Mathematical Recreations : Ramanujan's Nested Radical 4

I've previously discussed an interesting math problem posed by Srinivasa Ramanujan way back in 1911.

• the-biologist-is-in.blogspot.ca/2014/12/mathematical-recreations-ramanujans.html
• the-biologist-is-in.blogspot.com/2016/08/mathematical-recreations-ramanujans-2.html
• the-biologist-is-in.blogspot.com/2016/08/mathematical-recreations-ramanujans-3.html

Over the last three posts on the topic, I've explored my thoughts about this problem and then proved there are an infinite number of valid solutions (any value greater than three).

Since then I've been trying to figure out how to prove all values less than three are not valid solutions. I haven't figured out how to do this yet, but I have figured out how to prove a subset of values are not valid solutions. Any trajectory which reaches zero will then pass to less than zero and be invalid. I might formalize this statement once I've figured out if it can help me finish the overall solution. It might just be a blind alley...

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I haven't found anyone else working this problem in the way I have been. The closest I've found has been some comments below a YouTube video where a user talked about calculating through trajectories like I have been. They didn't suggest any sort of general solution to the problem, however.

I did find a mathematical paper using Ramanujan's solution to the problem as part of the title. The authors and reviewers of the paper assumed Ramanujan was correct and didn't test their assumption. I'm considering writing them a letter...


References:

• www.thefamouspeople.com/profiles/srinivasa-ramanujan-503.php
• My posts:
• the-biologist-is-in.blogspot.ca/2014/12/mathematical-recreations-ramanujans.html
• the-biologist-is-in.blogspot.com/2016/08/mathematical-recreations-ramanujans-2.html
• the-biologist-is-in.blogspot.com/2016/08/mathematical-recreations-ramanujans-3.html
• https://www.youtube.com/watch?v=r5BGIi84arY