I've previously discussed an interesting math problem posed by Srinivasa Ramanujan way back in 1911.
In that discussion I suggested there were probably multiple valid solutions to his puzzle, not just the solution he published (three). I even went a bit further to suggest there was an infinite number of valid solutions.
I still don't have a nice proof, but I have done some further calculations to help illustrate my thoughts. The following figure shows plots of subsequent radical values when initial values are set to various numbers. Initial values of [0, 1, 2, 2.9, 2.99, 2.999, 2.9999, 2.99999, 2.999999, 2.9999999, 2.99999999, 2.999999999, 2.9999999999, 2.99999999999, 2.999999999999, 2.9999999999999] result in the series of descending curves. Initial values of [3.1, 3.01, 3.001, 3.0001, 3.00001, 3.000001, 3.0000001, 3.00000001, 3.000000001, 3.0000000001, 3.00000000001, 3.000000000001, 3.0000000000001] result in the series of ascending curves. The closer these values are to 3, the further to the right in the figure they diverge from the single straight line resulting from an initial value of [3].
The calculated profile for every initial value below three drops down to just below zero and then rises towards an asymptote at zero. It surprised me that the fall of the profiles is slowed before they reach zero, as well as the behavior below zero, but neither result lead me to think there is a problem with intuitions about the problem. This more complicated behavior leads me to think that a proof showing every value less than three is invalid might be a bit difficult.
The behavior of the profiles starting above three are simple enough that I suspect there is a relatively simple proof of their general behavior. That is to say, I think it will be a relatively simple task to prove any values greater than three are valid solutions to Ramanujan's Nested Radical.
Stay tuned for further developments.
References
In that discussion I suggested there were probably multiple valid solutions to his puzzle, not just the solution he published (three). I even went a bit further to suggest there was an infinite number of valid solutions.
I still don't have a nice proof, but I have done some further calculations to help illustrate my thoughts. The following figure shows plots of subsequent radical values when initial values are set to various numbers. Initial values of [0, 1, 2, 2.9, 2.99, 2.999, 2.9999, 2.99999, 2.999999, 2.9999999, 2.99999999, 2.999999999, 2.9999999999, 2.99999999999, 2.999999999999, 2.9999999999999] result in the series of descending curves. Initial values of [3.1, 3.01, 3.001, 3.0001, 3.00001, 3.000001, 3.0000001, 3.00000001, 3.000000001, 3.0000000001, 3.00000000001, 3.000000000001, 3.0000000000001] result in the series of ascending curves. The closer these values are to 3, the further to the right in the figure they diverge from the single straight line resulting from an initial value of [3].
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| The top shows positive values. The bottom shows negative values. |
The calculated profile for every initial value below three drops down to just below zero and then rises towards an asymptote at zero. It surprised me that the fall of the profiles is slowed before they reach zero, as well as the behavior below zero, but neither result lead me to think there is a problem with intuitions about the problem. This more complicated behavior leads me to think that a proof showing every value less than three is invalid might be a bit difficult.
The behavior of the profiles starting above three are simple enough that I suspect there is a relatively simple proof of their general behavior. That is to say, I think it will be a relatively simple task to prove any values greater than three are valid solutions to Ramanujan's Nested Radical.
Stay tuned for further developments.
References
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