**Prove that the equation**

has an infinity of solutions in **Z**, the set of integers.

**Proof:**

**Solution by Andrei Stefanescu**

For any we will consider the numbers , and .

Note that the triplet is a solution of the equation, as

.

Since we can define this triplet for all k’s, there will be an infinity of solutions.

QED.

More later,

Nalin Pithwa

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Parametric form of solution is most dangerous one, since it asks you to explain “how you thought of this parametric form?”.

I would be like to know the thinking involved behind selection of this parametric form.

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There are questions in Math which have been or can be solved by a stroke of imagination. The proposed solution is one such. May I know whether you can think of any other approach ? Kindly share with us. – Nalin

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I can assure you that no magic is involved here. I failed to find the logical path of thoughts leading to this solution, that’s why I asked you for it. I would suggest you to think over it, till you are able to find the logical path of thoughts. See, section 1.3 in this file: https://gaurish4math.files.wordpress.com/2015/12/diophantine-equations-gaurish-rev4.pdf

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thanks…will certainly try and keep you posted 🙂

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In words of G. H. Hardy:

“A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”

A Mathematician’s Apology (London 1941).

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Thanks for a quotation from my all-time favourite mathematician

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