# Questions tagged [diophantine-equations]

Use for questions about finding integer or rational solutions to polynomial equations.

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### What is the importance of the Diophantine inequality $\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$?

In Humphreys' book Introduction to Lie algebras and Representation Theory(3rd printing, p.62), the Diophantine inequality $$\frac{1}{p} + \frac{1}{q} + \frac{1}{r} > 1$$ appeared while classifying ...
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### What are the integer solutions to $5x^3=y^2+1$?

I want to find the integer solutions to this Diophantine equation: $$5x^3=y^2+1$$ I have seen a lot of problems with monic variables, but not with a constant on the $x^3$ such as this. I know I can ...
43 views

### How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares?

How many ways can $2^{2012}$ be expressed as the sum of four (not necessarily distinct) positive squares? Thanks! For those curious, the solution which I have trouble comprehending is item 2 from the ...
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### How to avoid nested loops intelligently with the computer when solving Diophantine equations with finite range of variables?

I have 59 quadratic equations in 59 variables $\{x_1, ..., x_{59}\}$ and I'm interested in the integer solutions of this equation system, where all $x_i$ can have values only between $-100$ and $+100$....
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### How do you get the $6/5$ in Baker's explicit $abc$ conjecture?

I can't find Baker's "Experiments on the ABC conjecture" in which he gives the $6/5$ as the absolute constant. How do you get the $6/5$ as the constant? Baker's conjecture: For $a+b=c$, $(a,b,c)=1$...
### Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions. [duplicate]
Show that $x^3+3y^3+9z^3-9xyz=1$ has infinitely many integer solutions. I have found that (1,0,0) and (1,-18,12) are two solutions and tried (1,-18+n,12-n). There is a hint saying that I should try ...