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Results tagged with nt.number-theory
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user 48142
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
9
votes
What are the rational solutions to $y^4=x^3+x+1$?
This is not an answer, just an account of an attempt to solve the problem that didn't work. I think it sheds some light on the techniques that might be needed, which is why I am sharing it.
Although t …
- 19.9k
10
votes
Accepted
Integral points near elliptic curves
You can take $\theta = 0$, even $\theta = -1/6$ works.
Fix an integer $A \ne 0$ and an integer $B$. If $r$ is an integer, the elliptic curve $E : y^{2} = x^{3} + Ax + r^{2} A^{2}$ has the obvious poin …
- 19.9k
17
votes
Accepted
Existence of rational points on a generalized Fermat quartic
This question is amenable to the use of the Mordell-Weil sieve. (For a good introduction to this technique, see the paper here.) In this situation, there is a simple version of it (using a single prim …
- 20.3k
15
votes
Existence of rational points on some genus 3 curves
There are no rational solutions to curve (b). This curve has the automorphism $(x,y) \mapsto (x,-y)$ and the quotient is the genus one curve
$$ -yz + w^{2} = 0 \quad x^{2} + y^{2} + xz - z^{2} + yw = …
- 19.9k
4
votes
Mordell curves with large rank
Yes. Apparently Tom Womack used to maintain a webpage devoted to Mordell curves of high rank at the (no longer functional) page here although it can be viewed with the help of the Wayback Machine. Thi …
- 19.9k
11
votes
Accepted
Reference request for recurrence relation of division polynomials
These recurrences are stated explicitly in Weber's Lehrbuch der Algebra (published in 1908). See volume 3, section 58, page 200. Weber doesn't give a citation to them, so it's hard to know if they wer …
- 19.9k
5
votes
Accepted
Transforming the Kondo quintic $5T2$ into the Lehmer quintic $5T1$?
Regarding your first question, the answer is yes. In fact, your quartic Tschirnhausen formula shows that. In particular, this shows that the one-parameter Kondo quintic has a root in the splitting fie …
- 19.9k
26
votes
Accepted
Does the set of square numbers adhere to Benford's law in every base?
No. Benford's law works well for sequences that grow exponentially, and the squares grow too slowly.
In particular, fix a base $b \geq 3$, consider the case of $d = 1$, and choose $n = 2 \cdot b^{2k}$ …
- 19.9k
7
votes
Accepted
Squarefree parts of integers of the form $xy(x+2y)(y+2x)$
If a given squarefree integer $s$ is given, determining whether there is a pair of integers $(x,y)$ for which $s(x,y) = s$ is a problem about rational points on elliptic curves. In particular, there i …
- 19.9k
6
votes
Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$
Since $\mathbb{Z}_{(p)}$ can be thought of as a subring of $\mathbb{Z}_{p}$,
if two quadratic forms $Q_{1}$ and $Q_{2}$ are equivalent over $\mathbb{Z}_{(p)}$, then they must be equivalence over $\mat …
- 19.9k
2
votes
Accepted
Small Galois group solution to Fermat quintic
I have answers to your first two questions, and some insight into the third.
First, there is a quartic in the family above with Galois group contained in $A_{4}$. One example is found by taking $q = 1 …
- 19.9k
8
votes
Accepted
Trace-free basis for $\mathcal{O}_K$, $K$ a cubic field
No. There do not always exist such $\alpha$ and $\beta$. If $K$ is a cubic field and such $\alpha$ and $\beta$ exist, then for all $x \in \mathcal{O}_{K}$, ${\rm Tr}\left(\frac{1}{3} \cdot x\right) \i …
- 19.9k
2
votes
Algebraic numbers which prescribed degree which does not belong to some fields
Proposition 2 is false, although perhaps only for $n = 4$ (and $t = 2$ or $t = 3$). If $t = 2$, every algebraic number $\gamma$ of degree $4$ is contained in $K_{4}$.
Let $K$ be the Galois closure of …
- 19.9k
8
votes
Accepted
Calculating the explicit constant – Siegel zeros and class numbers
One place to find this worked out in detail is the paper "On the Siegel-Tatuzawa theorem" by Jeffrey Hoffstein (published in 1980 in Acta Arithmetica). Lemma 1 of that paper states that if $\chi$ is a …
- 19.9k
1
vote
Bounds on largest possible square in sum of two squares
Rather than discuss $\max b_{i}$, I'll discuss the equivalent question of bounding $\min a_{i}$. The ABC conjecture implies that for all $\epsilon > 0$, $\min a_{i} \gg (c^{2}+1)^{n/2 - 1 - \epsilon}$ …
- 19.9k
7
votes
Accepted
Question about iterations not divisible by infinitely many prime numbers
Yes. This follows from a result of Corrales-Rodrigáñez and Schoof (see the paper here) solving the support problem of Erdős.
In particular, suppose that there are only finitely many primes $p$ that do …
- 19.9k
13
votes
Accepted
A criterion for the equation $ax^n+bx+c=0$ not solvable by radicals via $a,b,c$ and $n$
No, the conjecture is false at least for $n = 5$. The irreducible quintic trinomial $f(x) = 85x^{5} - 4x + 1$ satisfies $\gcd(b,5ac) = \gcd(-4,5 \cdot 85 \cdot 1) = 1$. However, the Galois group of $f …
- 19.9k
4
votes
Accepted
Best error terms for functions related to square free numbers
As I say in the comment, the asymptotics for $M_{+}$ and $M_{-}$ follow directly from those for $M$ and $\hat{M}$. Therefore $M_{+}(x) = \frac{1}{2 \zeta(2)} x + \frac{1}{2} M(x) + O(x^{1/2})$ and $M_ …
- 2,723
5
votes
Accepted
Finding the $K=\mathbb{Q}(\sqrt{6})$-rational points on the twist of $X_{0}(26)$
I used Magma to point search on $C/K$ up to a height of $1000$ and it appears that $C(K) = \emptyset$. If that's true, then one can probably use the Mordell-Weil sieve to prove it. Here's a bit more d …
- 19.9k
5
votes
Accepted
Can each natural number be represented by $2w^2+x^2+y^2+z^2+xyz$ with $x,y,z\in\mathbb N$?
The answer to question 1 is yes - the other questions seem to me to be more difficult.
If $n$ is odd, then there are non-negative integers $w$, $x$ and $y$ so that $n = 2w^{2} + x^{2} + y^{2}$. One wa …
- 19.9k
32
votes
Only odd primes?
If $k$ is odd and not a perfect square, then the sets are disjoint. In particular, if $\alpha = \frac{k - \sqrt{k}}{\frac{k-1}{2}}$ and $\beta = \frac{k + \sqrt{k}}{\frac{k-1}{2}}$, then $\alpha$ and …
- 19.9k
7
votes
Accepted
Common Galois extension over $\mathbb Q $
If $k$ is odd, then yes. If $L'/\mathbb{Q}$ is a cyclic extension of degree $4$, choose an extension $M/\mathbb{Q}$ that is cyclic of degree $k$. Then the compositum $L'M/\mathbb{Q}$ will have ${\rm G …
- 19.9k
16
votes
Accepted
Looking for a "clever" argument for a $q$-series identity
Here's a proof that indicates a systematic method for proving such identities. Let $\eta(z) = q^{1/24} \prod_{n=1}^{\infty} (1-q^{n})$, with $q = e^{2 \pi i z}$. The identity you state in the equation …
- 10.6k
22
votes
Accepted
On Fibonacci numbers that are also highly composite
The largest highly composite Fibonacci number is $F_{3} = 2$.
If $p$ is a prime number, then either $p \mid F_{p-1}$ (if $p \equiv \pm 1 \pmod{5}$), $p \mid F_{p}$ (if $p = 5$), or $p \mid F_{p+1}$ (i …
- 19.9k
5
votes
Gaps between combinations of squares of integers
For any $\theta$ there is a constant $C_{\theta}$ and infinitely many $n$ for which $s_{n+1} - s_{n} \leq \frac{C_{\theta}}{\sqrt{n}}$.
Choose a rational approximation $\frac{h}{k}$ to $\theta$ so tha …
- 19.9k
16
votes
Accepted
Prove that $1$ is the sum of three tetrahedral numbers infinitely many different ways
There are infinitely many solutions. I'll show below that there are infinitely many positive integers $k$ for which $93k^{2} - 288k + 276 = z^{2}$ for some positive integer $z$. From such a $z$, we ge …
- 19.9k
9
votes
Curve with a rational point but no new points in number fields of low degree
Here's a family of examples that are geometrically irreducible. Let $p$ be a prime number and consider the modular curve $X_{1}(p)$. If $F$ is a number field, the points in $X_{1}(p)(F)$ are either no …
- 375
2
votes
Accepted
Imaginary quadratic fields with $\ell$-indivisible class number
Here's an elementary argument. For $\ell < 41$, $K = \mathbb{Q}(\sqrt{-163})$ works. For $\ell = 41$, $K = \mathbb{Q}(\sqrt{-3})$ works. Assume then that $\ell \geq 43$.
Choose an integer $1 \leq n \l …
- 19.9k
5
votes
Accepted
Why is this function a modular function of level $5$?
Here's a fairly straightforward way to show that $\phi$ is modular of level $5$ using Siegel functions.
Claim: The function $f(\tau)$ is a modular function for $\Gamma(5)$ if and only if $f(5\tau)$ is …
- 19.9k
48
votes
Accepted
How to prove that the solution to $x^{x+1}=(x+1)^{x}$ is transcendental?
The number $x$ is transcendental, and your Gelfond-Schneider argument almost works.
Suppose to the contrary that $x$ is algebraic. Then $x+1$ and $x/(x+1)$ are also algebraic, and so the Gelfond-Schne …
- 19.9k
7
votes
Euclid-style proof of Dirichlet’s theorem on primes in certain arithmetic progression
Yes. According to Paul Pollack's paper Hypothesis H and an impossibility theorem of Ram Murty, Murty gave an argument that an elementary Euclid-style proof is impossible when $a^{2} \not\equiv 1 \pmod …
- 19.9k
3
votes
Accepted
If $a_{g}(1)=g(x)$ and $a_{g}(r)=g(a_{g}(r-1))$ for $r>1$ then is it true that $\limsup\limi...
Yes, it is true, and I'm guessing it's well-known. (For example, it might be a theorem in Chapter 3 of Silverman's "The Arithmetic of Dynamical Systems", but I don't own that book yet.)
It is useful t …
- 19.9k
7
votes
Divisibility condition implies $a_1=\dotsb=a_k$?
Here's a a tweak of Seva's idea that gives a counterexample. Note that if $r$ is odd, then $2^{n}+1$ divides $2^{rn} + 1$.
Let $k = 6$, $a_{1} = 1$, $a_{2} = a_{3} = a_{4} = 2$, $a_{5} = a_{6} = 4$. T …
- 19.9k
16
votes
Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$
The equation you specify defines a surface $X$ in $\mathbb{P}^{3}$, and this surface is a K3 surface. It is conjectured that if $X$ is a K3 surface, there is a field extension $K/\mathbb{Q}$ over whic …
- 19.9k
10
votes
Accepted
Prime numbers in a sparse set
Yes, there is a $c > 1$ for which infinitely many numbers of the form $\lfloor k^{c} \rfloor$ are prime. The first result of this type was proven in Ilya Piatetski-Shapiro's Ph.D. thesis (written in 1 …
- 19.9k
27
votes
Accepted
Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?
Yes, it is true that $\{ x^{4} + y^{2} + z^{2} : x, y, z \in \mathbb{Z}[i] \} = \{ a + 2bi : a, b \in \mathbb{Z} \}$. Indeed, one can even take $x$ to be either $0$ or $1$ in all cases. Because $y^{2} …
- 19.9k
3
votes
Accepted
Properties of a certain sequence
Claim 1 is false. Let $\phi = \frac{1+\sqrt{5}}{2}$, $\overline{\phi} = \frac{1-\sqrt{5}}{2}$, $\lambda = \phi^{2}$ and $n_{0} = 1$.
I claim that $n_{k} = F_{2k+1}$, the $(2k+1)$st Fibonacci number fo …
- 19.9k
21
votes
Is there a real nonintegral number $x >1$ such that $\lfloor x^n \rfloor$ is a square intege...
At the request of the OP, I am turning my comment into an answer. It is possible to have $\lfloor x^{n} \rfloor$ close to a square for all positive integers $n$. For example, if $x = \frac{7 + 3 \sqrt …
- 19.9k
24
votes
Accepted
Norms in quadratic fields
This is false. The smallest counterexample is $d = 34$. Let $K = \mathbb{Q}(\sqrt{34})$. The fundamental unit in $\mathcal{O}_{K} = \mathbb{Z}[\sqrt{34}]$ is $35 + 6 \sqrt{34}$, which has norm $1$, an …
- 19.9k
10
votes
Accepted
Cusp forms with integer Fourier-coefficients
No. Take $k = 24$ and $N = 1$. Then $\Delta^{2} = q^{2} - 48q^{3} + 1080q^{4} + \cdots \in S_{K}(\Gamma_{1}(N),\mathbb{Z})$. However, if we write $\Delta^{2} = c_{1} f_{1} + c_{2} f_{2}$, where $f_{1} …
- 19.9k
16
votes
Accepted
$x^3+x^2y^2+y^3=7$, and solvable families of Diophantine equations
(a) No. There are no integer solutions. The curve $C$ you give has genus $3$ and it has an obvious automorphism $\phi(x,y) = (y,x)$. The quotient curve is an elliptic curve. In particular, if you let …
- 19.9k
23
votes
Accepted
Algorithmic (un-)solvability of diophantine equations of given degree with given number of v...
I'm going to take a stab at this. First (as mentioned in Andres Caicedo's answer to this question), Siegel proved in 1972 that there is an algorithm to determine whether a quadratic equation in any nu …
- 103
1
vote
Half integral weight modular forms that reduce to a nonzero constant modulo a given prime
If such a form does exist, then its level must be a multiple of $p$.
If $f = \sum a_{n} q^{n}$ is a half-integer weight modular form with integer coefficients with $a_{i} \equiv 0 \pmod{p}$ for all $ …
- 19.9k
14
votes
Number of real roots of irreducible polynomials that are solvable by radicals
Klueners and Malle have a database of number fields of degree $\leq 19$ that (tries) to include every Galois group and every possible signature. Examining this database shows that for composite $n$ wi …
- 19.9k
7
votes
Accepted
Criterion for generic polynomials
I believe there is no good way to determine in general if a polynomial $P(\mathbf{t},X)$ is generic. In fact, given a number field $K$ and a univariate polynomial $P(t,x) \in \mathbb{Q}[t,x]$, the pro …
- 19.9k
6
votes
Approximations to $\pi$
This is not exactly an answer to the stated question, but it's too long for a comment. Rather than the form given in the question, one could represent a number in the form $\frac{a + b \sqrt{d}}{c}$, …
- 19.9k
6
votes
Accepted
Diophantine equations that involve cubes and the volume of square frustums
The Problem 1 you specify defines a cubic surface in $\mathbb{P}^{3}$, and there is a lot known about the rational points on such a surface. (For example, Chapter 2 of the book "Rational and nearly ra …
- 19.9k
7
votes
Accepted
Can the Petersson inner product $\langle f(z), f(2z) \rangle$ be zero?
Yes, the Petersson inner product can be zero. In my paper "Explicit bounds for sums of squares (see Lemma 5) I show that if $f$ is a newform of level $N$ and $p$ is a prime that does not divide $N$, t …
- 19.9k
7
votes
An explicit formula for a cuspidal form of weight $2$ and arbitrarily large prime level
You ask for the explicit Fourier expansion of a weight $2$ cusp form of level $p$ for $p$ arbitrarily large. This suggests you're OK with only certain primes $p$. If $p \equiv 11 \pmod{12}$ is prime, …
- 19.9k
8
votes
Examples of models for modular curves
Here's an example. Let's take $\Gamma = \Gamma_{0}(4)$, and
$\Gamma' = \Gamma(2)$. We'll let $\alpha = \begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}$, so $\Gamma' = \alpha \Gamma \alpha^{-1}$. The func …
- 19.9k