Questions tagged [diophantine-equations]

Question 1

Let given $n \in \mathbb{Z}^+$ and equation $x^3 + y^3 + z^3 = n$ over $\mathbb{Q}$. Let $x = -\dfrac{4 a^3-4 nb^3+1}{3 b},y = \dfrac{a^3-nb^3+1}{3 b},z = \dfrac{a}{b}$, where $a,b\in \mathbb{Q}$, ...

Question 2

first time poster so I'm sorry if any of the formatting is slightly off. I am trying to use this equation to find the inverse of a 3x3 matrix. $$\mathbf A^{-1} = \frac{1}{det(\mathbf A)} \sum_{s=0}^{n-...

Question 3

Euler conjectured (and Gauss later proved) that: If $p\equiv 1\pmod 3$, $2$ is a cubic residue mod $p$ iff $p=x^2+27y^2$ for some integer $x$ and $y$. If $p\equiv 1\pmod 4$, $2$ is a quartic residue ...

Question 4

For prime $p>6$, I am trying to show that $p^2-9=y^3$ has no solution with elementary methods. Factoring $p^2-9$ and quadratic residues doesn't seem to work. It would be nice if I could factor $y^3+...

Question 5

A previous question asked for rational solutions of $x+y+\frac 1x+\frac 1y=2025$ without any positivity assumption. It has been closed since the absence of positive solutions is already known, but the ...

Question 6

I am interested in the proof or disproof of some conjectures about rational points and sections over $\mathbb{Q}$ for the following family of genus-3 hyperelliptic curves: $$ C_t: f(x,a)\, g(x,a)\, h(...

Question 7

Consider Oeis $A065387$: $a(n)=\sigma(n)+\phi(n)$, where $\sigma(n)$ represents the sum of all divisors of $n$ and $\phi(n)$ is Euler's totient function. I ask to prove that the assertion the ...

Question 8

Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{2004}$. The answer is $45$, but I don't know how it is $45$. How to do ...

Question 9

I am interested in finding all rational points on the hyperelliptic curve $$ C: f(x)\, g(x) = y^2, $$ where $$ f(x) = \left(625x^4 + 3100x^3 - 11344x^2 + 6200x + 2500\right), \quad g(x) = \left(961x^4 ...

Question 10

I am trying to prove that $(0,0,0)$ is the only integer solution of $$a^p+ p b^p+ (10p+1) c^p=0$$ I found this question on an Italian forum for the particular case in which $p=11$. The question was ...

Question 11

Let given $p,q \in \mathbb{Z}^+,r \in \mathbb{Z}$ and equation $p x^2 + q y^2 = z^3 + r$. If exist solutions of Pell equation $qb^2-3pa^2=r\pm1$, then $x = a(pa^2 - 3qb^2 + 3r)\\ y = b\\z = pa^2 + qb^...

Question 12

Given $f(x)=ax^2+bx+c$, where $a,b,c\in\mathbb{Z}$, $a>0$ is square-free. If the coefficients involve parameters, for example \begin{equation*} f(x)=5p^2x^2+2qx+1, \end{equation*} where $p,q$ are ...

Question 13

While playing with the identity $$ p(p+2)+1=(p+1)^2, $$ I noticed that for any integer $n$ and any integer $p$ (twin prime not actually needed here), $$ (n^2+p)(n^2+p+2)+1=(n^2+p+1)^2. $$ So the pair $...

Question 14

I've been exploring some symmetric Diophantine forms that blend additive and multiplicative structures, and one equation I found keeps bugging me: $$ x^3 + y^3 = (x + y)^m - (xy)^n, $$ with integer $x,...

Question 15

I am interested in finding the answer to the following problem: Given a monic polynomial with integer coefficients $P_n(x)=x^n+a_nx^{n-1}+\dots+a_1x+a_0$ of degree $n\ge2$, is it possible for the ...

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Questions tagged [diophantine-equations]

From sum of three cubes to Desboves/Selmer elliptic curve

Finding Inverse of a 3x3 Matrix with Cayley-Hamilton, and Diophantine Equation

Quintic residues and $p=x^2+125y^2$

Elementary way of proving that $p^2-9=y^3$ has no solution

Rational solutions for $x+y+\frac 1x+\frac 1y=2025$ without positivity assumption

Rational points and sections on a family of genus-3 hyperelliptic curves

An equivalent formulation for the twin prime conjecture

Find the number of ordered pairs of positive integers $(x, y)$ that satisfy the equation $\frac{1}{x} + \frac{1}{y} = \frac{1}{2004}$. [duplicate]

Rational points on a hyperelliptic curve defined by a product of two quartics

There are no integer solution to $a^p+ p b^p+ (10p+1) c^p=0$ for any $p$ prime

Parametrization $p x^2 + q y^2 = z^3 + r$

Criterion that $ax^2+bx+c=n^2$ has integer solutions

Extending the twin‑prime polynomial Diophantine pair $(n^2+p,n^2+p+2)$ to a triple

On the Diophantine equation $x^{3}+y^{3}=(x+y)^{m}-(xy)^{n}$ — finiteness and partial results

On the number of integer solutions of the diophantine equation $x^n+a_nx^{n-1}+\dots+a_1x+a_0=y^n$ for $n\gt1$

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