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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A system of nonlinear Diophantine equations whose positive solutions are not coprime
Consider the following system of Diophantine equations:
$$v_1k_1=k_1^3-k_2^3+k_3^3 \\
v_2k_2=k_1^3+k_2^3-k_3^3 \\
v_3k_3=-k_1^3+k_2^3+k_3^3 \tag{1}$$
where $v_1,v_2,v_3$ and $k_1,k_2,k_3$ are integer-...
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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 ...
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Does $A-A=\mathbb Q$ hold for $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$?
Let $A=\{x^4+y^4:\ x,y\in\mathbb Q\}$. Then
$$A-A:=\{a-b:\ a,b\in A\}=\{u^4+v^4-x^4-y^4:\ u,v,x,y\in\mathbb Q\}.$$
Motivated by Question 415482, here I ask the following question.
Question. Is it true ...
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Solution modulo $9$ of certain linear equation implies triviality modulo $3$
Question: Let $k \geq 2$ and $r \geq 4$ be two natural numbers. We are given eight integers $\nu_{ij} \geq 0$ for every $1 \leq i \leq k$ and $1 \leq j \leq r$ such that the following two conditions ...
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Are there integer solutions of $m^4+m^2n^2+n^4=k^2$?
The recent question about Sets of integers with same sum and same sum of reciprocals, with its wealth of solutions, raises naturally the question: Can we require all those numbers to be squares? Of ...
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System of linear diophantine equations with many small solutions?
Let $n$ be positive integer, $k$,$B$ fixed positive integers.
Let $f_i(x_1,x_2...x_n)$ be a system of $n-k$ linearly independent linear
equations over the integers.
Let $S(f_i,k,B)$ be the set of ...
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Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$
I came up with the following question on a facebook group: find the positive integer solutions of the equation $$\frac{x}{y+z}+\frac{y}{x+z}+\frac{z}{x+y}=4$$
Now clearly this is very difficult, ...
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A curious Diophantine problem
Let $a, b, c, d$ be positive integers where $\gcd(a, b)=\gcd(c, d)=\gcd(b, d)=\gcd(bd, ad+bc)=1$ and $\min(b, d)>1$. Is it possible to have
$$bd(ad+bc)^{2}\varphi(a)\varphi(c)=ac\varphi(bd)\varphi^{...
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What are the integer solutions to $2x^5+3y^5=5z^5$?
This equation has the obvious integer solution $(x,y,z)=(1,1,1)$. By Faltings's theorem, the equation has finitely many primitive integer solutions (those with $\gcd(x,y,z)=1$).
What is the complete ...
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Conjecture about some recurrent primes
I want to know if there are conjectures similar to this one, I know there is the Bell primes conjecture or Gardner conjecture (mentioned in this page https://en.wikipedia.org/wiki/Bell_number), but ...
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Density version of the Erdős-Graham conjecture
In 2003 E. S. Croot [Ann. of Math. 157(2)(2003), 545-556] proved the Erdős-Graham Conjecture which states that if $\{2,3,\ldots\}$ is partitioned into finitely many subsets then one of the subsets ...
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Methods of finding integer solutions beyond the reach of direct search
Consider a classical problem: given a polynomial Diophantine equation $P(x_1,\dots,x_n)=0$, determine whether it has an integer solution. While this problem is undecidable in general, we may still ...
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The new shortest open cubic equations
Section 8.3.2 of recent book [1] studies the following problem. Define the length of a Polynomial Diophantine equation as the sum of degrees of monomials plus sum of base 2 logarithms of the ...
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Can $9xy$ divide $1+x^2+x^3+y^2$?
Can $9xy$ divide $1+x^2+x^3+y^2$ for integers $x,y$? Equivalently, do there exist integers $x,y,z$ such that
$$
1 + x^2 + x^3 + y^2 + 9 x y z = 0 \quad ?
$$
This equation arises in the search for the ...
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What are the integer solutions to $x^4+2y^4=3z^4$?
This equation has the obvious integer solution $(x,y,z)=(\pm 1,\pm 1,\pm 1)$. By Faltings's theorem, the equation has finitely many primitive integer solutions (those with $\gcd(x,y,z)=1$).
What is ...
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On the equation $x^4+y^3+z^2+1=0$
Are there infinitely many triples of integers $(x,y,z)$ satisfying the equation
$$
x^4+y^3+z^2+1=0 \quad ?
$$
This is one of the simplest-looking equations (another one is famous $x^3+y^3+z^3=3$) for ...
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Can you solve the listed smallest open Diophantine equations?
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size is substitute 2 instead of all variables, absolute values instead of all coefficients, and ...
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Solutions of a quadratic Diophantine equation over algebraic integers
The problem is not very exactly Diophantine in the classical way. I am trying to find some algebraic integer solutions in a number ring to a quadratic equation over the same ring.
Precisely, let $\...
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Fermat degree FRM(n)
$\DeclareMathOperator\FRM{FRM}\DeclareMathOperator\frm{frm}$Assume $\ 3\le n\in\mathbb N$.
Question: What is the highest degree $\ f\in\mathbb N\ $ such that there does not exist any none-zero ...
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Solvability of two-variable quadratic equations with a parameter
(a) Prove that there exist infinitely many values of integer parameter $a$ such that equation
$$
2 x^2+a x y+y^2+1 = 0
$$
is solvable in integers $(x,y)$.
(b) The same question for a similar equation
$...
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On the smallest open Diophantine equations: beyond Hilbert's 10 problem
In 2018, Zidane asked What is the smallest unsolved Diophantine equation? The suggested way to measure size of the equation is substitute 2 instead of all variables, absolute values instead of all ...
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Integers $d$ for which the negative Pell equation is soluble for both $d$ and $2d$?
Let $\text{NPE}_d$ denote the negative Pell equation:
$$ x^2-dy^2=-1$$
Where $d$ is a given positive nonsquare integer and integer solutions are sought for x and y.
we know that (in this paper
archive)...
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Can you describe all rational solutions to these simple-looking equations?
Can you describe, in parametric form or in any other explicit way, all rational solutions to any of the following equations:
$$
y^2 + z^2 = x^3+1,
$$
$$
y^2 + z^2 = x^3-1,
$$
$$
y^2+x^2y+z^2+1=0.
$$
...
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Eisenstein triples (and triangles with rational sides and a rational-degree angle) in Pascal's triangle
This question leads to a follow-up: are there any Eisenstein triples (satisfying $a^2\pm ab+b^2=c^2$) in one row of Pascal's triangle apart from the following:
$\binom{23}{8}^2+\binom{23}{8}\binom{23}{...
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How to prove that this equation has no other integer solutions?
To find the integer solutions of an indeterminate equation and prove that there are no other solutions, where all variables are positive integers and n can be regarded as a constant, let's first ...
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How to prove this problem about ternary quadratic form?
Is this right? And how to prove it ?
For $n \equiv 1,2 \bmod 4$
$$ \Bigg|\ \mathbb Z^3\cap\Big\{(a_1,a_2,a_3)\ \Big|\
a_1^2+a_2^2+a_3^2=n \Big\}\Bigg| \\
= \frac12\Bigg|\mathbb Z^3\cap\Big\{(a_1,...
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Infinite number of decompositions into sum of four cubes
The context is the sum-of-four-cubes problem (see here).
I ask myself the following question (I asked a similar question on MSE). Do we know if any integer $k=a^3+b^3+c^3+d^3$ can be decomposed in an ...
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What is the smallest unsolved Diophantine equation?
If $P=\sum_{\bf i} a_{\bf i}x^{\bf i}\in {\mathbb Z}[x_1,\dots,x_d]$, let $|P|=\sum_{\bf i}|a_{\bf i}|x^{\bf i}$ and $h(P)=|P|(2,\dots,2)$, so that there is only a finite number of $P$ with $h(P)\leq ...
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On Mordell equation $y^2=x^3+k$ [closed]
Have the Mordell equation $y^2=x^3+k$ solved for all integer $k$ or not?
Please Could you tell me about a good review papers about such equation.
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The smallest solution to $2^{2k}-1=\text{powerful}$
Integer is powerful if all the exponents in its factorization are at least $2$.
Every powerful integer can be written in the form $a^2 b^3$.
For odd $k$, define $F(k)=2^{2k}-1=(2^k-1)(2^k+1)$.
This ...
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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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Which integers can be expressed as $P(t)^2 + Q(t)^2 + R(t)^5$?
Inspired by this article and that one, I have two questions:
(1) Is the question of whether every integer can be expressed in the form $x^2 + y^2 + z^5$ ($x$, $y$, $z$ in $\mathbb{Z}$) an open problem?...
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Interesting solutions of equation x^y = y^x [closed]
There is simple equation $x^y=y^x$. By taking logarithm we can see that it is equivalent to $\frac{\ln x}{x}=\frac{\ln y}{y}$. When we plot and inspect the function $f(x)=\frac{\ln x}{x}$, we can see ...
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Genus 0 curves on surfaces and the abc conjecture
One of the most obvious methods to prove that a given Diophantine equation $P(x_1, \dots, x_n)=0$ has infinitely many integer solutions is to find polynomials $P_1, \dots, P_n$ in one variable $u$, ...
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Diophantine equation of sixth degree [closed]
Show that the following equation admits infinitely many solutions:
$$x^3 + 2 y^6 - 2 z^6 = 1,\qquad \gcd(x,y)=\gcd(x,z)=\gcd(y,z)=1.$$
For example, $(79,5,8)$ is a solution .
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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 ...
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Numbers that can be written as a sum of three cubes in exactly one way (a^3 + b^3 + c^3)
Based on online info, it seems that most of these numbers have many solutions. Are there any that have only 1 known solution or only a few solutions?
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Papers related to a diophantine equations about Magic square of squares for $n=3$
The open problem of magic squares of squares explained here. Consider the following magic square of squares:
$$
\begin{aligned}
&a^2&b^2&&c^2\\\\
&d^2&e^2&&f^2\\\\
&...
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Solving a system of exponential Diophantine equations
I am trying to find solutions to the system $p^x+q=q^y+r=r^z+p$ where $p,q,r$ are primes and $x,y,z$ are integers greater than $2$.
So far, I have only found that at most one among $x$, $y$, and $z$ ...
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Integer solutions for a simple cubic
What are the integer solutions to $x^3 - 7xy + y + 1 = 0$? A computation only finds $(0, -1), (-1, 0), (-6, 5), (-49, 342)$. This is surprisingly few.
Are these all of them? Is there an algebraic ...
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On the Diophantine equation $2^a3^b + 4^c5^d = 2^b3^a + 4^d5^c$ [closed]
In this year's team selection test to the international mathematical Olympiad in Japan, a interesting diophantine equation appeared as the final problem of the second day.
Find all quadruples $(a,b,c,...
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Fermat last theorem : proof of a criterion by Cauchy
In 13 Lectures on Fermat's Last Theorem, Ribenboim states the following theorem (on page 7) attributed to Cauchy:
If the first case of Fermat's theorem fails for the exponent $p$, then the sum:
$$ 1^{...
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Large integral points on the quadratic twist $ D y^2=x^3+A x +B$
For integers $A,B,D$ and $D$ squarefree let $E : y^2=x^3+A x + B$
and $E_D$ be the quadratic twist of the elliptic curve $E$:
$$ E_D : D y^2=x^3+Ax +B$$
$E_D$ is isomorphic to $ E'_D : y^2=x^3+D^2 A ...
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Infinitely many coprime solutions of $F(x,y)= k(a_1 x + a_2 y)^2 z^2$?
This might be related to an open problem.
Let $F(x,y)$ be homogeneous degree 4 squarefree polynomial
with integer coefficients and
$h(x,y)=a_1 x + a_2 y$ and $\gcd(F,h)=1$ and $k$ be integer.
Consider ...
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The existence of solutions of a Diophantine exponential equation
Given a prime number $p$ and a positive integer $n$, I am interested in the (non)existence of positive integer solutions $x,x_0,\dots,x_{p^n}$ of the following Diophantine equation
$$p^x+p^n=\sum_{i=0}...
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Proving that there are no integral points on a union of hyperbolas
I have a curve C: (x^2±x-y^2+1)(x^2∓x-y^2) where x,y ∈ Z+ that I want to prove has no non-trivial integral points other than (0,0),(1,0),(0,±1). I am having a hard time coming up with a solution.
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Can you "slice" a triangular number into three equal slices?
Problem statement:
Does there exist positive integers $a<b<c$ such that
$$1 + 2 + \dots + (a-1) = (a+1) + \dots + (b-1) = (b+1) + \dots + c?$$
(Note that $a$ and $b$ are not in the sums.)
...
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On the integer solutions of the equation $y^2 = x^3 + n$
Let $n$ be a nonzero integer. I am interested in the integer solutions $(x, y)$ to the equation $y^2 = x^3 + n$.
Let $S$ be the set of all integer solutions $(x, y)$ to this equation.
I am wondering ...
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Is 36 a sum of 4 rational fourth powers?
Hasse principle is known to hold for homogeneous quadratic equations, but fail for some 3- and 4-variable cubics, such as $5x^3+4y^3+3z^3=0$ or $15x^3+10y^3+4z^3+3t^3=0$. These counterexamples are ...
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About the units in $\mathbb{Z}[\frac{1+\sqrt{d}}{2}]$
Let's consider $K=\mathbb{Q}[\sqrt{d}]$ where $d$ is positive and square free. It is well known that the ring of integers is
$$
{O}_{K}=\mathbb{Z}\left[\frac{1+\sqrt{d}}{2}\right]
$$ if $d=1 \mod 4$ ...
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