Questions tagged [diophantine-equations]
Diophantine equations are polynomial equations $F=0$, or systems of polynomial equations $F_1=\ldots=F_k=0$, where $F,F_1,\ldots,F_k$ are polynomials in either $\mathbb{Z}[X_1,\ldots,X_n]$ of $\mathbb{Q}[X_1,\ldots,X_n]$ of which it is asked to find solutions over $\mathbb{Z}$ or $\mathbb{Q}$. Topics: Pell equations, quadratic forms, elliptic curves, abelian varieties, hyperelliptic curves, Thue equations, normic forms, K3 surfaces ...
935 questions
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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(...
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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 ...
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Integers representable as binary quadratic forms
It is known that odd prime $p$ can be represented as $p=x^2+y^2$ if and only if $p \equiv 1$ mod $4$, represented as $p=x^2+2y^2$ if and only if $p \equiv 1$ or $3$ mod $8$, represented as $p=x^2+3y^2$...
- 25.7k
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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, $...
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Is the continued fraction of a constructible number special in some way?
Rationals have finite CF and quadratic have periodic CF. CF in turn can be represented in terms of the modular group SL2(Z), e.g. using the standard generators S(z)=-1/z and T(z)=z+1. On the other ...
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Representation of a number as a product of $\sqrt{n^2 + 1} + n$
Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and
$$
\prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
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Is 136 a difference of two rational fourth powers?
There is a rich literature that studies which small positive integers are the sums of two rational fourth powers, see e.g. Section 6.6 of Henri Cohen's book Volume I: Tools and Diophantine Equations. ...
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Representing $x^6-4$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^6-4$ is a sum of two squares of integers.
Equivalently, prove that $x^3-2$ and $x^3+2$ are simultaneously sums of two ...
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Representing $x^3-2$ as a sum of two squares
Prove that there exist infinitely many integers $x$ such that integer $P(x)=x^3-2$ is a sum of two squares of integers.
Ideally, I am looking for a proof method that also applies for other $P(x)$, ...
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Hilbert 10th problem for cubic equations
Hilbert 10th problem, asking for algorithm for determining whether a polynomial Diopantine equation has an integer solution, is undecidable in general, but decidable or open in some restricted ...
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Are these equations solvable in positive integers?
By Matiyasevich theorem, there is no algorithm to decide whether a given Diophantine equation $P(x_1,\dots, x_n)=0$ has a solution in positive integers. As suggested in What is the smallest unsolved ...
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Positive integers such that $(x+y)(xy-1)=z^2+1$
Do there exist positive integers $x,y,z$ such that
$$
(x+y)(xy-1)=z^2+1
$$
In my previous question Can you solve the listed smallest open Diophantine equations?, I discuss the smallest equations for ...
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The elliptic curve for $x_1^9+x_2^9+\dots+x_6^9 = y_1^9+y_2^9+\dots+y_6^9$
I. Theorem: "If there are $a,b,c,d,e,f$ such that,
$$a+b+c = d+e+f\tag1$$
$$a^2+b^2+c^2 = d^2+e^2+f^2\tag2$$
$$3u^3-3uv+w=-def\qquad\tag3$$
with $(u,v,w)$ as the symmetric polynomials $u=a+b+c,\; ...
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Algorithm for computing rational points if the rank of Jacobian is 0
Is there a general algorithm that can compute in finite time all rational points on any curve of genus $g\geq 2$ whose Jacobian has rank $0$?
If not, for what special cases such algorithm is known? ...
- 11.3k
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3-dimensional Boolean cube of Squares
Do there exist positive integers $A, B, C$ such that all seven numbers $$A, B, C, A+B, B+C, A+C, A+B+C$$ are perfect squares?
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What are the rational solutions to $y^4=x^3+x+1$?
What are the rational solutions to $y^4=x^3+x+1$?
This equation is interesting because it has substitution $y^2=z$ that reduces it to elliptic curve $z^2=x^3+x+1$. Sometimes, the existence of such ...
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Diophantine equation Oeis A159589
Considera the diophantine equation:
$y^2=x^2+(x+449)^2$.
Is there a method to solve this equation?
And why an Oeis sequence Is dedicated to this equation? Has this diophantine equation something ...
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Is there any formula to find number of Pythagorean triplets between two integers 2 and j, j>2?
Given $j \geq 5$, is there a formula for the number of Pythagorean triplets $(a, b, c)$ satisfying the constraint that $a, b, c \leq j$?
There exists at least one Pythagorean triplet for $j\geq5$; ...
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Which types of Diophantine equations are solvable?
Is there a list somewhere of which types of Diophantine equations are solvable, which types are not solvable, and which types are not known to be solvable or not? (When I say solvable, I mean that we ...
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On the Diophantine equation $x^{4}+y^{4}=z^p$
Do there exist integers $x,y,z$ with $xyz\neq 0$, such that
$$x^4 + y^4 = z^p$$
where $p\geq 5$ is some prime ?
If yes, are there infinitely many of them ? And if there exists infinitely many of ...
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Mahler's proof of $S$-unit equation
Many modern proofs of the (ineffective) finiteness of solutions of the $S$-unit equation $x+y=1$ use Roth's theorem. In particular it is used Lang's version of Roth's theorem which takes in account ...
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Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins
(Update):
Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as,
$$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^...
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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 ...
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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, ...
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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 ...
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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 ...
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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
$$
\quad \quad z^3 - z = x^4+y^4 \...
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How difficult is to find rational points on these genus 3 curves:
How difficult is to find all rational points on these genus 3 curves:
$$
(a) \quad \quad x^3 + y^3 x +y^2 - y = 0
$$
$$
(b) \quad \quad x^4 - y^3 + x y + x = 0
$$
Short motivation. Consider the ...
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answers
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Number of integers $x \leq B$ such that $f(x)\mid g(x)$ for coprime polynomials $f,g$
Let $f, g \in \mathbb{Z}[x]$ be coprime polynomials.
I am interested in an upper bound for
$$
N(B) = \# \{ x \in [-B, B] \cap \mathbb{Z}: f(x)\mid g(x) \}.
$$
I assume there must be something known ...
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$y^3=x^4+x$, and computing all rational points on rank $0$ Picard curves
What are the rational solutions to the equation
$$
y^3 = x^4 + x,
$$
in particular, are there any (finite) solutions other than $(x,y)=(0,0)$ and $(-1,0)$?
Context: This is the simplest-looking ...
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$y^3 = x^4 + x + 2$, and existence of rational points on rank 0 Picard curves
Do there exists rational numbers $x$ and $y$ such that
$$
y^3 = x^4 + x + 2 ?
$$
Context: There are a lot of publications about computing rational points on elliptic and hyperelliptic curves, and ...
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Regarding the Challenge Problem in 3Blue1Brown's most recent video: Will $\binom{x}{4}+\binom{x}{2}+1=2^k$ for $x>10$? [duplicate]
Link to the video here with timestamp
In deriving the formula for regions of Moser's Circle Problem, it observed that the formula
$$
F(x)=\binom{x}{4}+\binom{x}{2}+1
$$
achieves values that are equal ...
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Number of solution to homogeneous linear Diophantine equations
Let $T,M\in\mathbb{N}$ be fixed. Consider a linear Diophantine equation of the form
$a_1 x_1 + a_2 x_2 + … + a_n x_n = 0 $
with $a_i \in [-T,T] \subset \mathbb{Z}$. Is there an asymptotic formula to ...
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Existence of large integer solution for a simple-looking equation
Is it true that for every $k>0$ Diophantine equation
$$
y^2 + x^2y + z^2x + 1 = 0
$$
has an integer solution $(x,y,z)$ such that $\min\{|x|,|y|,|z|\}\geq k$?
Motivation: this equation arises in the ...
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Rational points on genus 3 curves defined by short equations
(a) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3-y=x^4-x.
$$
(b) Find all pairs of rational numbers $(x,y)$ such that
$$
y^3+y=x^4+x.
$$
If not a complete answer, I would be happy to ...
2
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On the equation $4y^p= x^2 + 3$
Do there exist some non-zero rational numbers $x, y$ such that $x \neq \pm y$ and
$$4y^p = x^2 + 3 \tag{1}$$
for some odd prime $p$?
If one lets $y=a/b$ and $x=c/d$ where $a, b, c, d$ are non-zero ...
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Question on a crucial lemma in Euler's approach to Fermat's Last Theorem for $n=3$
As many of you may know, the illustrious L. Euler put forward a proof of the case $n=3$ of Fermat's Last Theorem via infinite descent. The thing is that, at a certain point, he resorted to the ...
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Does $2$ variable linear Diophantine equation in $NC$ imply $2$ dimensional shortest vector is in $NC$?
Consider the Linear Diophantine in known $a,b,c\in\mathbb Z$
$$ax+by=c.$$
Above can be solve by Extended Euclidean which is not in $NC$ as far as we know. It is clear if Extended Euclidean is in $NC$ ...
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3
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Centered-hexagonal triangular squares
Is there a centered-hexagonal, triangular, square (apart from 0 and 1)?
In other words, is there a positive integer that is simultaneously
(1) a perfect square, $n^2$, $n \ge 2$,
(2) a triangular ...
- 2,370
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2
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Integer solutions of an exponential equation
How can I solve this equation?
$$7^{x} +2=y^{2}$$
$x$ and $y$ must be natural numbers.
- 3,065
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74
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Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed
The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
- 3,098
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votes
0
answers
101
views
Persistence of KAM tori as a function of dimension
I have tried posting this question in MSE, but I think it might be too technical so I'm trying again here.
In KAM theory one tries to describe the persistence of quasi-periodic motion when an ...
- 121
4
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1
answer
916
views
Does this conic have a rational point?
Consider the conic
$$C = \{X^2+uY^2+vZ^2=0\}\subset\mathbb{P}^2_{\mathbb{Q}(u,v)}$$
over the function field $\mathbb{Q}(u,v)$.
Does $C$ have a $\mathbb{Q}(u,v)$-rational point?
- 28.2k
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views
Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$?
Related to this question,
where Bogdan Grechuk suggested this question.
Q1 Is every even number greater than $44$ not divisible by $8$ of the form $x^2+y^2+z^4+t^4$...
- 25.4k
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vote
1
answer
364
views
Good references to study Baker's theory
I am studying diophantine equations and I need the theory of Bakers, Can you advise me about good books, or lectures on Baker's theory?
- 17
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Diophantine applications of Paramodularity
I’ve asked this question to quite a few people in person and so far haven’t seen a good answer... but I believe one should exist, so here goes!
Ok, we all know how to (roughly) prove Fermat’s Last ...
- 6,367
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1
answer
139
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Solutions to the Diophantine equation $a^xy+x=c$
Fix positive integers $a,c$ with $a>2$. Is it possible that the Diophantine equation
$$a^xy+x=c$$
has infinitely many solutions (in positive integers $x$ and $y$)?
- 801
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1
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Pythagorean triples and quadratic residues modulo primes
QUESTION. Are my following conjectures true? How to prove them?
Conjecture 1. For each prime $p>100$, there are $a,b,c\in\{1,\ldots,p-1\}$ such that
$$\left(\frac ap\right)=\left(\frac bp\right)=\...
- 801
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answers
235
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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 $...
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0
answers
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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: ...
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