# 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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### On the equation $7x^3 + 2y^3 = 3z^2 + 1$

The question is whether there exist integers $x,y,z$ such that $$7x^3+2y^3=3z^2+1.$$ After a similar equation On the equation $9x^3+y^3=z^2+3$ has been solved, this is one of the nicest cubic ...
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### Six consecutive positive integers with certain shape

Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ? If they exist, one of those six integers A will be the product of 2 and a square of ...
153 views

### Triangular repdigits

I would like to know whether $55$, $66$ and $666$ are the only triangular numbers that are repdigits, i.e., numbers at least $10$ whose digits w.r.t. base 10 all agree. In more sophisticated terms, I ...
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### Nice diophantine equations with large smallest solutions

Given a polynomial $P$ with integer coefficients in finitely many variables, we denote by $v(P)$ the product of the absolute values of the non-zero coefficients and the non-zero total degrees of the ...
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### On the full list of near-repdigit perfect powers

I'm interested in the full list of perfect powers ($a^b$ where $a, b \in \mathbb{Z}$, $a \ge 1$, $b \ge 2$) that are near-repdigit in base 10. A near-repdigit is a $k$-digit number where $k \ge 2$ and ...
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### Sum of three square is a square and sum of their product taken two at a time is also a square

Let $a^2 + b^2 + c^2 = X^2$ and $$(ab)^2 + (ac)^2 + (bc)^2 = Y^2$$ Such that $a,b,c,x,y$ are all non zero Integers. How to find All solutions ? Is there any parametrization which gives Infinitely ...
254 views

### $1 + 3 x^3 + x y^2 + 6 y z^2 = 0$ - the new shortest open cubic equation

Are there integers $x,y,z$ such that $$1 + 3 x^3 + x y^2 + 6 y z^2 = 0 \,\, ? \quad\quad (1)$$ If the length of an equation is the sum of degrees of monomials plus sum of logarithms of the ...
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