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 ...
297 questions with no upvoted or accepted answers
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Is the set of integers of the form $a/(b+c)+b/(a+c)+c/(a+b)$ computable?
The starting point of this question is the observation that the smallest positive integers $a,b,c$ satisfying
$$\frac{a}{b+c} + \frac{b}{a+c} + \frac{c}{a+b} = 4$$
are absurdly high, namely $$(...
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Can a number be palindromic in more than 3 consecutive number bases?
$2017:$ Was initially asked on MSE - but wasn't solved or updated there since.
Update $2019$: I've returned to this problem, made some progress and updated the post here. (I've basically rewritten ...
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The lonely molecule
Suppose $n$ air molecules (infinitesimal points) are bouncing around in
a unit $d$-dimensional cube, with perfectly elastic wall collisions.
Let $k=n^{\frac{1}{d}}$.
For example, in 3D, $d=3$, with $n=...
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Integer solution to $(3x-1)y^2 + x z^2 = x^3-2$
Do there exist integers $x,y,z$ satisfying
$$
(3x-1)y^2 + x z^2 = x^3-2 \quad ?
$$
Hilbert's 10th Problem is unsolvable in general, but is still open for cubic equations: it is unknown whether there ...
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Does every integer $n>1$ have the form $a^2+b^2+3^c+5^d$ with $a,b,c,d$ nonnegative integers?
Lagrange's four-square theorem states that every nonnegative integer is the sum of four squares. I have tried to replace two of the four squares by two powers. This leads to my following question: ...
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Conjecture on the existence of non-trivial integer solutions to $A^4+B^4+C^4=2 n^2 D^4$
I have been investigating the existence of non-trivial integer solutions to the Diophantine equation:
$A^{4}+B^{4}+C^{4}=2n^{2}D^{4}$
where $n$ is a square-free positive integer.
Through extensive ...
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Effective proofs of Siegel's theorem using arithmetic geometry
This is a speculation and perhaps naive. The theorem of Siegel that
There exist only finitely many integral points on a curve of genus $\geq 1$ over a number ring $\mathcal O_{K, S}$ where $S$ is a ...
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Is $y^2z+z^2y = x^3+x^2+3x-1$ solvable in integers?
As it is clear from the title, the question is whether there exist integers $x,y,z$ such that
$$
y^2z+z^2y = x^3+x^2+3x-1.
$$
My two-years-old notes claim that I checked it up to $|x|\leq 450,000,000$ ...
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12
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Existence of odd integer solutions to a system of elementary symmetric equations
Consider the following system of equations:
$$
\sigma_{2k}(x_1,x_2,\dots,x_{2n+1}) = (-1)^k \binom{n}{k}, \quad 1 \le k \le n,
$$
where $\sigma_k(x_1,x_2,\dots,x_{2n+1})$ denotes the $k$-th elementary ...
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Is every integer a difference of two powers?
True or false? (I don't know.) Every positive integer is the difference of two powers. Examples:
$ 1 = 3^2 - 2^3 $
$ 2 = 3^3 - 5^2 $
$ 3 = 2^7-5^3 $
$ 4 = 2^3-2^2 = 5^3-11^2 $
$ 5 = 2^5 - 3^3 $
...
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Why are solutions to $\sqrt[k]{x_1^k+x_2^k+x_3^k+x_4^k}$ for $k=2,3$ curiously smooth?
Given an integer solution $s_m$ to the system,
$$x_1^2+x_2^2+\dots+x_n^2 = y^2$$
$$x_1^3+x_2^3+\dots+x_n^3 = z^3$$
and define the function,
$$F(s_m) = x_1+x_2+\dots+x_n$$
For $n\geq3$, using an ...
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Growth of $n=n(k)$ for which there's a non-trivial solution to $x_1^k+\cdots+x_n^k=y^k$?
Walter Hayman just asked me the following question. What, if anything, is known about the growth of the function $n(k)$, where $k\geq1$ is an integer, and $n=n(k)\geq2$ is the smallest integer for ...
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Is it true that $\{x^4+y^3+z^2:\ x,y,z\in\mathbb Q_{\ge0}\}=\mathbb Q_{\ge0}$?
Let $\mathbb Q_{\ge0}$ be the set of all nonnegative rational numbers. I have the following conjecture based on my computation.
4-3-2 Conjecture. Each $r\in\mathbb Q_{\ge0}$ can be written as $x^4+y^3+...
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11
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answer
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Find all integer solutions to the following easy-looking Diophantine equations
In general, it is not clear What does it mean to solve an equation? in integers. In this question, let us assume that an equation
$$
P(x_1,\dots,x_n)=0
$$
is solved if we have proved that its integer ...
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Product of four consecutive primes plus $1$ equals square
Some days ago, I noticed that $3\cdot 5\cdot 7\cdot 11 +1=34^2$.
I am almost sure that if we denote four consecutive primes by $p, q, r, s$ then the equation
$$p\cdot q\cdot r\cdot s+1=x^2 \quad (...
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