Comments - Four Interesting Math Problems - Data Science Central2020-06-02T19:44:36Zhttps://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A687698&xn_auth=noRegarding nested square roots…tag:www.datasciencecentral.com,2018-02-01:6448529:Comment:6891682018-02-01T22:09:48.545ZVincent Granvillehttps://www.datasciencecentral.com/profile/VincentGranville
<p>Regarding nested square roots (problem #3) I proved earlier this morning that the distribution of 0's, 1's and 2's (for the vast majority of positive real numbers), respectively denoted as <em>p</em>(0), <em>p</em>(1), and <em>p</em>(2), is as follows:</p>
<p><a href="https://storage.ning.com/topology/rest/1.0/file/get/2773308965?profile=original" target="_self"><img class="align-center" src="https://storage.ning.com/topology/rest/1.0/file/get/2773308965?profile=RESIZE_480x480" width="393"></img></a></p>
<p>This is confirmed by empirical evidence. Note that <em>p</em>(0) + <em>p</em>(1) + <em>p</em>(2) =…</p>
<p>Regarding nested square roots (problem #3) I proved earlier this morning that the distribution of 0's, 1's and 2's (for the vast majority of positive real numbers), respectively denoted as <em>p</em>(0), <em>p</em>(1), and <em>p</em>(2), is as follows:</p>
<p><a href="https://storage.ning.com/topology/rest/1.0/file/get/2773308965?profile=original" target="_self"><img width="393" src="https://storage.ning.com/topology/rest/1.0/file/get/2773308965?profile=RESIZE_480x480" width="393" class="align-center"/></a></p>
<p>This is confirmed by empirical evidence. Note that <em>p</em>(0) + <em>p</em>(1) + <em>p</em>(2) = 1.</p>
<p>A <em>regular</em> number is a number satisfying this distribution, and the immense majority of numbers are regular numbers. Is Pi a regular number? Probably, and I am investigating this problem at this moment. I also created a general framework to study this kind of problems (mostly based on chaotic stochastic process theory), that also applies to the decimals of Pi and other popular numbers such as square root of 2, expressed in base 2 or 10 (or any base.) This could be a major step towards proving that the decimals of Pi are truly random - I once offered $500,000 to prove or disprove this fact. This is a 500 years old problem at least, and possibly the most challenging (unsolved) mathematical problem of all times.</p>
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