All Questions
46 questions
11
votes
1
answer
598
views
How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \Bigg|\ \mathbb Z^3\cap\Big\{(a_1,a_2,a_3)\ \Big|\
a_1^2+a_2^2+a_3^2=n \Big\}\Bigg| \\
= \frac12\Bigg|\mathbb Z^3\cap\Big\{(a_1,...
- 113
-5
votes
1
answer
150
views
On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
- 17
0
votes
2
answers
215
views
Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
2
votes
1
answer
313
views
On the Diophantine equation $2^a3^b + 4^c5^d = 2^b3^a + 4^d5^c$ [closed]
In this year's team selection test to the international mathematical Olympiad in Japan, a interesting diophantine equation appeared as the final problem of the second day.
Find all quadruples $(a,b,c,...
8
votes
3
answers
830
views
About the units in $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$
Let's consider $K=\mathbb{Q}[\sqrt{d}]$ where $d$ is positive and square free. It is well known that the ring of integers is
$$
{O}_{K}=\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\right]
$$ if $d=1 \mod 4$ ...
1
vote
1
answer
352
views
A Mordell equation $y^3=x^2+20$ [closed]
Recently I met a problem when I'm studying algebraic number theory.
Problem. Find all positive integer solutions of $y^3=x^2+20$.
I solved the situation when $x$ is an odd because the two ideals $(x+...
- 505
5
votes
1
answer
356
views
Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$
While working on finite order elements of $\operatorname{SO}_n$, I meet this question:
Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers.
As ...
- 3,432
0
votes
1
answer
152
views
Almost Pell type equation
Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
- 8,998
7
votes
3
answers
611
views
Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...
- 8,842
1
vote
0
answers
115
views
Integral points in smooth cubic curves
Let $X$ be a smooth affine cubic curve in $\mathbb A^2$ defined by $f(T_1,T_2)\in\mathbb Z[T_1,T_2]$ (of course $\deg(f)=3$ by definition), and
$$n(f, B)=\{(x_1,x_2)\in\mathbb Z^2| |x_1|\leq B, |x_2|\...
- 403
2
votes
0
answers
93
views
Integers solutions of products of truncated Riemann zeta functions
Let $n \in \mathbb{N}$ be a positive integer.
It is possible to prove that the equation $F_1(n)=m$ where $m \in \mathbb{Z}$ and
$$
F_1(n)=(1+2+\ldots+n)\cdot\left(\frac{1}{1}+\frac{1}{2}+\dots+\frac{1}...
- 1,343
1
vote
1
answer
364
views
Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
- 17
2
votes
0
answers
356
views
Classifying solutions of a certain Diophantine Equation
The following question arose from a problem I am working on.
Let $N, k$ be positive integers. Consider the Diophantine equation in $a, b, c$:
$$
\frac{1}{a} + \frac{N - 1}{b} = \frac{N^k}{c}
$$
with ...
- 791
7
votes
2
answers
602
views
Bounds on Bézout coefficients
Let $0<a_1 \le a_2 \le \cdots \le a_n$ be positive integers such that $a_1 + \cdots + a_n = m$ and $\gcd(a_1,\ldots,a_n)=1$. Let $\mathbf a :=(a_1,\ldots,a_n)\in\mathbb Z^n$ and $\mathbf x:=(x_1,\...
- 277
7
votes
2
answers
398
views
Given that $n > 3$ and $z$ is a Gaussian integer, when can $z^n \pm z$ be a rational integer?
I came across the following conjecture. If you have any thoughts on how to approach it, let me know.
Conjecture. For any integer $n > 3$ and any Gaussian integer $z$ that is not a unit, if $z^n - z$...
- 1,625
30
votes
2
answers
1k
views
Is equation $xy(x+y)=7z^2+1$ solvable in integers?
Do there exist integers $x,y,z$ such that
$$
xy(x+y)=7z^2 + 1 ?
$$
The motivation is simple. Together with Aubrey de Grey, we developed a computer program that incorporates all standard methods we ...
- 7,182
2
votes
0
answers
242
views
Solving $x^k+y^k+z^k=w^k$ non-trivially in strictly positive integers
Consider the equation $x^k+y^k+z^k=w^k$ in $x$, $y$, $z$ and $w$ with $k\in\mathbb{N}_{\geq2}$.
If we look for solutions that are strictly positive and non-trivial i.e. $x\neq-y$, $x\neq w$ etc... ...
- 4,862
5
votes
0
answers
284
views
On $w^4+x^4+y^2+z^2$ over a number field
In 1921 Siegel confirmed a conjecture of Hilbert by proving that for any number field $K$ each element of
$$K_{\geq0}=\{a\in K:\ \sigma(a)\geq0\ \mbox{for all}\ \sigma\in\mathrm{Gal}(K/\mathbb Q)\}$$ ...
- 15.6k
8
votes
0
answers
245
views
Hilbert 10th problem for genus 2 equations
Hilbert 10th problem, while undecidable in general, remains open for 2-variable equations: we do not know if there is an algorithm that, for polynomial $P(x,y)$ with integer coefficients, decides ...
- 7,182
19
votes
1
answer
679
views
Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?
Recall that the ring of Gaussian integers is
$$\mathbb Z[i]=\{a+bi:\ a,b\in\mathbb Z\}.$$
Clearly
$$(a+bi)^2=a^2-b^2+2abi\ \ \mbox{and}\ \ (a+bi)^4=(a^2-b^2)^2-4a^2b^2+4ab(a^2-b^2)i.$$
Question. Is it ...
- 15.6k
6
votes
3
answers
605
views
Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$
Consider the equation $$\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}.$$
Of course, there are solutions to this like $(a,b,c) = (9,8,6)$.
Is there any known approximation for the ...
- 749
9
votes
0
answers
328
views
Cohn's eight diophantine equations
Today I was reading J.H.E. Cohn's Eight diophantine equations (1966). On p. 158
he comes across the equation $y^2 = a^3 + 3a$ for odd values of $a$ and writes that this is equivalent to $x^3 + (x+1)^3 ...
- 32.6k
1
vote
0
answers
135
views
$n$-variable polynomials modulo $p$
The Hasse-Weil bound implies that for any 2-variable polynomial $P(x,y)$, there exists approximately $p$ solutions in $\mathbb{F}_p$ of $P(x,y) \equiv a \pmod p$ for sufficiently large $p$, and any ...
- 141
0
votes
1
answer
320
views
find all of rational solutions of the quartic equation?
Consider the equation $a^4+6v^2a^2-8a+v^4=0$ over the rationals.
Note that the following are solutions: $(a,v)=(1,1),(0,0),(2,0)$. Are there any other rational solutions?
4
votes
2
answers
709
views
On the product $\prod_{k=1}^{(p-1)/2}(x-e^{2\pi i k^2/p})$ with $x$ a root of unity
Let $p$ be an odd prime. Dirichlet's class number formula for quadratic fields essentially determines the value of the product $\prod_{k=1}^{(p-1)/2}(1-e^{2\pi ik^2/p})$. I think it is interesting to ...
- 15.6k
5
votes
0
answers
230
views
Diophantine applications of Paramodularity
I’ve asked this question to quite a few people in person and so far haven’t seen a good answer... but I believe one should exist, so here goes!
Ok, we all know how to (roughly) prove Fermat’s Last ...
- 562
1
vote
1
answer
209
views
Representing integers efficiently with quadratic polynomials
For every large enough $T$ given four integers $a,b,c,d$ with absolute value less than $T^2$ are there integers $w_1,x_1,y_1,z_1,w_2,x_2,y_2,z_2$ with absolute value less than $T$ such that
$$w_1x_1+...
- 13.9k
6
votes
1
answer
705
views
Extension of Erdos-Selfridge Theorem
Erdos and Selfridge open their paper "The Product of Consecutive Integers is Never a Power" (1974) with the theorem
$\text{Theorem 1:}$ The product of two or more consecutive positive integers is ...
- 429
0
votes
1
answer
394
views
Diophantine equations and modular forms
Let $D$ be a square-free positive integer which is the fundamental discriminant of a real quadratic field. Consider the following quadratic form $$Q_{D}(x,y)=x^2+Dy^2.$$
My questions are :
What is ...
- 347
0
votes
1
answer
346
views
Upper bounds for solutions to a Pell-like equation
Let $N$ be a fixed positive integer that is not a square and $m$ be any nonzero integer. Let $x$ and $y$ be positive integers that solve $$x^2 - N y^2 = m^2$$ with $x + y$ minimal (in light of the ...
- 2,283
18
votes
1
answer
1k
views
Is $x^2+x+1$ ever a perfect power?
Using completing the square and factoring method I could show that the Diophantine equation $x^2+x+1=y^n$, where $x,y$ are odd positive and $n$ is even positive integers, does not have solution, but ...
- 841
2
votes
1
answer
470
views
Non-negative integer solutions of x^2+y^3=n
I have the next equation: $x^2+y^3=n$. Where n is a positive integer constant.
I want to know the exact number of non-negative integer solutions.
Also I want to know what are those solutions. How ...
- 133
5
votes
2
answers
822
views
On the Diophantine equation $x^2 = y^p + 2^{r}z^p$ where $p\geq 7$ is an odd prime and $r \geq 2$
It is known that the only nonzero pairwise coprime integer solutions to the above Diophantine equation are for $r=3$, for which $(x, y, z) = (3,1,1)$ and $(-3,1, 1)$. (Cohen, Number Theory Volume 2: ...
user83236
13
votes
1
answer
1k
views
On cubic reciprocity for $x^3+y^3+z^3 = 996$?
I. The Diophantine equation,
$$x^3+y^3+z^3 = 3w^3\tag1$$
with $x\geq y \geq z$ and $w=1$ has only two known solutions, namely $1,1,1$ and $4,4,-5$. Are there larger ones? As Noam Elkies points out ...
- 12.6k
2
votes
1
answer
352
views
How can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution for every prime $p>3$?
Without applying Fermat's Last Theorem, how can one show that the hyperelliptic curve $y^2 = x^{p} + \frac{1}{4}$ has only one positive rational solution $(x,y) = (0, \frac{1}{2})$ for ever prime $p \...
user83236
11
votes
2
answers
576
views
Equation $x^2=y^p + 1$
can you help me please for solving this diophantine equation : $x^2=y^p+1$
and if you can give me a general method to studying such equation : $x^2=y^p+t$
Thanks
- 143
48
votes
4
answers
4k
views
Fermat's last theorem over larger fields
Fermat's last theorem implies that the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}$ is finite.
Is the number of solutions of $x^5 + y^5 = 1$ over $\mathbb{Q}^{\text{ab}}$ finite?
Here $\...
- 11.3k
0
votes
1
answer
285
views
The number of solutions of a Diophantine equation [closed]
Is $\lim_{n \rightarrow \infty} |\{(x,y) \in \mathbb{Q}(\zeta_n)^2 : y^3 = x^3 + x + 1\}| < \infty ?$ where $\zeta_n$ is a primitive $n$-th root of unity.
That is, I am asking whether the number ...
- 11.3k
4
votes
0
answers
332
views
Diophantine equations over cyclotomic fields
Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...
- 11.3k
-3
votes
1
answer
193
views
$p=4x^2+27y^2$,with $p$ a prime [closed]
p is a prime ,on what condition the Diophantine equation is solvable.what is it Linear expression ,for example ,$x^2+3y^2=p$, $p=3k+1$ ,$x^2+5y^2=p$ ,
$p=1,9\pmod{20}$.
- 13
28
votes
6
answers
2k
views
Patterns among integer-distance points
Mark each point of $\mathbb{N}^2$ ($\mathbb{N}$ the natural numbers) if its
Euclidean distance from the origin is an integer. One obtains a plot like this, symmetric about the $45^\circ$ diagonal.
...
- 151k
8
votes
1
answer
1k
views
Fundamental units of imaginary quartic fields
Let $F/{\mathbb Q}$ be an imaginary quartic extension (i.e. the degree $[K:{\mathbb Q}]=4$ and no embedding of $K$ in ${\mathbb C}$ has its image inside the real numbers). Then the unit group of the ...
- 3,399
3
votes
1
answer
380
views
Do the solutions to the unit equation lie dense in the complex numbers
Let $S\subset \overline{\mathbf{Q}}\subset \mathbf{C}$ be the set of solutions to the unit equation, i.e., $S$ consists of algebraic integers $a$ such that $a$ and $1-a$ are units in the ring of ...
- 33
5
votes
5
answers
829
views
For any $n$, does there exist a number field with at least $n$ solutions to the unit equation
Let $n$ be a positive integer.
Does there exist a number field $K$ such that the number of solutions of the unit equation $$a+b =1, \quad a,b\in O_{K}^\ast$$ is at least $n$? Can we write down such a ...
- 225
27
votes
2
answers
4k
views
Fermat's Last Theorem in the cyclotomic integers.
Kummer proved that there are no non-trivial solutions to the Fermat equation FLT(n): $x^n + y^n = z^n$ with $n > 2$ natural and $x,y,z$ elements of a regular cyclotomic ring of integers $K$.
I am ...
- 621
27
votes
4
answers
11k
views
Is there an elementary way to find the integer solutions to $x^2-y^3=1$?
I gave this problem to my undergraduate assistant, as I saw that Euler had originally solved it (although I am having trouble finding his proof). After working on it for two weeks, we boiled the hard ...
- 18.7k