All Questions
Tagged with nt.number-theory diophantine-equations
787 questions
7
votes
1
answer
751
views
On the shortest open cubic equation
The question is: are there any integers $x,y,z$ such that
$$
1+4 x^3+x y^2+2 y z^2 = 0 \quad\quad\quad\quad (1)
$$
The motivation is: Define the length of a polynomial $P$ consisting of $k$ monomials ...
- 7,182
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
8
votes
4
answers
870
views
A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
- 7,182
3
votes
2
answers
226
views
Integer solutions to $x^2 + x + 1 = y^z$? [duplicate]
In the context of finite projective planes I am interested in the Diophantine equation $\frac{x^3-1}{x-1} = y^z$, which is also written as $x^2 + x + 1 = y^z$, for $z>1$. I stumbled by accident on ...
- 1,245
4
votes
0
answers
78
views
Repeated values of a monomial
Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
- 1,990
5
votes
1
answer
262
views
Radicands of square roots of the 2020s, written in simplest radical form
As of the time of writing, the current decade is the 2020s. An interesting property of this decade is that there are 3 years that satisfy the property that the square-free part (https://oeis.org/...
- 425
4
votes
0
answers
394
views
Are there integers $x,y,z$ such that $(x+1)y^2-xz^2=x^3+2x+2$?
Is equation
$$
(x+1)y^2-xz^2=x^3+2x+2
$$
solvable in integers?
Motivation: For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, define ...
- 7,182
1
vote
0
answers
208
views
Are there integers $x,y,z$ such that $1 + x - x^3 + x^2 y^2 + z + z^2 = 0$?
In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any ...
- 7,182
0
votes
1
answer
160
views
Diophantine equations involving recurrence sequences
I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
- 31
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
24
votes
2
answers
2k
views
Are (55, 165, 495, 1485) and (286, 1716, 10296, 61776) the only geometric sequences of length 4 among non-trivial binomials?
Let's define non-trivial binomial coefficients as values of $\binom{n}{k}$, where $n$ and $k$ are positive integers such that $2 \le k \le \frac{n}{2}$. (Therefore, $6$ is the smallest non-trivial ...
- 425
12
votes
1
answer
993
views
General solution of the quartic $a^4+b^4=c^4+d^4$?
The background to the question:
$$a^4+b^4=c^4+d^4 \label{1}\tag 1 $$
Tito Piezas, Tomita & others have recently given some parametric solutions on Math stack exchange & Math overflow. In math ...
- 127
1
vote
1
answer
151
views
$(2^a-1)+b^2=2^c$ [closed]
$31+15^2=256$.
Are there infinitely many solutions to:
$(2^a-1)+b^2=2^c$ with a,b,c positive integer and a,b,c different each other.
- 11
13
votes
2
answers
1k
views
On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$A^4+B^4 = C^4+D^4$$
we use Euler's method. Let,
$$(p+q)^4+(r-s)^4=(p-q)^4+(r+s)^4$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
- 12.6k
2
votes
0
answers
184
views
Will Coppersmith's method work for this bivariate modular polynomial shape?
I have a bivariate modular polynomial of shape
$$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$
where
$q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$,
$g(x)\in\mathbb Z[x]$ is of degree four and
$f(...
- 13.9k
3
votes
0
answers
176
views
For which primes $p$ in $\mathbb Z$ is $p\omega$ the sum of two cubes in $\mathbb Q(\omega)$?
This is related to an earlier question I posed —"Possible extensions of a conjecture …". Now that my note arXiv:2309.00162 has appeared I'll use it as a reference.
Elementary results(along ...
- 5,422
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
0
votes
0
answers
178
views
Elementary method for finding integer solutions for certain types of elliptic curve
There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $...
- 193
9
votes
1
answer
637
views
Representing $x^6-4$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^6-4$ is a sum of two squares of integers.
Equivalently, prove that $x^3-2$ and $x^3+2$ are simultaneously sums of two ...
- 7,182
18
votes
2
answers
2k
views
What is the taxicab number for rational fourth powers?
The taxicab number is the smallest integer that can be expressed as a sum of two positive integer cubes in two different ways, and it is equal to $1729=12^3+1^3=10^3+9^3$. There are generalizations to ...
- 7,182
5
votes
0
answers
454
views
Is 136 a difference of two rational fourth powers?
There is a rich literature that studies which small positive integers are the sums of two rational fourth powers, see e.g. Section 6.6 of Henri Cohen's book Volume I: Tools and Diophantine Equations. ...
- 7,182
4
votes
1
answer
408
views
Are these equations solvable in positive integers?
By Matiyasevich theorem, there is no algorithm to decide whether a given Diophantine equation $P(x_1,\dots, x_n)=0$ has a solution in positive integers. As suggested in What is the smallest unsolved ...
- 7,182
9
votes
1
answer
467
views
Positive integers such that $(x+y)(xy-1)=z^2+1$
Do there exist positive integers $x,y,z$ such that
$$
(x+y)(xy-1)=z^2+1
$$
In my previous question Can you solve the listed smallest open Diophantine equations?, I discuss the smallest equations for ...
- 7,182
3
votes
1
answer
115
views
3-dimensional Boolean cube of Squares
Do there exist positive integers $A, B, C$ such that all seven numbers $$A, B, C, A+B, B+C, A+C, A+B+C$$ are perfect squares?
- 109k
5
votes
3
answers
606
views
Can you describe all rational solutions to these simple-looking equations?
Can you describe, in parametric form or in any other explicit way, all rational solutions to any of the following equations:
$$
y^2 + z^2 = x^3+1,
$$
$$
y^2 + z^2 = x^3-1,
$$
$$
y^2+x^2y+z^2+1=0.
$$
...
- 7,182
14
votes
1
answer
612
views
What are the rational solutions to $y^4=x^3+x+1$?
What are the rational solutions to $y^4=x^3+x+1$?
This equation is interesting because it has substitution $y^2=z$ that reduces it to elliptic curve $z^2=x^3+x+1$. Sometimes, the existence of such ...
- 7,182
5
votes
1
answer
279
views
Diophantine equations involving the difference between perfect square and perfect cube
(a) Do there exist infinitely many triples $(x,y,z)$ of integers with $z\neq 0$ such that
$$
z(x^3-y^2) = x+1.
$$
(b) The same question for
$$
z(x^3-y^2) = y+1.
$$
In other words, are there infinitely ...
- 7,182
3
votes
2
answers
191
views
What can be said about the cube-free part of $x^3 -3xy^2 +y^3$?
For $x$ and $y$ in $\mathbf{Z}$, not both zero, let $cfp(x,y)$ be the cube-free part of $x^3 -3xy^2 + y^3$ (normalized to be $> 0$). One sees:
(#) $cfp(x,y)$ is either a product of primes $p$, with ...
- 5,422
1
vote
2
answers
643
views
Describe all integer/rational solutions to $x^3+y^3+z^3+t^3+s^3=0$
The question is in the title.
Equation $\sum_{i=1}^n x_i^3 = 0$ has no non-trivial integer solutions for $n=3$. For $n=4$, there are known descriptions of all integer/rational solutions, see
Choudhry, ...
- 7,182
6
votes
0
answers
321
views
A generalization of the Diophantine $m$-tuple problem
Are there distinct positive integers $a_1，a_2，a_3，b_1，b_2，b_3$ such that $a_ib_j+1$ is a perfect square number for each $i,j$ ($1\leq i,j\leq3$)?
I asked the following question in a group, and ...
- 61
4
votes
0
answers
307
views
Equations involving sum of fourth powers
Do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - 1 = x^4+y^4 \neq 0 \tag{$a$} \quad ?
$$
Also, do there exist rational numbers $x,y,z$ such that
$$
\quad \quad z^3 - z = x^4+y^4 \...
- 7,182
1
vote
0
answers
152
views
How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
- 7,182
1
vote
2
answers
127
views
Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...
- 3,625
0
votes
2
answers
228
views
$y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves
What are the rational solutions to the equation
$$
y^3 = x^4 + x,
$$
in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$?
Context: This is the simplest-looking ...
- 7,182
3
votes
0
answers
511
views
Regarding the Challenge Problem in 3Blue1Brown's most recent video: Will $\binom{x}{4}+\binom{x}{2}+1=2^k$ for $x>10$? [duplicate]
Link to the video here with timestamp
In deriving the formula for regions of Moser's Circle Problem, it observed that the formula
$$
F(x)=\binom{x}{4}+\binom{x}{2}+1
$$
achieves values that are equal ...
- 315
5
votes
0
answers
180
views
Existence of large integer solution for a simple-looking equation
Is it true that for every $k>0$ Diophantine equation
$$
y^2 + x^2y + z^2x + 1 = 0
$$
has an integer solution $(x,y,z)$ such that $\min\{|x|,|y|,|z|\}\geq k$?
Motivation: this equation arises in the ...
- 7,182
0
votes
2
answers
278
views
Solvability of two-variable quadratic equations with a parameter
(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation
$$
2 x^2+a x y+y^2+1 = 0
$$
is solvable in integers $(x,y)$.
(b) The same question for a similar equation
$...
- 7,182
2
votes
1
answer
587
views
On the equation $4y^p= x^2 + 3$
Do there exist some non-zero rational numbers $x, y$ such that $x \neq \pm y$ and
$$4y^p = x^2 + 3 \tag{1}$$
for some odd prime $p$?
If one lets $y=a/b$ and $x=c/d$ where $a, b, c, d$ are non-zero ...
- 1,019
2
votes
1
answer
258
views
An arithmetic problem involving a system of equations
Fix a positive integer $r$. Describe the solutions to the system of equations given by:
$$\begin{equation}\sum_{1\leq i\leq r}X_i^2\equiv0\pmod{X_k}(1\leq k\leq r)\end{equation}$$
Example: In the case ...
- 1,433
0
votes
0
answers
74
views
Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed
The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
- 3,098
10
votes
2
answers
1k
views
Integer solutions of an exponential equation
How can I solve this equation?
$$7^{x} +2=y^{2}$$
$x$ and $y$ must be natural numbers.
- 103
4
votes
1
answer
915
views
Does this conic have a rational point?
Consider the conic
$$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$
over the function field $\mathbb{Q}(u,v)$.
Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
- 8,998
0
votes
1
answer
204
views
Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
- 7,182
1
vote
0
answers
129
views
Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$?
Related to this question,
where Bogdan Grechuk suggested this question.
Q1 Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$...
- 25.4k
3
votes
0
answers
235
views
Is $16a+5$ always of the form $x^2+y^2+z^4$?
Working over the integers.
For $a$ up to $10^7$, $16a+5$ is always of the form $x^2+y^2+z^4$.
Q1 Is $16a+5$ always of the form $x^2+y^2+z^4$?
Heuristic argument:
For prime $p=4b+1$, both of $p$ and $...
- 25.4k
3
votes
0
answers
257
views
Are all odd integers greater than $599$ of the form $x^2+y^2+z^4+t^4$?
For $a \le 10^7$, the equation over integers $4a+1=x^2+y^2+z^4+t^4$
has solutions.
Q1 Is it true that all integers of the form $4a+1$
are also of the form $x^2+y^2+z^4+t^4$?
Heuristic argument: ...
- 25.4k
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
6
votes
1
answer
669
views
Hilbert's tenth problem for equations with finitely many solutions
Is there a known example of a set $S$ of Diophantine equations such that
$S$ is computable;
it is a theorem that every equation in $S$ has (at most) finitely many solutions;
the function that maps an ...
- 82.7k
1
vote
0
answers
98
views
Hardness of solving $0=\sum_{i=1}^k \operatorname{linear}_i(x_1,\ldots,x_n)^D$ over the rationals
This is related to cryptography and this question
and another question.
In short, we are asking about decomposing multivariate polynomial
as sum of perfect powers of linear polynomials.
Working over $\...
- 25.4k
2
votes
0
answers
87
views
Complexity of finding solutions of trapdoored polynomial?
Related to this question Cryptography signature scheme based on hardness of finding points on varieties.
Working over $K=\mathbb{Q}[x_1,...,x_n,y_1,...y_m]$.
By abuse of notation, for polynomial $f$, ...
- 25.4k