{"id":8461,"date":"2026-10-08T10:01:26","date_gmt":"2026-10-08T17:01:26","guid":{"rendered":"https:\/\/theoryofeverything.org\/theToE\/?p=8461"},"modified":"2026-10-08T11:26:32","modified_gmt":"2026-10-08T18:26:32","slug":"the-collatz-conjecture","status":"publish","type":"post","link":"https:\/\/theoryofeverything.org\/theToE\/2026\/10\/08\/the-collatz-conjecture\/","title":{"rendered":"The Collatz Conjecture"},"content":{"rendered":"\n

For more information on the Collatz Conjecture, see this Wikipedia (WP) post<\/a>. The Mathematica Notebook (NB) is here<\/a>. The images in this file have Tooltip information on the vertices (e.g. n with an indicator if it is prime or the prime factors of n, and the max trajectory or height). It also includes the “shortcut version” of the Collatz Conjecture which also divides the 3n+1 by 2 thereby avoiding the even height increases.<\/p>\n\n\n

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The WP image with vertices only<\/figcaption><\/figure>\n<\/div>\n\n\n

In the images below, the horizontal colored lines generate a diagonal line of the same color from each n on that line. The Black diagonal line at the bottom are the lowest height n’s (which are simple factors of 4 after n=1 & 2).<\/p>\n\n\n\n

The following images are graphs related to the Collatz Conjecture (click the PNG file to open the SVG in a new tab):<\/p>\n\n\n

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The x<\/em> axis represents starting number, the y<\/em> axis represents the highest number reached during the chain to 1 (up to with n<10,000). This plot shows a restricted y<\/em> axis: some x<\/em> values produce intermediates as high as 2.7\u00d7107<\/sup> (for x<\/em> = 9663)<\/figcaption><\/figure>\n<\/div>\n\n
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Same as above, but with n<100,000<\/figcaption><\/figure>\n<\/div>\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

For more information on the Collatz Conjecture, see this Wikipedia (WP) post. The Mathematica Notebook (NB) is here. The images in this file have Tooltip information on the vertices (e.g. n with an indicator if it is prime or the prime factors of n, and the max trajectory or height). It also includes the “shortcut … Continue reading The Collatz Conjecture<\/span> →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[],"class_list":["post-8461","post","type-post","status-publish","format-standard","hentry","category-physics"],"aioseo_notices":[],"aioseo_head":"\n\t\t\n\t\n\t\n\t\n\t\n\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t\n\t\t