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Five Fives Fancy Formula (Posted on 2012-12-17) Difficulty: 3 of 5
Find all real x satisfying this equation:

√(5+√(5-√(5+√(5-√(5+x))))) = x

Prove that there are no others.

No Solution Yet Submitted by K Sengupta    
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re: another form of the answer - nope | Comment 3 of 4 |
(In reply to another form of the answer by Larry)

Repeatedly squaring and taking care of the 5's gives

x = (5-((5-(x-5))-5)) - 5

Putting (1858)^1/8 into the RHS gives 2.562304215 (which differs in the seventh decimal place.)  Raising this to the eighth power gives 1857.999223

My guess based on simpler equations like √(5-√(5+x)) = x
is that the squared out form will be a degree 32 polynomial with 31 real but extraneous roots. The root that works will likely involve a 32nd root but not as simply as n^(1/32)

  Posted by Jer on 2012-12-18 13:59:18

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