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 ...
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Solving system of linear diophantine equations with exponential coefficients over the integers
In general, solving a system of linear diophantine equations over the integers is polynomial time solvable on the size of the coefficients of the equations.
I am interested in an extension of this ...
-1
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On the full list of near-repdigit perfect powers
I'm interested in the full list of perfect powers ($a^b$ where $a, b \in \mathbb{Z}$, $a \ge 1$, $b \ge 2$) that are near-repdigit in base 10. A near-repdigit is a $k$-digit number where $k \ge 2$ and ...
2
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2
answers
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Sum of three square is a square and sum of their product taken two at a time is also a square
Let $a^2 + b^2 + c^2 = X^2$ and
$$(ab)^2 + (ac)^2 + (bc)^2 = Y^2$$
Such that $a,b,c,x,y$ are all Integers
How to find All non trivial solutions ?
Is there any parametrization which gives many ...
6
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225
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$1 + 3 x^3 + x y^2 + 6 y z^2 = 0$ - the new shortest open cubic equation
Are there integers $x,y,z$ such that
$$
1 + 3 x^3 + x y^2 + 6 y z^2 = 0 \,\, ? \quad\quad (1)
$$
If the length of an equation is the sum of degrees of monomials plus sum of logarithms of the ...
8
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2
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Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
Do there exist positive integers $m$, $n$, $p$, $q$ such that $m>1$, $p\neq q$, $p$ and $q$ divide $mn^2 - 1$, and $mn$ divides $p - q$?
It seems numerically up to $n \leq 10^6$ that for $m=3$ or $...
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Finding integral points of quadric without degree 1 terms
I consider for some $n\in\mathbb{N}$ the index set $I\subset\binom{n}{2}$ the following polynomial $p_I\in\mathcal{R}:=\mathbb{R}[x_1,...,x_n]$ with
$$p_I(x_1,...,x_n)=\sum_{\lbrace i,j\rbrace \in I}(...
29
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Can $9xy$ divide $1+x^2+x^3+y^2$?
Can $9xy$ divide $1+x^2+x^3+y^2$ for integers $x,y$? Equivalently, do there exist integers $x,y,z$ such that
$$
1 + x^2 + x^3 + y^2 + 9 x y z = 0 \quad ?
$$
This equation arises in the search for the ...
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On the (hyper?)elliptic curve $y^2=x^2-x^3z^2+z-1$
The question here is if there exists $x,y,z\in\mathbb Z$ such that$$y^2=x^2-x^3z^2+z-1\label{1}\tag{1}$$other than the trivial solution$$x=0,y^2+1=z\text{ for all }y\in\mathbb Z\label2\tag2$$I know ...
6
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2
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Difficult elliptic curves for $a^4+b^4+c^4+d^4 = (a+b+c+d)^4$?
Similar to the case $x^4+y^4+z^4 = 1$ discussed in this MO post, define the system,
$$x^4+y^4+z^4+1 = (x+y+z+1)^4\tag1$$
$$\frac{x^2+x+1}{(x+y+1)(x+z+1)}=u\tag2$$
$$\frac{y^2+y+1}{(y+z+1)(y+x+1)}=v\...
12
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On the equation $9x^3+y^3=z^2+3$
The question is whether there exist integers $x,y,z$ such that
$$
9x^3+y^3=z^2+3.
$$
This is one of the nicest (if not the nicest one!) cubic equations for which I do not know whether integer ...
4
votes
4
answers
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A cubic equation, and integers of the form $a^2+192b^2$
This question resembles my previous question A cubic equation, and integers of the form $a^2+32b^2$ , but seems to be more difficult.
We are trying to determine whether there are any integers $x,y,z$ ...
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A parametric elliptic curve for $x^4+y^4+z^4 = 1$?
Noam Elkies found that $x^4+y^4+z^4 = 1$ has infinitely many rational points $xyz \neq 0$ using an elliptic curve. We use a different approach that will produce pairs of solutions and a parametric ...
5
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5
answers
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The diophantine equation $ \sum_{n=1}^{N} \frac{1}{x_{n}} = \prod_{k=1}^{N} \left(1-\frac{1}{x_{k}} \right) $
Background
I wonder if there are any rational numbers such that their Egyptian fraction (sum) representations are equal to their Egyptian product analogue. In other words, I am curious1 about ...
1
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0
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110
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Situations where the number of solutions to a linear Diophantine equation is always even
I have a number theory situation that I hope someone will recognize as a known situation and can direct me to some relevant papers in the literature.
This came out of some numerical experiments run by ...
2
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0
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Question on digital sum of the square of $n$
If we set $f(n)=$ the digital sum of $n$,for example, $f(2024)= 2+0+2+4= 8$.
Are there any $n>375501$ in solutions to the equation $f(n^2)=9,$ except $n=10k$, $n=10^a+10^b+1$, $n=5 \cdot 10^a+1$ or ...
2
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answers
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The connection of Faltings height and Tate module
Suppose $K$ is a number field, $S$ a finite set of places of $K$, and $A$ is an abelian scheme over $\mathcal O_{K,S}$. I want to ask is there some connections between the Faltings' height $h_F(A)$ ...
1
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On the equation $q(\mathbf{x}) = 1$ for $q$ a quadratic form
Let $q(\mathbf{x}) = q(x_1, \cdots, x_n)$ be a quadratic form with integer coefficients. For $n \geq 3$, is there a reasonable theory for the set of integer solutions to the equation
$$\displaystyle q(...
6
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On the shortest open cubic equation
The question is: are there any integers $x,y,z$ such that
$$
1+4 x^3+x y^2+2 y z^2 = 0 \quad\quad\quad\quad (1)
$$
The motivation is: Define the length of a polynomial $P$ consisting of $k$ monomials ...
1
vote
1
answer
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A Mordell equation $y^3=x^2+20$ [closed]
Recently I met a problem when I'm studying algebraic number theory.
Problem. Find all positive integer solutions of $y^3=x^2+20$.
I solved the situation when $x$ is an odd because the two ideals $(x+...
8
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4
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A cubic equation, and integers of the form $a^2+32b^2$
I am trying to determine whether there are any integers $x,y,z$ such that
$$
1+2 x+x^2 y+4 y^2+2 z^2 = 0. \quad\quad\quad (1)
$$
It is clear that $x$ is odd. We can consider this equation as quadratic ...
3
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2
answers
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Integer solutions to $x^2 + x + 1 = y^z$? [duplicate]
In the context of finite projective planes I am interested in the Diophantine equation $\frac{x^3-1}{x-1} = y^z$, which is also written as $x^2 + x + 1 = y^z$, for $z>1$. I stumbled by accident on ...
4
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Repeated values of a monomial
Let $H,M\geq 1$ and let $h_0$ and $m_0$ be fixed integers with $(h_0,m_0)\in [H,2H]\times[M,2M]$. Let $\alpha$ be a positive real number. I'm trying to find an upper found for the number of integer ...
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Radicands of square roots of the 2020s, written in simplest radical form
As of the time of writing, the current decade is the 2020s. An interesting property of this decade is that there are 3 years that satisfy the property that the square-free part (https://oeis.org/...
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Are there integers $x,y,z$ such that $(x+1)y^2-xz^2=x^3+2x+2$?
Is equation
$$
(x+1)y^2-xz^2=x^3+2x+2
$$
solvable in integers?
Motivation: For a polynomial $P$ consisting of $k$ monomials of degrees $d_1,\dots,d_k$ and integer coefficients $a_1,\dots,a_k$, define ...
1
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0
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Are there integers $x,y,z$ such that $1 + x - x^3 + x^2 y^2 + z + z^2 = 0$?
In my previous question Can you solve the listed smallest open Diophantine equations? I discuss the smallest equations (in some well-defined sense) for which it is not known whether they have any ...
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1
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Diophantine equations involving recurrence sequences
I am working on a Diophantine equation by using transcendental and reduction methods given by Baker and Davenport. However, when I read some papers i don't understand the reduction step, for example ...
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Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$
While working on finite order elements of $\operatorname{SO}_n$, I meet this question:
Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers.
As ...
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Are (55, 165, 495, 1485) and (286, 1716, 10296, 61776) the only geometric sequences of length 4 among non-trivial binomials?
Let's define non-trivial binomial coefficients as values of $\binom{n}{k}$, where $n$ and $k$ are positive integers such that $2 \le k \le \frac{n}{2}$. (Therefore, $6$ is the smallest non-trivial ...
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1
answer
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General solution of the quartic $a^4+b^4=c^4+d^4$?
The background to the question:
$$a^4+b^4=c^4+d^4 \label{1}\tag 1 $$
Tito Piezas, Tomita & others have recently given some parametric solutions on Math stack exchange & Math overflow. In math ...
1
vote
1
answer
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$(2^a-1)+b^2=2^c$ [closed]
$31+15^2=256$.
Are there infinitely many solutions to:
$(2^a-1)+b^2=2^c$ with a,b,c positive integer and a,b,c different each other.
13
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2
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On Euler's elliptic curve for $A^4+B^4 = C^4+D^4$?
To solve,
$$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$$
and define $p = (a^3 - b),\, q = a y,\, r = b (a^3 - b),\, s = y.\,$ The equation above transforms to ...
2
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0
answers
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Will Coppersmith's method work for this bivariate modular polynomial shape?
I have a bivariate modular polynomial of shape
$$f(x,y)=x^2y-g(x)\equiv 0\bmod q$$
where
$q=(2p-1)(2p+1)$ is a product of two primes $2p-1$ and $2p+1$,
$g(x)\in\mathbb Z[x]$ is of degree four and
$f(...
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For which primes $p$ in $\mathbb Z$ is $p\omega$ the sum of two cubes in $\mathbb Q(\omega)$?
This is related to an earlier question I posed —"Possible extensions of a conjecture …". Now that my note arXiv:2309.00162 has appeared I'll use it as a reference.
Elementary results(along ...
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1
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Almost Pell type equation
Consider the following Diophantine equation
$$
2x^2-Ny^2 = -1.
$$
where $N$ is an integer. Is there any result expressing the values of $N$ for which the above equation admits an integral solution?
0
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0
answers
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Elementary method for finding integer solutions for certain types of elliptic curve
There are some problems in high school Olympiad that ask to find integer solutions of the form $Q(x^2) = dy^2 (*)$ where $Q$ is a quadratic polynomial and $d$ is an absolute constant and quite often, $...
2
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0
answers
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Is the continued fraction of a constructible number special in some way?
Rationals have finite CF and quadratic have periodic CF. CF in turn can be represented in terms of the modular group SL2(Z), e.g. using the standard generators S(z)=-1/z and T(z)=z+1. On the other ...
4
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1
answer
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Representation of a number as a product of $\sqrt{n^2 + 1} + n$
Question. Do there exist two multisets $A, B$ consisting of positive integer numbers such that $|A|$ and $|B|$ have different parity and
$$
\prod_{n\in A}(n + \sqrt{n^2 + 1}) = \prod_{m\in B}(m + \...
8
votes
1
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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 ...
16
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2
answers
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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 ...
5
votes
0
answers
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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. ...
5
votes
1
answer
363
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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 ...
4
votes
2
answers
291
views
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? ...
9
votes
1
answer
435
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 ...
3
votes
1
answer
102
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?
5
votes
1
answer
355
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:
$$
y^2 + z^2 = x^3+1,
$$
$$
y^2 + z^2 = x^3-1,
$$
$$
y^2+x^2y+z^2+1=0.
$$
...
0
votes
0
answers
81
views
Generalized Jacobi-Madden equation
I already posted this question here, but received no answers other the some useful cases.
The Jacobi-Madden equation
$ a^4+b^4+c^4+d^4 = (a+b+c+d)^4 $
has an infinitude of integer solutions with all ...
14
votes
1
answer
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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 ...
1
vote
0
answers
117
views
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 ...
1
vote
0
answers
83
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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 ...
5
votes
1
answer
253
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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 ...