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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10 votes
2 answers
778 views

Is there a two-variable prime-representing polynomial (in the sense of Jones-Sato-Wada-Wiens)?

In the math.se question Proof of no prime-representing polynomial in 2 variables, Alon Amit asks if Ribenboim's claim that a prime-representing polynomial (a Diophantine polynomial in which the ...
2 votes
2 answers
714 views

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 non zero Integers. How to find All solutions ? Is there any parametrization which gives Infinitely ...
8 votes
3 answers
580 views

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 $...
11 votes
4 answers
1k views

Six consecutive positive integers with certain shape

Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ? If they exist, one of those six integers A will be the product of 2 and a square of ...
3 votes
1 answer
144 views

Triangular repdigits

I would like to know whether $55$, $66$ and $666$ are the only triangular numbers that are repdigits, i.e., numbers at least $10$ whose digits w.r.t. base 10 all agree. In more sophisticated terms, I ...
5 votes
1 answer
414 views

Nice diophantine equations with large smallest solutions

Given a polynomial $P$ with integer coefficients in finitely many variables, we denote by $v(P)$ the product of the absolute values of the non-zero coefficients and the non-zero total degrees of the ...
-1 votes
0 answers
67 views

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 ...
1 vote
1 answer
97 views

Size of sets associated to Gaussian integers

Given a non-zero Gaussian integer $z$, we define the set $\mathcal S(z)$ containing all solutions of $ab+cd=z$ satisfying $\min(\vert a\vert,\vert b\vert)>\max(\vert c\vert,\vert d\vert)$ with $a,b,...
6 votes
0 answers
244 views

$1 + 3 x^3 + x y^2 + 6 y z^2 = 0$ - 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 ...
3 votes
1 answer
832 views

Find all integer solutions to the following easy-looking Diophantine equations

In general, it is not clear What does it mean to solve an equation? in integers. In this question, let us assume that an equation $$ P(x_1,\dots,x_n)=0 $$ is solved if we have proved that its integer ...
29 votes
1 answer
1k views

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 ...
0 votes
0 answers
34 views

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}(...
16 votes
2 answers
1k 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 ...
0 votes
0 answers
110 views

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 votes
2 answers
448 views

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 votes
1 answer
556 views

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
421 views

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 $x,y,z$ ...
8 votes
4 answers
837 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 votes
1 answer
376 views

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 ...
69 votes
3 answers
7k views

Can you solve the listed smallest open Diophantine equations?

In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
30 votes
5 answers
3k views

Parametric solutions of Pell's equation

Given a positive integer $n$ which is not a perfect square, it is well-known that Pell's equation $a^2 - nb^2 = 1$ is always solvable in non-zero integers $a$ and $b$. Question: Let $n$ be a ...
15 votes
6 answers
8k views

Methods for solving Pell's equation?

It is known that the minimum solution of Pell's equation $x^2-dy^2=\pm1$ can be found from the continued fraction expansion of $\sqrt d$. Are there other methods for finding the minimum (or any other) ...
6 votes
1 answer
676 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 ...
20 votes
3 answers
2k views

what is the maximum number of rational points of a curve of genus 2 over the rationals

Conjecturally, there exists an integer $n$ such that the number of rational points of a genus $2$ curve over $\mathbf{Q}$ is at most $n$. (This follows from the Bombieri-Lang conjecture.) We are ...
5 votes
5 answers
723 views

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 ...
6 votes
3 answers
1k views

Complexity of solving systems of linear diophantine equations

It is "well known" that a matrix system $Ax=b$ where $A\in \Bbb Z^{m\times n}$, $x\in \Bbb Z^n,b\in\Bbb Z^m$ for some $m,n \in \Bbb N$, can be solved in polynomial time, using Smith/Hermite Normal ...
1 vote
0 answers
111 views

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 ...
5 votes
1 answer
362 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: $$ y^2 + z^2 = x^3+1, $$ $$ y^2 + z^2 = x^3-1, $$ $$ y^2+x^2y+z^2+1=0. $$ ...
2 votes
0 answers
197 views

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 votes
1 answer
250 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 ...
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 ...
2 votes
0 answers
112 views

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 $h_F(A)$ ...
15 votes
2 answers
2k views

sum of three cubes and parametric solutions

The first paragraph in the following link asserts that the equation $x^3+y^3+z^3=2$ has finite many parametric solutions over $\mathbb{Q}$, i.e., there are finite many polynomial triples $(x(t),y(t),z(...
1 vote
0 answers
133 views

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(...
1 vote
1 answer
311 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+...
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 ...
3 votes
2 answers
218 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 ...
4 votes
0 answers
77 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 ...
5 votes
1 answer
255 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/...
4 votes
0 answers
371 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 ...
1 vote
0 answers
202 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 ...
0 votes
1 answer
139 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 ...
5 votes
1 answer
333 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 ...
12 votes
1 answer
847 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 ...
5 votes
2 answers
572 views

Egyptian fractions similar to Erdos-Straus conjecture

It is known that the Erdos-Straus conjecture is about writing $4/n$ as three unit fractions. My question is whether it is known that if $a>4$ $$ \frac an=\frac1{x_1}+\frac1{x_2}+\cdots+\frac1{x_k} $...
0 votes
1 answer
146 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?
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 ...
1 vote
1 answer
145 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.
5 votes
6 answers
3k views

How many cubes are the sum of three positive cubes?

Are there infinitely many integer positive cubes $x^3 = a^3 + b^3 + c^3$ that are equal to the sum of three integer positive cubes? If not, how many of them are there?
2 votes
0 answers
154 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(...

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