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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On the equation $7x^3 + 2y^3 = 3z^2 + 1$

The question is whether there exist integers $x,y,z$ such that $$ 7x^3+2y^3=3z^2+1. $$ After a similar equation On the equation $9x^3+y^3=z^2+3$ has been solved, this is one of the nicest cubic ...
Bogdan Grechuk's user avatar
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 ...
Tong Lingling's user avatar
3 votes
1 answer
153 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 ...
Jens Reinhold's user avatar
6 votes
1 answer
437 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
70 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 ...
Bubbler's user avatar
  • 99
2 votes
2 answers
740 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 ...
Guruprasad's user avatar
7 votes
0 answers
254 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 ...
Bogdan Grechuk's user avatar
8 votes
3 answers
633 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 $...
Noname's user avatar
  • 191
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}(...
Jens Fischer's user avatar
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 ...
Bogdan Grechuk's user avatar
0 votes
0 answers
111 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 ...
CrSb0001's user avatar
  • 143
6 votes
2 answers
452 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\...
Tito Piezas III's user avatar
12 votes
1 answer
609 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 ...
Bogdan Grechuk's user avatar
4 votes
4 answers
427 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 $x,y,z$ ...
Bogdan Grechuk's user avatar
7 votes
1 answer
387 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 ...
Tito Piezas III's user avatar
5 votes
5 answers
728 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 ...
Max Muller's user avatar
  • 4,565
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 ...
James McLaughlin's user avatar
2 votes
0 answers
199 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 ...
Tong Lingling's user avatar
2 votes
0 answers
114 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 $h_F(A)$ ...
Richard's user avatar
  • 523
1 vote
0 answers
134 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(...
Stanley Yao Xiao's user avatar
6 votes
1 answer
690 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 ...
Bogdan Grechuk's user avatar
1 vote
1 answer
318 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+...
jdhejw's user avatar
  • 317
8 votes
4 answers
841 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 ...
Bogdan Grechuk's user avatar
3 votes
2 answers
220 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 ...
Maarten Havinga's user avatar
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 ...
Joshua Stucky's user avatar
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/...
William Hu's user avatar
4 votes
0 answers
373 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 ...
Bogdan Grechuk's user avatar
1 vote
0 answers
203 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 ...
Bogdan Grechuk's user avatar
0 votes
1 answer
141 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 ...
Omega's user avatar
  • 31
5 votes
1 answer
334 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 ...
WhatsUp's user avatar
  • 3,232
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 ...
William Hu's user avatar
12 votes
1 answer
856 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 ...
David's user avatar
  • 99
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.
stuttgart's user avatar
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 ...
Tito Piezas III's user avatar
2 votes
0 answers
155 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(...
Turbo's user avatar
  • 13.7k
3 votes
0 answers
162 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 ...
paul Monsky's user avatar
  • 5,412
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?
Puzzled's user avatar
  • 8,852
0 votes
0 answers
160 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, $...
jackdean's user avatar
  • 151
2 votes
0 answers
72 views

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 ...
Lucian Ionescu's user avatar
4 votes
1 answer
196 views

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 + \...
Pavel Gubkin's user avatar
8 votes
1 answer
606 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 ...
Bogdan Grechuk's user avatar
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 ...
Bogdan Grechuk's user avatar
5 votes
0 answers
442 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. ...
Bogdan Grechuk's user avatar
4 votes
1 answer
379 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 ...
Bogdan Grechuk's user avatar
4 votes
2 answers
301 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? ...
Bogdan Grechuk's user avatar
9 votes
1 answer
442 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 ...
Bogdan Grechuk's user avatar
3 votes
1 answer
104 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?
Fedor Petrov's user avatar
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: $$ y^2 + z^2 = x^3+1, $$ $$ y^2 + z^2 = x^3-1, $$ $$ y^2+x^2y+z^2+1=0. $$ ...
Bogdan Grechuk's user avatar
0 votes
0 answers
82 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 ...
user967210's user avatar
14 votes
1 answer
589 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 ...
Bogdan Grechuk's user avatar

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