**48**

votes

**4**answers

2k views

### When has the Borel-Cantelli heuristic been wrong?

The Borel-Cantelli lemma is very frequently used to give a heuristic for whether or not certain statements in number theory are true.
For example, it gives some evidence that there are finitely many ...

**13**

votes

**1**answer

457 views

### When complex conjugation lies in the center of a Galois group

Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...

**14**

votes

**3**answers

2k views

### A variant of Goldbach Conjecture

I'm asking if this variant of weak Goldbach's Conjecture is already known.
Let $N$ be an odd number. Does there exists prime numbers $p_1$, $p_2$ and $p_3$ such that $p_1+p_2-p_3=N$? Ideally, can we ...

**11**

votes

**2**answers

476 views

### distribution of $\sqrt{-1} \mod p$

While reading up on quadratic reciprocity, I learned that if $p = 4k+1$ then $-1$ has a square root in $\mathbb{Z} / p \mathbb{Z}$.
Let $r_p$ be an integer with $0\leq r_p < p$ and $r_p^2 \equiv ...

**4**

votes

**2**answers

598 views

### Is every positive integer a sum of at most 4 distinct quarter-squares?

There appears to be no mention in OEIS: Quarter-squares, A002620. Can someone give a proof or reference?
Examples:
quarter-squares: ${0,1,2,4,6,9,12,16,20,25,30,36,...}$
2-term sums: ${2+1, 4+1, ...

**16**

votes

**1**answer

586 views

### Question on the irrationality of $e$

I was surprised that the numbers $\pi$, $\ln{(2)}$, $\zeta{(2)}$, and $\zeta{(3)}$ can be shown to be irrational in what seems to be "three-lined proofs" (as identified here on Overflow: Establishing ...

**7**

votes

**1**answer

217 views

### how do automorphisms of elliptic curves act on the Tate module?

Let $E/k$ be an elliptic curve over some algebraically closed field $k$ of characteristic $p\ge 0$. It's known that $Aut(E)$ acts faithfully on the Tate module $T_\ell(E)$ ($\ell\ne p$) with ...

**5**

votes

**1**answer

140 views

### On one class of Somos-like sequences

This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer?
Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence ...

**8**

votes

**1**answer

202 views

### Integers $d$ for which the Negative Pell equation is soluble for both $d$ and $2d$?

Let $\text{NPE}_d$ denote the negative Pell equation:
$$ x^2-dy^2=-1$$
Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y.
we know that (in this paper):
...

**1**

vote

**0**answers

261 views

### Generalization of proposition of Granville related to abc conjecture

Related to this question.
For polynomial $f$, let $rad(f)$ denote the radical of $f$,
the product of irreducible factors.
Suppose that $G(x,y) \in \mathbb{C}[x,y]$ is homogeneous
without any ...

**6**

votes

**2**answers

586 views

### Counterexample to Proposition of Granville related to abc conjecture

Looks like there is counterexample to Proposition related to
abc conjecture. Confusion is likely.
From RATIONAL AND INTEGRAL POINTS ON QUADRATIC TWISTS OF A GIVEN HYPERELLIPTIC CURVE, Andrew ...

**4**

votes

**0**answers

137 views

### Simultaneously using the real and 2adic norms

In the book Modern Computer Arithmetic, there is a section that talks about division with remainder and such in a way that exploits the interplay between the real and 2-adic norms; e.g. the linked-to ...

**16**

votes

**1**answer

923 views

### Has this strong number theoretic conjecture of Euler been proved, and where could I find such a proof?

Polya cites this work of Euler as an example of a conjecture which Euler considered impossible to doubt, and yet still needing a demonstration.
It is on pages 90-98 of "Induction and Analogy in ...

**5**

votes

**1**answer

116 views

### Bounding a Sum of Adjoint L-Function Values

Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with ...

**4**

votes

**3**answers

218 views

### Is there a generalization of the “characteristic polynomial” to other split/quasi-split algebraic groups?

Let $G = GL_n$ over a field $F$, and let $\gamma \in G(F)$ be a semisimple element. The characteristic polynomial $c_\gamma(t)$ of $\gamma$ encodes a fair bit of information about $\gamma$. ...

**2**

votes

**1**answer

71 views

### A question about decomposing mod 2 modular forms of level p^2

Fix an odd prime $p$. Each $f \in \mathbb{Z}/2[[x]]$ can be written as $f_{+} + f_{-} + f_0$ where each exponent k of $x$ appearing in $f_{+}$ (resp. $f_{-}$, $f_0$) has Legendre symbol $(k/p)$ equal ...

**2**

votes

**2**answers

105 views

### On successive minima and basis of a lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let
$\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$
be a ...

**3**

votes

**0**answers

94 views

### Small polynomial roots modulo prime powers

Let $f(x) \in \mathbb{Z}[x]$ have no rational roots,
$n>1,q=p^n$ for prime $p$.
Let $r$ be the smallest root of $f(x)=0$ modulo $q$.
Smallest mean smallest absolute value when lifted
to the ...

**-2**

votes

**1**answer

123 views

### On odd perfect numbers $N$ given in the Eulerian form $N = {q^k}{n^2}$, Part II

I posted this question on MSE two days ago, but did not receive any responses. I have cross-posted it on MO, hoping it gets more attention here and that it is appropriate for this site.
A positive ...

**1**

vote

**1**answer

101 views

### What is the best known lower bound for: $\max_{2\leq i\leq p-1}(ord_n(i)) $?

Given an in integer $n$, and let $p$ be its smallest prime divisor (you can assume that $p$ is very large ). Let $ord_n(i)$ denotes the order of $i$ as an element of $\Bbb Z_n^*$ the multiplicative ...

**11**

votes

**1**answer

253 views

### 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

112 views

### examples of non-unique factorisation in cyclotomic fields [closed]

I was looking into cyclotomic extensions of the natural numbers, and I found that extending the naturals with the 23rd root of unity caused the ring to no longer be a UFD. In other words, there now ...

**2**

votes

**0**answers

202 views

### Quadratic Twists on Elliptic Curves- John Coates [closed]

I was actually facing difficulty in Lemma 2.4 of Section 2 (Generalization of Birch's Lemma).-Quadratic Twists on Elliptic Curves- Proceeding of London Mathematical Society. I don't understand why ...

**0**

votes

**1**answer

114 views

### Algebraic Hecke characters with a given infinite part

I'm needing to find out if there exists an algebraic Hecke character for a number field F, $\phi: \mathbb{A}_F \rightarrow \mathbb{C}$, for a fixed infinite part $\phi_\infty$ and a fixed component ...

**2**

votes

**1**answer

134 views

### Estermann-Weil bound for Kloosterman sums

Let $m\geq 2$ be a positive integer and consider the Kloosterman sums
$$
\mathrm{Kl}(a,b,m)=\sum_{\substack{1\leq x\leq m\\ \gcd(x,m)=1}}\exp\left(\frac{2\pi i}{m}(ax+b\bar{x})\right),
$$
where ...

**2**

votes

**0**answers

190 views

### Algebraic integer with conjugates on the unit circle

Let $\alpha$ be an algebraic integer on the unit circle in $\mathbb{C}$ such that all the conjugates of $\alpha$ lie on the unit circle. Does it follow that $\alpha$ is a root of unity?

**8**

votes

**1**answer

196 views

### Dedekind-finite arithmetic vs natural numbers arithmetic

It is known that the Dedekind-finite cardinals are closed under addition and multiplication, so one may do arithmetic in them, as opposed to only natural numbers.
How much can those two arithmetics ...

**4**

votes

**2**answers

271 views

### Find all possible rational values of a parametric quartic such that it is reducible

Description: Given the following parametric quartic polynomial
$y^4 - 28 z y^3 - 14 (656 - 328 z + 83 z^2) y^2 +
4 z (-20464 + 10232 z + 3409 z^2) y +
91 (62208 - 62208 z + 41504 z^2 - 12976 z^3 ...

**2**

votes

**0**answers

127 views

### Normal basis in cyclotomic number fields

Let $p$ be an odd prime integer and let $\zeta$ be a primitive $2p$-th root of unity. Does $\alpha=1+\zeta+\zeta^{-1}+\dots+\zeta^{\frac{p-1}{2}}+\zeta^{-\frac{p-1}{2}}$ generates a normal basis of ...

**5**

votes

**0**answers

164 views

### How can I effectively compute tetration mod a?

Is there some general techique to compute tetration and pentation mod some number?
$m \uparrow^2 n\mod a$ and $m \uparrow^3 n\mod a$
I know about Euler's theorem to compute $m \uparrow n\mod a$, ...

**0**

votes

**1**answer

91 views

### Dirichlet series without order term

is there a name in use for Dirichlet series without the order term, analogously to Laurent or Puiseux polynomials? Is there work known about such expressions?
$D(s) = \sum_{0<n<N}a_n/n^s$
The ...

**6**

votes

**4**answers

495 views

### Some Non-Trivial Algebraic(Rational) Number

Every problem about algebraic-ness (rational-ness) of numbers that I have seen is in one of the below types:
The number is algebraic(rational) and proving that it is algebraic(rational) is trivial, ...

**19**

votes

**1**answer

747 views

### Steinhaus's Easter Egg Problem

The following is the text of Steinhaus's so-called Easter egg problem. According to this article of Roman Duda, this was recorded in the New Scottish Book around Easter 1955 and "Steinhaus offered an ...

**5**

votes

**0**answers

81 views

### Constants for Rosser's Sieve

I am trying to apply Iwaniec's formulation of Rosser's sieve (here) to obtain nontrivial lower bounds for almost-primes in various sequences. These sequences have sieve dimension 1 (if $g(p)$ is the ...

**2**

votes

**0**answers

140 views

### Diophantine equations and the numbers $4,7,8$

Consider the diophantine equation
$$
x^n+y^n+z^n=k\cdot xyz,
$$
where $n,x,y,z$ are positive integers. Now consider $k\in\left\{4,7,8\right\}$. It is well-known or easily provable that for $n=1$ and ...

**9**

votes

**6**answers

671 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.

**3**

votes

**1**answer

247 views

### A question on Pythagorean triples

We say that $(a,b,c) \in \mathbb{N}^3$ is a Pythagorean triple if $a^2 + b^2 = c^2$. Is there a characterization of those Pythagorean triples $(a,b,c)$ for which $ab$ is a square-residue modulo $c^2$?
...

**10**

votes

**3**answers

358 views

### The diameter of a certain graph on the positive integers

Let $G(n)$ be the graph whose vertices are the positive integers $1,2,3,4, \ldots, n$ two of which are joined by an edge if their sum is a square. Is the diameter of this graph 4 for all sufficiently ...

**10**

votes

**0**answers

125 views

### Hyperelliptic curves over $\mathbb{Q}$ with a $\mu_p^2$ subgroup in their Jacobian

Given a prime number $p>2$, I'm looking for a smooth projective hyperelliptic curve $C$ defined over $\mathbb{Q}$ whose Jacobian $J(C)$ has a subgroup isomorphic to $\mu_p^2$ as a ...

**5**

votes

**0**answers

75 views

### A Generalized Wiener-Ikehara Theorem with multiple poles on the line

One version of the Wiener-Ikehara Theorem says that if
$$
f(s) = \sum \frac{a(n)}{n^s}
$$
is a Dirichlet series with nonnegative coefficients that converges absolutely for $\text{Re}(s) > 1$ and ...

**4**

votes

**1**answer

215 views

### The horizontal distribution of zeros of $\zeta^\prime(s)$

I have a question about a detail in the proof of Proposition 1.6 in "The horizontal distribution of zeros of $\zeta^\prime(s)$", K. Soundararajan, Duke J. Math. vol. 91 1998.
Throughout I will ...

**1**

vote

**3**answers

131 views

### Decidability of sum of powers exponential diophantine equation

I want to ask about decidability of exponential Diophantine equation:
$z_12^{\eta_1} + \ldots + z_n2^{\eta_n} = z$, where $z_i,\eta_i,z \in \mathbb{Z}$, and $\eta_i$ - are variables.
Can we find ...

**0**

votes

**3**answers

168 views

### Counting zero-sum free sequences of a given length in $\mathbb{Z}_n$

Let $n$ and $d$ be positive integers. Define $\alpha_n^d$ to be the number of vectors $(x_1, x_2, \cdots, x_d)$ in $\mathbb{Z}_n^d$ such that given any subset $S$ of $\{ 1, 2, 3, \cdots d\}$, ...

**4**

votes

**1**answer

146 views

### Subgroups of $Sp_{2g}$ giving rise to Shimura data

Consider the Shimura datum $(GSp_{2g},\mathcal{H}_g)$. Let $G$ be a reductive $\mathbb{Q}$-subgroup of $Sp_{2g}$. I want to know under what condition there exists a point $x\in\mathcal{H}_g$ such that ...

**4**

votes

**1**answer

312 views

### Counting number of points in a lattice with bounded sup norm

Let $\Lambda$ be a lattice in $\mathbb{R}^n$. For $\bar{x} \in \mathbb{R}^n$, let
$\| \bar{x} \| = max_{1 \leq i \leq n} \{ |x_i| \}$, i.e. the sup norm. Let $\lambda_1, ..., \lambda_n$
be a ...

**12**

votes

**1**answer

393 views

### Natural probability on integers

This is a follow-up to this classical question asked recently here: we know (e.g. using the second Borel-Cantelli Lemma) that no probability measure on $\mathbb{Z}$ has the property that $n\mathbb{Z}$ ...

**8**

votes

**2**answers

848 views

### divisible by all standard prime numbers

This question is about prime numbers in nonstandard models of Peano Arithmetic. Every such model looks like N+AxZ, where A is a dense linear order without end points.
There are many nonstandard ...

**4**

votes

**1**answer

275 views

### Do we know any bound on $lcm(2^1-1, 2^2-1,…,2^n-1)$?

We know that lcm(1,...n) is approximately $e^n$ and and also we know that $gcd(2^a-1, 2^b-1)=2^{gcd(a,b)}-1$.
I wonder if there exists an upperbound/lowerbound/approximation for $lcm(2^1-1, ...

**0**

votes

**0**answers

77 views

### Rationality of intersection of algebraic groups

Suppose that $G$ (defined over $\mathbb{Q}$) and $H$ (defined over $\mathbb{R}$) are two algebraic subgroups of a larger algebraic group defined over $\mathbb{Q}$. Assume that $G(\mathbb{R})$ and ...

**0**

votes

**1**answer

157 views

### Normal basis with cyclotomic units

Let p be an odd prime integer and let $\zeta$ be a primitive p-th root of unity.
Let $\alpha$ be a non-trivial cyclotomic unit of $\mathbb Q(\zeta)$, i.e. an element of the form ...