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Octic Sum Settlement (Posted on 2015-02-06) Difficulty: 3 of 5
The roots of the equation x3 - x + 1 = 0 are P, Q, R.

Find P8 + Q8 + R8.

No Solution Yet Submitted by K Sengupta    
Rating: 4.0000 (1 votes)

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Some Thoughts re: Answer only (possible spoiler) | Comment 2 of 3 |
(In reply to Answer only (possible spoiler) by broll)

So annoying to get an integer answer without an explanation!

However, there is a similar problem, with x3 - x - 1 = 0, leading to a well-known sequence.

Now for x3 - x - 1 = 0, the roots are: x~~1.3247,x~~-0.66236-0.56228i, x~~-0.66236+0.56228i,

while for x3 - x + 1 = 0, the roots are:
x~~-1.3247, x~~0.66236-0.56228i, x~~0.66236+0.56228i, so there's just a sign change in the real part.

Since we are raising to an even power, the result is the same for either equation, see:

http://mathworld.wolfram.com/PerrinSequence.html

http://en.wikipedia.org/wiki/Perrin_number



  Posted by broll on 2015-02-07 04:22:17
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