{"id":325,"date":"2026-09-04T12:00:48","date_gmt":"2026-09-04T12:00:48","guid":{"rendered":"https:\/\/mathrelated.co.uk\/?p=325"},"modified":"2026-09-29T19:35:23","modified_gmt":"2026-09-29T19:35:23","slug":"mathematical-discovery","status":"publish","type":"post","link":"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/04\/mathematical-discovery\/","title":{"rendered":"Mathematical discovery"},"content":{"rendered":"\n<p class=\"wp-block-paragraph\">There are many ways that Mathematical discovery can happen.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Doing an Undergraduate degree or Masters or PhD are ways that discoveries can be made.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Working in Algebra for many years and asking \u201cwhat if\u201d is another way.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">My discovery, depending on your point of view, is one or more of the following:<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>Discovered a strong probable prime or \u2018is prime\u2019 test for a particular set<\/li>\n\n\n\n<li>\u201cfixed the b\u201d so same base used all the time for a prime test of Proth numbers<\/li>\n\n\n\n<li>Discovered a set in which a probable prime test behaves strongly.<\/li>\n<\/ul>\n\n\n\n<p class=\"wp-block-paragraph\">Thought it worth describing the process that has allowed that to happen and will talk subjectively about that next. <br>( Do jump ahead to the Conjectures at the bottom of this post if you are interested more in that )<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Have spent many years working in the set<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2217<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mrow><mi>t<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1+(-1+2^t)*(2^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Documented lots of algebraic observations, and through them, was able to come up with an order conjecture involving a least common multiple.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Nothing too exciting so far.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Through thinking about order of two elements in particular, began to see the importance of thinking lengthwise about things with decreasing and increasing values of t.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This is in contrast to thinking about isomorphism primarily, where we tend to view order as an important separator and then move laterally between [groups].<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Went so far as to propose a new definition \u201c<a href=\"https:\/\/mathrelated.co.uk\/index.php\/2026\/08\/14\/defining-a-super-element\/\" target=\"_blank\" rel=\"noopener\">sturdy element<\/a>\u201d to facilitate this thinking.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Tabulated [a lot] of group examples using those elements.<br>( See other posts on this site )<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Adjusted the set definition slightly from<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2217<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mrow><mi>t<\/mi><mo>+<\/mo><mn>1<\/mn><\/mrow><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1+(-1+2^t)*(2^{t+1})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">to nearby sets<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This was a key step<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Broke away from considering only sets we can describe as Proth numbers and considered other sets.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Still thinking about order and patterns and came up with some new <a href=\"https:\/\/mathrelated.co.uk\/index.php\/2026\/09\/03\/conjectures-1\/\" target=\"_blank\" rel=\"noopener\">conjectures<\/a>.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">To try and add a few chapters to my draft book, returned to sets of Proth numbers but considered slightly different powers.<br><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This was another key step.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">During this time was always asking \u201cwhat if\u201d and \u201csuppose\u201d type questions of what I was seeing.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>This supposing and enquiring was another key step.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Spotted something interesting when working with groups in set<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2<\/mn><mi>t<\/mi><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u2217<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mrow><mi>t<\/mi><mo>+<\/mo><mn>3<\/mn><\/mrow><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><\/mrow><annotation encoding=\"application\/x-tex\">1+(-1+2^t)*(2^{t+3})<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Factored the order of a couple of groups to see if there was anything to see.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Made a supposition and tested it for a couple of examples.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Found that the pattern did not apply in all cases.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Asked the question did it only apply to primes.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>There was the discovery [ see conjecture next ]<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<figure class=\"wp-block-image size-full\"><img loading=\"lazy\" decoding=\"async\" width=\"611\" height=\"77\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap37divided4sufficient3conjectureZupdatedPminus1powertest-1.jpeg\" alt=\"\" class=\"wp-image-406\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap37divided4sufficient3conjectureZupdatedPminus1powertest-1.jpeg 611w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsturdyChap37divided4sufficient3conjectureZupdatedPminus1powertest-1-300x38.jpeg 300w\" sizes=\"auto, (max-width: 611px) 100vw, 611px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n\n\n\n<p class=\"wp-block-paragraph\">Created a script to test things out.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>Used a computer algebra package to run the script to see it work with larger examples.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><br>The largest prime found [9769 digits] using this probable prime test as a filter is next.<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mo form=\"prefix\" stretchy=\"false\">\u2212<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2<\/mn><mn>16224<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mn>16227<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>P<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">1+(-1+2^{16224})\u00d7(2^{16227}) ~ ~ is ~ ~ Prime<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">Looked for a further set with something that looked algebraic<\/p>\n\n\n\n<figure class=\"wp-block-image size-large\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"88\" src=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-4-1024x88.png\" alt=\"\" class=\"wp-image-328\" srcset=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-4-1024x88.png 1024w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-4-300x26.png 300w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-4-767x66.png 767w, https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/conjecture2plus3probableC-4.png 1221w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n\n\n\n<p class=\"wp-block-paragraph\">The largest prime found [386 digits] using this probable prime test as a filter is next.<\/p>\n\n\n\n<div class=\"wp-block-math\"><math display=\"block\"><semantics><mrow><mn>1<\/mn><mo>+<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><mn>1<\/mn><mo>+<\/mo><msup><mn>2<\/mn><mn>639<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mo>\u00d7<\/mo><mo form=\"prefix\" stretchy=\"false\">(<\/mo><msup><mn>2<\/mn><mn>642<\/mn><\/msup><mo form=\"postfix\" stretchy=\"false\">)<\/mo><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>i<\/mi><mi>s<\/mi><mtext>&nbsp;<\/mtext><mtext>&nbsp;<\/mtext><mi>P<\/mi><mi>r<\/mi><mi>i<\/mi><mi>m<\/mi><mi>e<\/mi><\/mrow><annotation encoding=\"application\/x-tex\">1+(1+2^{639})\u00d7(2^{642}) ~ ~ is ~ ~ Prime<\/annotation><\/semantics><\/math><\/div>\n\n\n\n<p class=\"wp-block-paragraph\">The early build up to my discovery involved tabulating over 100 groups.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\">This tabulation is documented in the early chapters of my <a href=\"https:\/\/mathrelated.co.uk\/wp-content\/uploads\/2026\/09\/mathsuperChap0toVer0dot14.pdf\" rel=\"nofollow\">(draft) book<\/a>.<\/p>\n\n\n\n<p class=\"wp-block-paragraph\"><\/p>\n","protected":false},"excerpt":{"rendered":"<p>There are many ways that Mathematical discovery can happen. Doing an Undergraduate degree or Masters or PhD are ways that discoveries can be made. Working in Algebra for many years and asking \u201cwhat if\u201d is another way. My discovery, depending on your point of view, is one or more of the following: Thought it worth [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,5,6],"tags":[8,14,13],"class_list":["post-325","post","type-post","status-publish","format-standard","hentry","category-abstractalgebra","category-mathematics","category-numbertheory","tag-group","tag-prime","tag-probableprime"],"_links":{"self":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/325","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/comments?post=325"}],"version-history":[{"count":11,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/325\/revisions"}],"predecessor-version":[{"id":422,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/posts\/325\/revisions\/422"}],"wp:attachment":[{"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/media?parent=325"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/categories?post=325"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathrelated.co.uk\/index.php\/wp-json\/wp\/v2\/tags?post=325"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}