Phi: (1Âħsqrt(5))/2 =~ {1.6180339887499, -0.6180339887499}
How?
Suppose the RHS was equal to 0 instead of 1.
New equation: 89x^12 - 144x^11 = 0
x^11*(89x -144) = 0
Then x could be either 0 or 144/89
144/89 = 1.61797752808989 which is close to phi.
Other clue: 89 and 144 are both adjacent Fibonacci numbers.
Plug it in, both answers check.
Plus Descartes Rule of Signs says there can be at most 1 positive root and 1 negative root, so the other 10 solutions must be complex.
Edited on May 10, 2023, 9:22 am
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Posted by Larry
on 2023-05-09 14:42:22 |
Guess and check
