# 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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### Will an integer combination of some number of copies of the set of powers of 2 and the set of powers of 3 always have natural density 0?

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### Looking for a better way to solve quadratic integer equation with 2 variables [closed]

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### On the equations $(x^n+1)(y^n+1)=z^2+1$ and $(x^n-1)(y^n-1)=z^2+1$

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### Pythagorean triples and quadratic residues modulo primes

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### Is $xz+1 $ a proper divisor of $a_3z^3+a_2z^2+a_1z+1$ finitely often?

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### Are the nonnegative rationals diophantine with only two quantifiers?

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### Positive integers $m$ such that $2m^2-1=x^4+y^4$ for some $x,y\in\{0,1,\ldots\}$ with $|x-y|>1$

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### Diophantine consequences of the Buzzard–Diamond–Jarvis conjecture

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### Positive integers written as $x^4+T(y)^2+T(z)+2^w$ with $x,y,z,w\in\mathbb N$ and $T(n)=n(n+1)/2$

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### Existence of $\{0,1\}$-solution to a system of linear equations with coefficients in $\{0,1\}$

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### Relation between stacky curves and “M-curves”

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### Is decidability reducible to unique decidability (perhaps in multilinear polynomial situations)?

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### Why wolfram alpha gives integers solutions for some equations of the form $ x^3 +(k\times10^n)^3 + z^3=0 $?

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### Is there an integer-valued quadratic polynomial $P(x,y)$ such that $\{P(x,y)+2^k:\ x,y\in\mathbb Z\ \text{and}\ k\in\mathbb N\}=\mathbb Z^+$?

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### Natural numbers in the form $\lfloor\frac{a^3+b^3}2+\frac{c^3+d^3}6\rfloor$

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### On the equation $y^2 = x^3 - z^3$ [closed]

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### Is $\prod_{k=1}^nk^{\sigma(k)}$ a square or a cube for some $\sigma\in S_n$?

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### Is there a permutation $\tau\in S_n$ with $\tau(1)^{\tau(2)}+\cdots+\tau(n-1)^{\tau(n)}+\tau(n)^{\tau(1)}$ a square?

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### Variation in decidability of diophantine equations with field extension

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### On the equations $x^yy^z=z^x$ and $w^x+x^y+y^z=z^w$

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### Diophantine equation $x^3+y^9=z^6-1$

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### Status of $x^3+y^3+z^3=6xyz$

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### Quadratic Diophantine equations with all values prime

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### Finding number fields over which Diophantine equations are solvable

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### Is there a simple proof that $Ax^3+By^3=C$ has only finitely many integer solutions

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### The “stubborn” solutions to sums of three cubes

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### $n$ variables Diophantine

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### Mordell like equation [closed]

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### How to solve special Diophantine equation systems (which one can solve by hand) with the computer?

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### Finding Pythagorean quadruples on a given plane?

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### Action of the symmetric group $S_3$ on an elliptic curve $E$ defined over $\mathbb{Z}$

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### Does the equation in positive integers $(n,y)\,\prod_{k=1}^n(p_k^2-1)=y^2\,$ only have the solution $(3,24)$?

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### Relaxation of Waring's problem $n=\sum_{i=1}^{k+1} a_i x_i^k$

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### Does the diophantine equation $\,\prod_{k=1}^n(p_k^{x_k}-1)=y^2\,$ have always at least a solution for $\,n\gt2\,$?

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### Diophantine equation $3(a^4+a^2b^2+b^4)+(c^4+c^2d^2+d^4)=3(a^2+b^2)(c^2+d^2)$

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### Restricted divisor summatory function

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### When there are infinite many primitive solutions of Vinogradov system?

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### A conjectural limit involving primorial and factorial

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### How can the number of rational points depend on the choice of height function?

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### Cubic surface in $\mathbb{P}^3$ singular along a line

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### Software for $S$-unit equation

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### Are the numbers $\varphi(n^2)\sigma(n^2)\ (n=1,2,3,\ldots)$ pairwise distinct?

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### How to solve this equation $a^2+3b^2c^2=7^c$

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### Is it true that $\{x^4+y^2+z^2:\ x,y,z\in\mathbb Z[i]\}=\{a+2bi:\ a,b\in\mathbb Z\}$?

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### Genus $0$ algebraic curves integral points decidable?

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### A certain Pell Equation

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### Counting solutions of a equation involving prime powers

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### Efficiently find at least one lattice point on hyperbola of equation $axy+bx+cy+d=0$

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### Maximum number of integer solutions with some size constraints to bivariate polynomials?

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