## On the Diophantine equations

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• Is it true that for every given set of integers (p,q,r) with min(p,q,r) 1 and max(p,q,r) 2 equation of the form a^p + b^q = c^r has no more than finite number
Message 1 of 3 , Feb 20, 2008
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Is it true that for every given set of integers (p,q,r) with min(p,q,r)>1 and max(p,q,r)>2 equation of the form

a^p + b^q = c^r

has no more than finite number of solutions in positive integers (a, b, c)?

Regards,

Andrey

[Non-text portions of this message have been removed]
• ... I mean distinct positive integers (a, b, c). Indeed, e.g., (n^2-1)^3 + (n^2-1)^2 = (n^3-n)^2 form an infinite number of solutions for (p,q,r) = (3,2,2).
Message 2 of 3 , Feb 20, 2008
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> Is it true that for every given set of integers (p,q,r) with min(p,q,r)>1 and max(p,q,r)>2 equation of the form
>
> a^p + b^q = c^r
>
> has no more than finite number of solutions in positive integers (a, b, c)?

I mean distinct positive integers (a, b, c).

Indeed, e.g., (n^2-1)^3 + (n^2-1)^2 = (n^3-n)^2 form an infinite number of solutions for (p,q,r) = (3,2,2).

Best regards,

Andrey

[Non-text portions of this message have been removed]
• ... See the Fermat-Catalan conjecture. http://mathworld.wolfram.com/Fermat-CatalanConjecture.html
Message 3 of 3 , Feb 21, 2008
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Andrey Kulsha wrote:
> Is it true that for every given set of integers (p,q,r) with min(p,q,r)>1 and max(p,q,r)>2 equation of the form
>
> a^p + b^q = c^r
>
> has no more than finite number of solutions in positive integers (a, b, c)?
>

See the Fermat-Catalan conjecture.

http://mathworld.wolfram.com/Fermat-CatalanConjecture.html
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