# A cute number theory question for Math Olympiads and IITJEE Mathematics

Prove that the equation $x^{3}+y^{3}+z^{3}=2$ has an infinity of solutions in Z, the set of integers.

Proof:

Solution by Andrei Stefanescu

For any $k \in Z$ we will consider the numbers $x_{k}=1+6^{3k+1}$, $y_{k}=1-6^{3k+1}$ and $z_{k}=-6^{2k+1}$.

Note that the triplet $(x_{k},y_{},z_{k})$ is a solution of the equation, as $(1+6^{3k+1})^{3}+(1-6^{3k+1})^{3}+(-6^{2k+1})^{3}$ $= 1 + 3.6^{3k+1}+3.6^{6k+2}+6^{9k+3}+1-3.6^{6k+2}-6^{9k+3}-6^{6k+3}$ $= 2+ 6^{6k+3}-6^{6k+3}=2$.

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

QED.

More later,

Nalin Pithwa

## 6 thoughts on “A cute number theory question for Math Olympiads and IITJEE Mathematics”

1. gaurish

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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• nkpithwa

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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• gaurish

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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2. nkpithwa

thanks…will certainly try and keep you posted 🙂

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• gaurish

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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3. nkpithwa

Thanks for a quotation from my all-time favourite mathematician

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