answers

92 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?

**18**

votes

**2**answers

543 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 ...

**5**

votes

**1**answer

129 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

356 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 ...

**11**

votes

**2**answers

426 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 $a(t)$, ...

**1**

vote

**1**answer

87 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 $\rho$ ...

**-6**

votes

**0**answers

66 views

### Continuity of Real line [closed]

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 .

**0**

votes

**0**answers

87 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 ...

**0**

votes

**0**answers

59 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 ...

**4**

votes

**0**answers

160 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 ...

**3**

votes

**1**answer

164 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

182 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.

**0**

votes

**2**answers

176 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 ...

**1**

vote

**0**answers

75 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 ...

**3**

votes

**2**answers

192 views

### Proving Legendre's Sum of 3 Squares Theorem via Geometry of Numbers

Every positive integer can be written as the sum of 4 squares $n = a_1^2 + a_2^2 + a_3^2 + a_4^2$ however, if we only allow sum of 3 squares some numbers have to be left out:
$n = a^2 + b^2 + c^2$ ...

**6**

votes

**1**answer

234 views

### What is the exact statement about uniform boundedness of rational points on curves of genus greater than one? Singular points can be unbounded

According to several sources, it is conjectured (or at least believed)
that the rational points of curves over the rationals of genus $g > 1$
are uniformly bounded by $g$. E.g. here p. 1.
Assuming ...

**3**

votes

**2**answers

249 views

### Primes $P_{2n-1}$ that are $2$ mod $3$

Are infinitely many primes $P_{2n-1}$ expressible as $3k-1$?
The primes $P_{2n-1}$ are every other prime beginning with $2$: $2,5,11,17,23,31,\cdots$. The first few are of the form $3k-1$, but $31$ ...

**5**

votes

**1**answer

145 views

### Hecke characters and Conductors

Motivation: Let $\ell$ be an odd prime. There is a conductor-preserving correspondence between primitive Dirichlet characters of order $\ell$
and cyclic, degree $\ell$ number fields $K/\mathbb{Q}$.
...

**6**

votes

**0**answers

196 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 ...

**2**

votes

**0**answers

79 views

### Factorization of linear recurrences

For each (commutative unitary) ring $R$, let $\mathfrak{R}(R)$ be the set of all linear recurrences over $R$, that is, the set of all sequences $(a(n))_{n \geq 0}$ in $R$ such that
$$a(n+k) = r_1 ...

**3**

votes

**0**answers

88 views

### Is the reciprocal of this Kummer theorem true?

Le p be an odd prime, let $\zeta$ be a primitive p-th root of unity, let K denote the p-th cyclotomic field. Let us define a singular element of K as an integral element $\alpha$ of K, not divisible ...

**12**

votes

**0**answers

251 views

### No Siegel-Landau zeros for $\mathrm{GL}(n)$

The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:
There ...

**1**

vote

**0**answers

72 views

### Is there a “complete” Sidon sequence?

A sequence of natural numbers $(a_n)$ with the property that all pairwise sums of elements are distinct is called a Sidon sequence and it is proved there are at most $s(n)\sim\sqrt n$ elements of ...

**3**

votes

**2**answers

222 views

### Congruent numbers and elliptic curves

Who first explicitly stated the link between $N$ being a congruent number and the existence of rational points of infinite order on $y^2=x(x^2-N^2)$?

**1**

vote

**0**answers

106 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

151 views

### Is the difference of these two real-rooted functions real-rooted?

During our on-going search of approximations to the Riemann $\Xi(z)$ function, we discovered a family of functions $W_n(z)$ as shown in (1).
Our final goal is to prove that:
Proposition 1: ...

**4**

votes

**2**answers

199 views

### How to get a good upper bound on $\sum_{1 \lt d, d|n} \phi(d)/\log d$?

I'm actually interested in a slightly smaller quantity, but I'm willing to accept the following simplification, especially if there are small error terms.
Let's start with $ n \gt 1$, Euler's totient ...

**8**

votes

**2**answers

316 views

### Characterization of finite groups using sum of the orders of their elements

Notation: If $G$ is a finite group, $o(g)$ denotes the order of the element $g\in G$.
Motivation: Some finite groups could be uniquely determined by the size of the group. For example given a prime ...

**3**

votes

**0**answers

255 views

### Fermat-Wiles “first case” in extensions of cyclotomic fields

I fell on the following fact : let p be an odd prime, let K denote the p-th cyclotomic field, let L be an extension of K with finite degree not divisible by p, and assume that the prime ideal $(1 - ...

**5**

votes

**1**answer

192 views

### The number of integral solutions to $x^2+y^2-az^2=0$

I think this must be well-known (and probably not hard to prove either), but I cannot find a reference: for a (positive) rational number $a$, the number of integral solutions to the equation
$$ ...