How to crack this one ?

Prove that the equation

\[6(6a^2 + 3b^2 + c^2) = 5n^2\]

has no solutions in integers except $a = b = c = n = 0$.

## APMO 1989

- nafistiham
**Posts:**829**Joined:**Mon Oct 17, 2011 3:56 pm**Location:**24.758613,90.400161-
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### APMO 1989

\[\sum_{k=0}^{n-1}e^{\frac{2 \pi i k}{n}}=0\]

Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

Using $L^AT_EX$ and following the rules of the forum are very easy but really important, too.Please co-operate.

### Re: APMO 1989

The first thing that came to my mind after seeing $3,6$ and the squares is It should work I believe.

"Everything should be made as simple as possible, but not simpler." - Albert Einstein

- samiul_samin
**Posts:**1004**Joined:**Sat Dec 09, 2017 1:32 pm

### Re: APMO 1989

Solved here.nafistiham wrote: ↑Mon Apr 09, 2012 9:16 pmHow to crack this one ?

Prove that the equation

\[6(6a^2 + 3b^2 + c^2) = 5n^2\]

has no solutions in integers except $a = b = c = n = 0$.