# Tagged Questions

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 $\mathbb{Q}$. ...

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### Integer Solutions of $x+y^n = y + x^m$ for $n < m$

I found 8 of them and believe there is no more: $$2+3^2=3+2^3$$ $$2+6^2=6+2^5$$ $$6+15^2=15+6^3$$ $$3+16^2=16+3^5$$ $$3+13^3=13+3^7$$ $$2+91^2=91+2^{13}$$ $$5+280^2=280+5^7$$ $$30+4930^2=4930+30^5$$ ...
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### Is the following consequence of the Lang conjecture known?

This came up in a discussion with a colleague of mine, who studies PDEs. He was asking for a function $f \colon \mathbb{N} \rightarrow \mathbb{N}$ such that, for all but finitely many $n$, the ...
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### Solutions of the equation $X^4-DY^4=z^4$

In order to obtain infinite integer non trivial solutions of the equation $X^4-DY^4=Z^4$ (all numbers natural) we do the following. We set $X=(r_1●p+p)$, $Y=p$, $Z=(r_1●p)$, $D=(4r_1^3+6r_1^2+4r_1+1)$...