# 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 ...

**10**

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### Universality of $y^4-x^3$ mod $p$

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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)

**5**

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### Counting primitive solutions to a diophantine inequality

**2**

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### Is there a general method for solving Diophantine equations of this type?

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### Find the positive integers $x^3+y^3=3z^3$ [closed]

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### trivial solutions for Diophantine equations

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### What is the smallest unsolved diophantine equation?

**2**

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### Diophantine equation for generating computably enumerable set

**21**

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### Can we write each positive rational number as $\frac1{p_1-1}+\ldots+\frac1{p_k-1}$ with $p_1,\ldots,p_k$ distinct primes?

**0**

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### On segments of the series $\sum_p\frac1{p-1}$

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### Is it true that $\sum_{k=m}^n\frac{\sigma(k)}k\not\in\mathbb Z$ for all derangements $\sigma\in S_n$ and $1\le m\le n$?

**4**

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### Is there a permutation $\pi\in S_n$ with $\sum\limits_{0<k<n}\frac1{\pi(k)^2-\pi(k+1)^2}=0$ for each $n>7$?

**1**

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### Derangements and unit fractions

**6**

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### Can the partition function $p(n)$ take perfect power values?

**19**

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### Permutations $\pi\in S_n$ with $\sum_{k=1}^n\frac1{k+\pi(k)}=1$

**2**

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### Showing a rational polynomial is non injective

**17**

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### Does the equation $(xy+1)(xy+x+2)=n^2$ have a positive integer solution?

**2**

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### A question regarding Goormaghtigh conjecture

**3**

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### On the existence of integer square root of a $3 \times 3$ positive definite matrix

**5**

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### Linear diophantine quasivariety having a unique solution

**9**

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### Does $2^x-3p^y=5$ (with $p$ an odd prime) have only finitely many positive integer solutions?

**5**

**1**answer

### Number of integer solutions of a linear equation under constraints

**7**

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### Rational perfect power values of $y(y+1)$

**1**

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### Hyperelliptic curves imply FLT-like results

**5**

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### No rational points on $x^n+a=y^2$ for all $n>4$"?

**1**

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### On $x^4+16z^n=y^2$ and $x^4+z^n=y^2$

**3**

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### A specific Diophantine equation restricted to prime values of variables.

**3**

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### Solving elliptic equation in rational functions

**5**

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### What is the time complexity for solving Diophantine equations of degree 2?

**2**

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### A generalization of Bernoulli's inequality and what does it application for?

**1**

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### On the diophantine equations $x^n+n=y^m$ and $x^n-n=y^m$

**7**

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### $(2x^2+1)(2y^2+1)=4z^2+1$ has no positive integer solutions?

**0**

**1**answer

### Solutions to linear equations from recurrence sequences with no repeated roots

**5**

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### Isomorphism classes of lattices

**6**

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### Solutions to the Diophantine equation $x^2+3y^2+3z^2=n$

**5**

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### On the Diophantine equation $x^{4}+y^{4}=z^p$

**5**

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### Does Fermat's last theorem hold in the Grothendieck ring of the ordinals?

**9**

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### Enquiry on a Diophantine problem

**3**

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### Density version of the Erdos-Graham conjecture

**4**

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### Number of nontrivial integral solutions to $f(x)=f(y)$

**0**

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### Solutions to exponential diophantine: 2^a + 3^b = 2^c + 3^d [duplicate]

**2**

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### Can we efficiently factor $n$ given that $n=pq$ where $p,q$ are primes satisfying $p=a^2+b^2, q=2ab+1$ for some $a,b$

**1**

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### Dimension of $S$-units over $\mathbb{C}[x]$

**2**

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### Does each integer have the form $x^4-y^3+z^2$ with $x,y,z$ positive integers?

**3**

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### Solution to an exponential Diophantine equation

**5**

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### Is every integer $n>1$ the sum of two triangular numbers and two powers of $5$?

**15**

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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?

**0**

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### Mathematical Aspects of Hectoc-type Puzzles

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### Solutions to diophantine equation

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