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robve · 02-24-2021, 03:42 PM

Valentin, nice result and in-depth investigation.

Sorry for this long reply:

As a kind suggestion and to offer some respectful constructive feedback: criticizing how we post our results is not helpful. Most of us don't have a lot of time to work on fun stuff. We cannot delay our posts to the end of the week, which is bad for two reasons: 1) it looks like we are just summarizing what other people already posted, and 2) it is not competitive to post our replies very late (because you hinted at some competition for working on these concoctions). If you want more participants and if you want everyone to post in a more organized way then simply do not hint at a competition and produce a Hall Of Fame outcome. It was fun to work on this, but I am not so sure I want to do this again.

Obviously, searching online or looking at other posts totally spoils the fun working on this, at least for me. All results and updates I posted are solely mine! Please note that I mentioned e in my post as the likely result for large trials. I gave no further explanation, as I started thinking about a theoretical result on the expected number of trials for a sum of random variables to exceed a threshold, but that knowledge is no longer at the top of my head, so I let it go and moved on to the other concoctions. Now that the first concoction has ended I verified my gut feeling about this. The result is known to converge to e as the expected value: $$ \mathbb{E}[X] = 1 + \sum_{k \geq 1} \Pr[X > k] = 1 + \sum_{k \geq 1} \frac{1}{k!} = e $$ where X is the number of trials you need for the sum to exceed 1. Indeed, a Buffons Needle-like or Monte Carlo approach to estimate e.

Please note that I posted my initial results early and added some new results as I went back to work on the concoctions. I think that most of us approach it that way, because "aha!" moments and inspiration are not constrained to a single day or hour or even a week, arriving with a sort of Gaussian distribution in our heads between your initial post and the deadline, rather than early or late. We also have a day job to take care of first and foremost.

I also thought it would be fine to have my posts combined into one post, deciding to update my initial post with EDIT, which seems reasonable and fair as it indicates what I added or changed. For example, once I went back to work on this I found I misunderstood one formula, corrected my program, and produced the result that you probably looked for so that felt satisfying. In that case it is easy to see that 1/A is the result, which works for any A not only 2021 because 2021 dominates the first primes in the series. Basically, the sum is approximately over terms x^n/(A^n+x^n) and converges to 1/A quickly when 1<x<<A. I don't need to program that further to understand it.

Also, why would you criticize posting extra code like BASIC and Python bad, when posted in addition to HP PRIME code? HP PRIME is not banned, which I had asked. If banned then that should be made more clear and I will no longer participate because I do not own a physical HP-71B or other vintage HP calculator (though I will be on the lookout for a used HP-71B that works when available at a reasonable price.)

To try and run my HP PRIME programs and learn more about the HP PRIME as I go, I spent hours typing them into the HP PRIME on that tiny key pad. I felt that using an virtual calculator on my laptop isn't what you meant these concoctions to be for, since the intent is to actually use the calculators instead of letting them collect dust.

On the subject of calculators collecting dust, the first calculator I used was a HP-45 in the 80s that my Dad owned and cherished. Not sure if he still actively uses it, but he still cannot part with his HP-45!

- Rob