# 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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### Could someone check this direct proof of Fermat'sLast Theorem? [closed]

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### Solutions to the Diophantine equation $a^xy+x=c$

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### The number of perfect squares which can occur in an arithmetic progression of length n

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### Why is this “the first elliptic curve in nature”?

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### Chinese remaindering to solve solvable diophantine equations

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### Recursively obtained hard Diophantine equation for “Baseless numbers”

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### Methods of sheaf theory for solving Diophantine equations

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### Are twin primes the only solution to this equation?

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### On weaker forms of the abc conjecture from the theory of Hölder and logarithmic means

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### Natural number solutions for equations of the form $\frac{a^2}{a^2-1} \cdot \frac{b^2}{b^2-1} = \frac{c^2}{c^2-1}$

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### On a structural decomposition of polynomials based on integral roots

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### When is $\phi(a^n+b^n+c^n)=0\mod n$?

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### $3$-ranks of elliptic curves and representations $p=ax^3+by^3$

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### Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples - $\mathsf{II}$

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### Small linear relations in unbalanced diophantine equations from primitive Pythagorean triples

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### Small linear relations between primitive Pythagorean triples $\mathsf{II}$

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### Cohn's eight diophantine equations

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### FLT and integral points on elliptic curves

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### Why $n$ or $n+1$ has the form $x^4+T_y+T_z$?

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### Number of integer solutions to quadratic polynomial with integer coefficients

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### Write $n^2$ as $x^2+y^2+2\times4^z$ or $x^2+y^2+5\times 4^z$

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### $n$-variable polynomials modulo $p$

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### Write $p^2$ as $x^2+2y^2+3\times 2^z$ with $x,y,z$ nonnegative integers

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### Diophantine equation $10^n-a^3-b^3=c^2$

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### Simplest diophantine equation with open solvability

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### Solutions in primes of the equation $\,3p^2+q^2=r^2+3$

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### Does this equation have more than one integer solution?

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### On Kellner's result and the Erdos-Moser equation

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### Which Hilbert's 10th polynomials are known to have solutions?

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### Request for an exact formula related to a partition in number theory

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### Around the diophantine equation $\frac{a}{2b+3c}+\frac{b}{2c+3a}+\frac{c}{2a+3b}=\text{odd integer}$, over positive integers

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### find all of rational solutions of the quartic equation?

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### On variants of the abc conjecture in terms of Lehmer means

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### Are there any references in the literature relating to work on finding a Diophantine equation representing abc

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### Sum from combinatorics on nonnegative integer numbers

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### Diophantine representation of the set of prime numbers of the form $n²+1$

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### Integer points of one Mordell equation

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### Research work on $ax^n-by^m=1$

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### Normalising Beal's conjecture

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### Prove $\frac{\text{Area}_1}{c_1^2}+\frac{\text{Area}_2}{c_2^2}\neq \frac{\text{Area}_3}{c_3^2}$ for all primitive Pythagorean triples

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### A question about integer triples

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### On sparse $0/1$ linear equations solvable with compressed sensing

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### Diophantine equations that involve cubes and the volume of square frustums

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### Solutions of a Diophantine equation with large common divisors

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### Different solution of power Diophantine equation based on constant term

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### Diophantine equation in Laurent polynomials

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### Quadratic factors of $l_1(x,y)^3+l_2(x,y)^3+l_3(x,y)^3-n$

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### Rational Diophantine set for the non-squares

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### On a variant of Brocard's problem using the definition of Pochhammer symbols

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